Discoveries

Every claim of the tree on one dated, tagged line with its witness, one section per topic, filtered by tag, topic and date.

Since 1104 claims

Acoustic barrier

2026-08-28 Conjecture The level trend of the self-similar acoustic barrier is fineness, not self-similarity: a random tile at matched solid fraction and cell size, three seeds, r = 4, also falls with level and blocks more at every L2 and L3 - random against self-similar at L2 / L3: carpet 0.264 / 0.074 against 0.390 / 0.128, net 0.229 / 0.099 against 0.467 / 0.215, tree 0.221 / 0.066 against 0.781 / 0.666, void 0.211 / 0.104 against 0.270 / 0.235 - with a seed spread of 0.006 to 0.037.

Atlas

2026-09-08 Verified mrlygame quests replay outside the engine: following the quest draw order through board, mask, rulebook, score and pivot options with a shift-and-add or FFT count reproduces mrlygame::quest cell for cell on seeds 1, 4, 7 and 9 under the default config, the four cheapest quests, each a single attempt at canvas climb 1 without the copy path; the other 60 quests, 37 of them re-attempts, rest on the same draw order and a stepper checked against next_grid on 24 random cases. Witness: lab/rs/still-hunt.
2026-09-08 Verified Over 64 default quests and 4626 frames, 609 pass the ring cut (peak-ring share at least three times a random field share at the same ring, wavelength at most twice the mask side, peak ring at or below half the field): 426 of 1043 basic-path frames, 76 of 92 copy-path frames, 107 of 3491 side-3 frames; 3 of the 64 settled stills pass, all on side-3 masks; the cut also admits envelope peaks, 149 of the 562 hits past generation 0 peaking at ring 1 or 2 of their field, 34 at a wavelength longer than the crop, among them 66 of the 73 copy-path hits, the copy path being one quest whose masks are its own 45-cell canvas on a 64 field; none of the 176 resolved drawn-mask hits sit at ring 1 or 2; the full-square peak leaves the ring cut for a Fourier corner in 376 of 4561 frames. Witness: lab/rs/still-hunt.
2026-09-08 Verified At canvas cap 512 and mask cap 128, seeds 65 to 68, the copy path yields 40 ring-cut frames of 51 and the one drawn-mask still, seed 66, a copy mask of side 75 on a board of 5 by 5 tiles, peaks at ring 10 of field 512, wavelength 51 cells or 0.68 mask sides, inside the lobe 8 to 13; seed 67 with a copy mask of side 121 on a board of 3 by 3 tiles peaks at the tile fundamental ring 4, one ring below its lobe 5 to 8, in 11 of its 27 hits, at ring 13 in 5 and elsewhere from 8 to 93 in the rest. Witness: lab/rs/still-hunt.
2026-09-08 Proved Sublattice closure: when every mask offset lies in dZ^2 and zero is outside birth, every outer-totalistic rule carries a frame A (x) B on the d side torus to A' (x) B, with A' one step of the same rule on the side torus under the mask divided by d, because the count at a cell in position p of its tile is B(p) times the quotient count and a cell at an empty position reads count zero; verified frame by frame on 16 runs of the mini preset, all 16 rules, the carpet-tile seed at tessellation 3 under its copied-inverted mask of index 9. Witness: lab/rs/life-atlas.
2026-09-08 Verified On the mini preset, 848 runs on canvas 27 to 64 generations, 103 settled frames (34 distinct up to the dihedral group and torus translation) tested at every torus shift and both cuts 3 and 9: in place 72 are confined to one block of the cut, 4 are full tilings, 1 is the board footprint and the 3 proper cuts are the settled runs of the index-9 sublattice mask, no index-1 frame cutting properly; after a torus shift 6 index-1 frames cut properly, all at cut 9 and tessellation 3, the 16-cell still of the corner seed (four 2x2 blocks at spacing 3, shift (1, 1), outer fill 4) on Moore under B357/S357, B3/S3 and B3/S23 and on the two-bar and cross masks under B3/S23, and the 36-cell still of the two-bar seed on the cross mask under B3/S23 (shift (1, 0), outer fill 6); index-2 and index-3 masks cut properly nowhere. Witness: lab/rs/life-atlas.
2026-09-08 Verified Mini fate table (dead, still, loop, timeout): number rules 106, 24, 56, 132; design rules 448, 2, 0, 27, the rolls off leaving at most one count of the nine design sequences inside budget 8, the budget of 45 of the 53 mask cells, the other 8 being tessellation-3 copy masks of budgets 26 to 80; B3/S23 31, 17, 4, 1; the 160 timeouts hold no mover, a timeout being a lower bound on the period and never a class. Witness: lab/rs/life-atlas.
2026-09-08 Verified The median level set of the visit counts cuts properly in 9 of 848 runs in place and 10 after every torus shift, eight of them sublattice-lemma runs and the others the cross-tile heat frame of one index-1 run and the two-bar seed under the cross mask and B3/S23; on the 103 settled runs the proper heat sets coincide with the proper frames, so the heatmap is a design no more often than the frame it summarises. Witness: lab/rs/life-atlas.
2026-09-08 Refuted The lobe law on the sequence rules of mrlygame: the peak ring sits in the mask first negative lobe in 52 of 176 resolved drawn-mask frames (crop at least four mask sides, generation above 0, 15 chapters of 13 quests, correlated frames); witness seed 3, chapter 1, generation 1, mask side 23, peak ring 11 against lobe 13 to 22; the lobe centre reads 0.64 mask sides at the median while the hits wavelength over mask side fills every quarter bin from 0 to 2, the cut itself; the copy path 35 of 73 is one quest whose lobes sit at rings 2 to 4 of a 64 field, the envelope scale of its 45-cell crop. Witness: lab/rs/still-hunt.
2026-09-08 Refuted The seed-comb product law: on the 176 resolved frames (15 chapters, comb spacing 1.08 to 5.69 rings, median 5.22, 22 rows on one-tile boards) the peak ring is within half a ring of a comb multiple in 77 against 52.9 by chance, the chance confirmed by a uniform-ring null at 52 and a permutation null at 53, and on both comb and lobe in 22; the 77 counts correlated frames, not independent trials; witness seed 3, chapter 1, generation 1, peak ring 11 against the multiple 10.04 of comb 5.02 and outside lobe 13 to 22. Witness: lab/rs/still-hunt.
2026-09-08 Refuted That a tessellated seed still peaks on the seed comb: of 21 stills from tessellation 3 on canvas 27 (comb 9), the 5 comb hits are two lattice-mask runs of index 3 and 9 and three four-cell frames of peak share 0.06 on the index-2 corner mask, while every Moore still peaks at ring 1, 4 or 6; the witness is the two-bar seed at tessellation 3 under Moore and B3/S23, still at generation 53, ring 4 against comb 9 and lobe 9. Witness: lab/rs/life-atlas.

Automata

2026-09-02 Proved Wolfram rule N and the design bang dim 3, code N are one subset of {0,1}^3 under (x0, x1, x2) = (l, c, r) with corner i = 4 x0 + 2 x1 + x2, so every design invariant is a rule invariant; recomputed on all 2048 rule cells. Witness: lab/rs/automata-census.
2026-09-02 Verified The elementary rules fall into 160, 88, 22 and 14 classes under reflection, Wolfram equivalence, the cube group and the cube group with complement, by orbit walk and by Burnside alike; Wolfram equivalence and the cube group are incomparable inside the order-96 group and meet exactly in the reflection, rule 137 lying in the Wolfram class of 110 and outside its cube orbit. Witness: lab/rs/automata-census.
2026-09-02 Proved The cube group is not a dynamical symmetry of the elementary automata: surjectivity is not a cube invariant, exactly one of the 22 classes being mixed, the 24-rule orbit of 30 with 16 surjective rules and 54 among the 8 that are not, while reversibility is constant on all 22 classes, the six reversible rules 15, 51, 85, 170, 204, 240 being the orbit of the identity, and both properties are constant on all 88 Wolfram classes. Witness: lab/rs/automata-census.
2026-09-02 Verified The 24 rules in the cube orbit of 110 share popcount 5, degree 3, genus compound, the Walsh amplitude profile and non-surjectivity, only their single-seed diagrams separating all 24; the multiset of Walsh amplitudes by weight is a cube invariant on all 256 rules while the signed level sums are not. Witness: lab/rs/automata-census.
2026-09-02 Verified Rules 60 and 102 draw the plane designs bang dim 2, code 13 and bang dim 2, code 14 cell for cell from one seed, and rule 90 draws bang dim 2, code 13 in the sheared frame j = (t + i)/2, each unique among the fill-3 codes to level 8 by two renderers. Witness: lab/rs/automata-census.
2026-09-02 Proved The single-seed diagram of rule 150 is an XOR substitution, row 2t being row t spread by two and row 2t + 1 that row xor its two unit shifts, and its first 2^k rows hold P(k) = 2^k F(k+2) live cells with B(k) adjacent pairs under P' = 4P - 2B and B' = 2P - 2B, so the growth exponent is log2(1 + sqrt 5); the recurrence and closed form are already A087206's and A071053's and are re-derived here. Witness: lab/rs/automata-census, A087206, A071053.
2026-09-02 Verified One live cell under the 256 rules gives 143 distinct space-time diagrams, 89 up to reflection, class sizes {1: 118, 2: 13, 4: 4, 8: 4, 16: 4}, identical at pad T and 2T for T = 64, 128, 256; equal occurring key implies equal diagram with 0 failures over 65536 ordered pairs, the key measured on the padded line takes 152 values by a boundary artefact, and on the cropped window exactly 143. Witness: lab/rs/automata-census.
2026-09-02 Proved In every dimension the level-1 side-3 tile of the design "not every coordinate odd" is the 3^dim - 1 Moore neighbourhood and the tile of "at most one odd coordinate" fills 2^(dim-1) (dim + 2); as popped masks the two agree iff dim <= 2, 20 against 26 cells at dim = 3, and in the plane the Moore mask is bang dim 2, code 7. Witness: lab/py/life-census, lab/py/sibling-census.
2026-09-02 Verified B3/S23 is the dim = 9 design of fill 140 of 512, so Langton's lambda is 140/512 by definition, with GF(2) degree 8 on 184 monomials, Walsh level sums 140, 308, -224, -896, -168, 840, 448, -224, -196, -28 on the 0/1 form and genus compound; over the 2^18 life-like rules the fill takes 479 of 513 values with 34 gaps, the genus splits 2044 isotropic, 4 axial only and 260096 compound, exactly 8 rules are affine, and the degree obeys deg(B, S) = deg B when B = S and max(deg B, 1 + deg(B xor S)) otherwise, giving 2 * 4^d rules of degree at most d. Witness: lab/py/life-census.
2026-09-02 Proved The nine-cell Moore XOR B1357/S02468 is rule 150 tensor rule 150, every time slice the outer product of two rule 150 rows and its population A071053 squared, which is A246035; the named replicator B1357/S1357 is the eight-cell XOR with kernel (1/x + 1 + x)(1/y + 1 + y) - 1, A160239, and equals the outer product with the centre copy removed only at t = 2^j. Witness: lab/py/life-census, A246035, A160239.
2026-09-02 Proved The decoupling lemma: an automaton whose dependency set, centre included, generates a sublattice of index k is k interleaved copies of the index-1 rescaled automaton under the same rule, so the 1D parity tile at side 2r + 1 decouples iff r is even and the Cantor tower iff its level is even; over the base-2 masks at dim = 1, 2, sides 3 to 9 and levels 1 to 3, the 195 distinct masks split 95 of index 1, 70 of index 2, 18 of index 4 and 12 rank-deficient. Witness: lab/py/sibling-census.
2026-09-02 Proved B3/S23 is not a rows-then-columns composite of two elementary rules in either order; the 65536 ordered pairs give 32260 distinct nine-input rules, none equal to Life, and exactly 10 life-like composites, all affine or thresholds at 0 or 9. Witness: lab/py/sibling-census.
2026-09-02 Proved Cantor-Life, B3/S23 on the eight-cell level-3 Cantor mask at offsets +-5, +-7, +-11, +-13, has no still life under 4 cells at any width, {0, 5, 7, 12} being minimal, and its nine-cell XOR has period dividing 256 on the ring of 1024 by Frobenius, exactly 256 on generic soups; every seed of width at most 14 dies (6113), stills (2003) or oscillates (76, periods 2, 3, 4, 6), and no mover is found there. Witness: lab/py/sibling-census.
2026-09-02 Proved Menger-Life, B3/S23 on the 20-cell Menger mask, leaks out of every plane holding a dead cell of Moore count 3, and a plane 2 x 2 block stacks as rule 90 along the normal with population 4 * 2^popcount(t); over 101 outer-totalistic rules and 200 random 5^3 seeds each, no mover appears. Witness: lab/py/sibling-census.
2026-09-02 Proved Under the composite 110.110 the bounding box of every finite pattern grows without bound, its upper-left corner moving (-1, -1) each generation; the population growth itself is only recomputed, not proved. Witness: lab/py/sibling-census.
2026-09-02 Refuted "Fredkin's replicator is the mod-2 sum of the nine Moore cells": the named replicator B1357/S1357 is the eight-cell sum with kernel (1/x + 1 + x)(1/y + 1 + y) - 1, A160239, while the nine-cell sum is B1357/S02468, A246035, and the two agree at no generation past t = 0 except through the removed centre copy at t = 2^j. Witness: lab/py/life-census, A160239, A246035.
2026-09-19 Verified Over the 195 base-2 masks at dim 1, 2, sides 3 to 9 and levels 1 to 3 the lattice index splits 16 of index 1, 11 of index 2 and one empty mask at dim 1 and 79, 59, 18 and 11 at dim 2, the both-even mask bang dim 2, code 1 at level 2 being an index-4 witness. Witness: lab/py/sibling-census.
2026-09-19 Conjecture On the level-3 Cantor mask from soups of 1024 cells at three densities, B36/S23 behaves like B3/S23 with activity lingering at density 0.5, B2/S sustains a soup near 0.18, B3/S012345678 freezes near 0.65 at the densities 0.25 and 0.5, and B3678/S34678 dies, fills the ring or holds a long period at a density between 0.3 and 0.9. Witness: the soup runs of lab/py/sibling-census, which carries no witness line for these four rules.
2026-09-19 Verified Of the 101 Menger rules on a 32^3 torus at 200 generations the 22 quiet in all four runs are every rule of birth {5} or {6} together with B4/S5 and B4/S6. Witness: lab/py/sibling-census.
2026-09-19 Verified Of the 69 still lifes among the 20200 Menger seed fates, 55 sit under B5/S34, B6/S34 and B56/S34. Witness: lab/py/sibling-census.

b-visibility

2026-08-28 Conjecture The gasket's b-visible density is (1 - 2^(b-1)/3^b) Prod_{p odd}(1 - p^(-(b+1))), not (8/9)/zeta(b+1) for every b >= 1: the local factor at 2 is 2^(b-1)/3^b, exact to the integer at b = 1, 2, 3 on every level 3..12, and the ratio to 1/zeta(b+1) is (1 - 2^(b-1)/3^b)/(1 - 2^(-(b+1))), which is 8/9 at b = 1 ((2/3)/(3/4)) and at b = 2 ((7/9)/(7/8)) only - 368/405 = 0.9086420 at b = 3, 2336/2511 = 0.9303067 at b = 4, climbing to 1; the two formulas part at b = 3, 0.8395292 against 0.8212786, and exact enumeration reads 0.8427119 at level 12, falling about 0.0022 a level toward the former; b = 1 is the proved 16/(3 Pi^2).

Boolean complexity

2026-08-28 Verified Geometry under-determines Boolean complexity at dim = 4: bang dim 4, code 27 and bang dim 4, code 281 share genus, GF(2) degree, popcount and the fill polynomial 4k^4 - 4k^3 + k^2 and split six of seven complexity measures; across the 424 classes (22 + 402) 92 groups hold two or more classes and 279 measure splits occur. Witness: lab/py/boolean-measures, mrlymath::bang::counting::sequence.
2026-08-28 Verified Sensitivity and block sensitivity separate exactly once in the dim <= 4 catalog: s = bs on all 22 classes at dim = 3 and on 401 of 402 at dim = 4, the exception bang dim 4, code 7128 with s = 2, bs = 3, orbit 24; C = bs on all 424 classes; exactly two classes meet deg = s^2, bang dim 4, code 855 and bang dim 4, code 1911, the second the AND-of-ORs that Huang 2019 names tight for s(f) >= sqrt(deg(f)). Witness: lab/py/boolean-measures.

Carpet star

2026-08-28 Conjecture The carpet-star decay coefficient is not exactly -1/8: it depends on the registration and band-normalization frame, the family frame through L = 400 giving slope -0.1242 and the ideal frame through the same L = 400 giving -0.18 to -0.19 and still drifting; only the (log L)/L decay order survives the frame change.

Cobham

2026-09-14 Proved Multiplicative independence is a property of a pair and never of a triple: dependence is an equivalence relation on the integers >= 2 whose classes are the powers of one least member, so three bases that are pairwise dependent are jointly dependent and a third base adds no hypothesis a pair does not carry; the invariant is the dependence-class partition and not the tuple. The two degenerate readings are settled the same way: base 1 is not a base, k-recognizability being defined for k >= 2 only, and bases 2 and 4 are one base, so Cobham's hypothesis fails there and so does its conclusion, the base-4 design {0, 1} being 4-recognizable, hence 2-recognizable, infinite and of density (1/2)^level at level level, hence not ultimately periodic. Witness: cobham.md:9 to cobham.md:15, Durand and Rigo Definition 1.1 and Remark 1.2 read at source, Bes on Buchi read at source.
2026-09-14 Proved Every proper one-dimensional design is base-locked, exactly. For F inside {0, ..., base-1} with 0 in F and 1 < card F < base, the set S_F of integers whose digits all lie in F is infinite, since d base^k lies in it for every nonzero d in F, and has density (card F / base)^level at level level, which tends to 0; an infinite ultimately periodic set has positive density, so S_F is not ultimately periodic, and Cobham's theorem forbids any base multiplicatively independent of base. The two hypotheses are the two exclusions and nothing else: card F > 1 removes F = {0} and card F < base removes the full digit set, and those are the only two semilinear designs at dim = 1. Witness: cobham.md:26, Bes Theorem 24 and Durand and Rigo Theorem 1.1 read at source.
2026-09-14 Proved A necessary condition on the digit set, sharp enough to settle every gasket. If S_F is semilinear then card F = base^d for an integer 0 <= d <= dim, and S_F lies in a finite union of d-dimensional affine subspaces: a linear set v + N c_1 + ... + N c_r whose generators span dimension e meets [0, N)^dim in Theta(N^e) points, so a semilinear set counts Theta(N^d) in the box with d the largest span dimension among its constituents, while the design counts (card F)^level exactly at side base^level, forcing card F = base^d. Consequence with no geometry: the base-2 gasket {(0,0), (0,1), (1,0)} has card F = 3, not a power of 2, so it is not semilinear and not 3-recognizable. Witness: cobham.md:32 and cobham.md:38.
2026-09-14 Proved A sufficient condition, and the two conditions agree at base = 2, dim = 2. Call F a block design when the dim coordinates split into a zero set Z and d blocks with F = {v : v_i = 0 on Z, and v_i = v_j whenever i and j share a block}; then card F = base^d and S_F = N c_1 + ... + N c_d with c_t the 0/1 indicator of block t, one linear set, hence semilinear and recognizable in every base. At base = 2, dim = 2 the characterization is complete: of the eight designs containing 0, the five of cardinality 1, 2, 2, 2, 4 are exactly the block designs and are semilinear, and the three of cardinality 3 are excluded by the count. Witness: cobham.md:33 and cobham.md:34.
2026-09-14 Proved The base-3 gasket is not semilinear and so not 2-recognizable. Its box count is 3^level at side 3^level, so d = 1 and a semilinear version would lie in finitely many lines; but it contains P_t = (3^t, 3^(t^2)) for every t >= 2, whose consecutive slopes are exactly s_t = 3^(t^2 - t) (3^(2t+1) - 1)/2, strictly increasing, so the P_t sit in strictly convex position, no three are collinear, and covering n of them costs at least n/2 lines. Hence no automaton reading base-2 digits enforces the base-3 gasket's digit rule. Witness: cobham.md:37.
2026-09-14 Proved Five of this tree's instruments, checked one by one on the page that carries each, split as follows under a second independent base; the audit covers those five and asserts nothing about the instruments it did not check. Survives: the box bound of the coprimality sieve, N*_level(m) <= (base+1)^dim fill^level m^(-alpha) with alpha = log(fill)/log(base), which is pure counting on S_level and passes to any subset by monotonicity, the Chebyshev sum built on it still converging when alpha > 1. Dies: the fill law of method.md, an identity on a Kronecker power in one base; the transfer matrix and its Perron root, the vertex shift of beneath.md, the even slice matrix M_even of cuts.md and the Collatz-Wielandt brackets of crop.md, each a finite automaton over the digits of one base; the carry automaton of cuts.md, finite only because x -> (x + dim)/3 contracts on integer carries inside one base, while a machine reading base-2 digits and tracking base-3 digits is base conversion; and the character contraction abs(Sum) <= fill - 2 + 2 cos(pi/(2 base)), which is the statement that the level-level transform factors over digit positions in one base. A sieve needs an upper bound and an equidistribution: two bases hand over the first and destroy the second. Witness: cobham.md:40 to cobham.md:46.
2026-09-14 Proved The two-set transversality theorem does not iterate, for an elementary reason: A cap B need not be invariant under either map. With A the middle-thirds set, T_3-invariant, and B = [0, 1), T_2-invariant, the intersection is A, which is not T_2-invariant, since 1/4 = 0.020202..._3 lies in it and T_2(1/4) = 1/2 = 0.1111..._3 does not. The m-fold bound is proved instead by rewriting the intersection as one slice of the product A_1 x ... x A_d inside T^d. Witness: cobham.md:52, Corso and Shmerkin 2024 Theorem 1.15 and Corollary 1.17 read at source.
2026-09-14 Proved What product designs do give is one budget per axis, not one budget in total. If every F_i is a product G_i^(1) x ... x G_i^(dim) across the dim axes in pairwise independent bases, each A_i is the product of its axis sets, the intersection is the coordinatewise intersection, upper box dimension is subadditive on products, and Corso and Shmerkin Corollary 1.17 applies on each axis, giving dim-upper_B(cap_i A_i) <= sum_(j=1)^dim max(0, sum_i dim_H A_i^(j) - (m-1)), upper box on the left and Hausdorff on the right. Witness: cobham.md:61.
2026-09-14 Verified The joint census of the base-2 gasket {(x, y) : x AND y = 0} against the base-3 gasket, C(N) = card(A cap B cap [0, N)^2), reads C(3^m) = 1, 3, 7, 19, 45, 111, 241, 467, 1175, 2443, 5285, 11939, 25281, 53477, 109001, 231737, 498083, 1077727, 2179165, 4372741, 9051805, 18107943, 37126191, 75050077, 151133095 at m = 0..24. Each level is computed by a pruned walk over the base-3 exponents whose cut is a proved bit-length bound on the remaining addition, and rebuilt for m <= 6 by scanning every pair in the box against both digit rules with one shared membership routine; the two agree at every level and the axis bound is asserted at each. Witness: cobham.md:77, lab/py/two-base-gasket verbs terms and control.
2026-09-14 Proved C(3^m) >= 2^(m+1) - 1, and that already breaks the naive planar budget. On the axis x = 0 membership in the base-2 gasket is automatic and membership in the base-3 gasket asks the base-3 digits of y to lie in {0, 1}, which 2^m values below 3^m satisfy; the axis y = 0 gives another 2^m; the origin is the only overlap. Hence the counting exponent is at least log_3 2 >= 0.630929, against a budget log_2 3 + 1 - 2 <= 0.584963, the lower bound truncated down and the budget rounded up. The real gaskets carry the same excess: the left edge lies in the real base-2 gasket and the real base-3 gasket meets it in a Cantor set of dimension log_3 2. Witness: cobham.md:78 and cobham.md:79, lab/py/two-base-gasket verbs terms and budget.
2026-09-14 Proved Budget zero is dimension zero and not finiteness: {2^n} has counting exponent 0, its count below N being at most log_2 N + 1, and is infinite, so a transversality bound of zero never closes a question that asks for a finite list. Finiteness in two bases is reached only at the bounded-digit-sum corner, by Senge and Straus 1973 ineffectively through Thue-Siegel-Roth and by Stewart 1980 effectively through Baker; that corner is not a design, its set is not closed under changing one digit and its count below b^k is O(k^c). Witness: cobham.md:67 and cobham.md:68, both statements read at source in the survey of Bugeaud, Cipu and Mignotte.
2026-09-14 Proved The three-base thin set has real upper box dimension zero, by the three-set theorem and not by any pair. With A_3, A_5, A_7 the closed subsets of the circle whose base-3, base-5 and base-7 digits lie in {0,1}, {0,1,2} and {0,1,2}, Corso and Shmerkin 2024 Corollary 1.17 at d = 3 asks for pairwise multiplicatively independent p_j >= 2, closed T_(p_j)-invariant A_j and affine g_j, and gives dim-upper_B(g_1(A_1) cap g_2(A_2) cap g_3(A_3)) <= max{s - 2, 0} with s = sum_j dim_H A_j; the bases are distinct primes, each set is closed and invariant by its digit rule, the g_j are the identity, and s - 2 = -0.121889 rounded up, so the bound is 0 and upper box dimension is nonnegative. The conclusion is about the three real sets and about upper box dimension, and it does not transfer to the integer set. Witness: cobham.md:90, lab/rs/three-base-thin verb budget, Corso and Shmerkin 2024 Corollary 1.17 read at source.
2026-09-14 Proved The third base is not redundant, and the redundancy is sharp in both directions. Upward, the three pair budgets dim_i + dim_j - 1 read 0.313536 at (3, 5), 0.195505 at (3, 7) and 0.247182 at (5, 7), each rounded up and each positive, so the two-set bound returns nothing on any pair. Downward, the pair (3, 5) alone is infinite: Erdos, Graham, Ruzsa and Straus 1975, quoted as Theorem 1.8 of Burrell and Yu 2021, give infinitely many integers with base-p digits <= A and base-q digits <= B whenever A/(p-1) + B/(q-1) >= 1, and 1/(3-1) + 2/(5-1) = 1.000000 exactly. The other two pairs miss the criterion by one digit each, both reading 0.833333, and both reach 1.000000 when the base-7 bound rises from 2 to 3; the criterion is sufficient and not necessary, so neither pair is claimed finite. Witness: cobham.md:88 and cobham.md:91 and cobham.md:92, lab/rs/three-base-thin verb budget, Burrell and Yu 2021 Theorem 1.8 read at source.
2026-09-14 Verified The members of the three-base thin set below 7^17 = 232630513987207 are 0, 1, 3186, 3187, 20007 and nothing else, found in 333 nodes. The walk enumerates the base-3 side as subset sums of distinct powers of 3 from the top power down and cuts a branch by a proved bound: once the powers 3^k and above are chosen the remaining addition is at most (3^k - 1)/2, so with j least such that 5^j > (3^k - 1)/2 the high part floor(n / 5^j) of every reachable n is one of two consecutive integers, and the branch dies when neither has all its base-5 digits <= 2; base 7 cuts the same way and membership in the base-3 set is never tested. Witness: cobham.md:94, lab/rs/three-base-thin verb seven 17.
2026-09-14 Verified The same five members and no sixth below 3^80000, a height of 38170 decimal digits, in 1710789 nodes and 61.48 s on about 3 GB. The node count grows near 21 level at height 3^level and the stored powers cost Theta(level^2) bits, so memory and not the clock is what stops the census. Witness: cobham.md:95, lab/rs/three-base-thin verb reach 80000.
2026-09-14 Verified The pruned walk is checked against a direct scan of every integer below 10^8 against all three digit rules, at base-7 bound 2 and again at base-7 bound 3, and the two agree in both. Witness: cobham.md:96, lab/rs/three-base-thin verb control.
2026-09-14 Verified The budget changes sign one digit away, and on the far side it is a heuristic already in print. By Lucas's theorem binomial(2k, k) is prime to p exactly when every base-p digit of k is below p/2, which reads <= 1 at 3, <= 2 at 5 and <= 3 at 7, so raising the base-7 bound from 2 to 3 gives {k : binomial(2k, k) is prime to 105}, OEIS A030979, whose triple budget log_3 2 + log_5 3 + log_7 4 - 2 reads 0.025951 rounded up against -0.121889 for the thin set; A030979 read at source records a prize for settling whether it is finite, names it as Erdos problem 376, and quotes a heuristic of Pomerance giving about x^0.02595... terms up to x, which is the same number as the budget. The same walk at base-7 bound 3 rebuilds all 23 terms A030979 publishes, counts 1374 members below 10^70, exactly the length of the table that entry calls complete to 10^70, and counts 216020 below 10^140; the effective exponents log(count)/log(height) read 0.044828 and 0.038103, truncated down, both above 0.025951 and falling. Witness: cobham.md:97 and cobham.md:98 and cobham.md:99, lab/rs/three-base-thin verbs budget, control, ten 70 and ten 140, OEIS A030979 read at source.
2026-09-14 Verified The twenty-five terms C(3^m) = 1, 3, 7, 19, 45, 111, 241, 467, 1175, 2443, 5285, 11939, 25281, 53477, 109001, 231737, 498083, 1077727, 2179165, 4372741, 9051805, 18107943, 37126191, 75050077, 151133095 at m = 0..24 are printed from scratch in this run by the verb terms and again by the verb hankel, which recomputes the census before its algebra, and the two agree term for term with each other and with the page. Both call the same pruned walk, so that agreement is determinism and not corroboration: the one independent rebuild is the verb control, which tests every pair of [0, 3^m)^2 against both digit rules with a separate routine and agrees at every m <= 6. Witness: lab/py/two-base-gasket verbs terms, hankel, control; cobham.md:77.
2026-09-14 Verified No linear recurrence with constant coefficients of order r <= 12 fits all twenty-five terms of C(3^m). Two exact tests over the rationals agree. The Hankel determinants of the matrix with entries a[i+j], computed by fraction-free Bareiss over Z, are nonzero at every size k = 1..13, reading 1, -2, 4, 8, 5360, -1259267712, -5516990041856, 112485023878830080, 5697070341654551514880, -1283183611235610612560681984, 512735847124145678895522067154944, -30665906970422092677442692752788846592, -148892102950447887517893509783802772470337536; a recurrence of order r would force every Hankel determinant of size above r to vanish, so det H_13 != 0 alone kills every order at most 12. Independently, for each order r = 1..12 the linear system a(n) = c_1 a(n-1) + ... + c_r a(n-r) taken over every one of the 25 - r available equations is inconsistent by Gauss-Jordan over Q, order by order with no order left consistent and none underdetermined. Order 12 is the largest the data can test: it leaves 13 equations against 12 unknowns, one spare, while order 13 leaves 12 equations against 13 unknowns, one short of determined, so that system is consistent for trivial reasons and tests nothing; 12 is exactly the bound twenty-five terms carry and not a choice. The two solvers behind the negative have their own generator: the verb selftest fits Fibonacci first at order 2 with x^2 - x - 1, 2^n + 3^n + 1 first at order 3 with dominant root 3.0000000, and n^3 + 2^n at no order below 5 and at 5 with (x - 1)^4 (x - 2), and checks bareiss against five determinants computed by hand, eleven checks in 0.05 s. Witness: lab/py/two-base-gasket verbs hankel, 3 min 15 s at HI = 24, and selftest.
2026-09-14 Verified The sequence 1, 3, 7, 19, 45, 111, 241, 467, ... is absent from the local OEIS dump, and so are eight simple transforms of it: the sequence from its second term, the first differences, the partial sums, a(n) - 1, a(n) + 1, (a(n) - 1)/2, 2 a(n), and the halved first differences. Each was searched as its seven-term window starting at the second term. Witness: grep of the local dump research/data/oeis/stripped, nine patterns, zero hits.
2026-09-14 Verified The decision rule is calibrated on one-base counts before any two-base object is read, and neither one-base control is clean. The base-3 design {0, 1} puts a maximum at 5.719220 against 2 pi / ln 3 = 5.719202, error 0.000%, at 1.7e7 times the median power of the band [0.5, 14] under the linear detrend and 1.9e7 under the cubic; the base-5 design {0, 1, 2} puts one at 3.903959 against 2 pi / ln 5 = 3.903963, error 0.000%, at 4.0e6 times the median. At 3^44 the base-3 control's nearest maximum to 2 pi / ln 5 sits at error 1.115% and 35.9 times the median under the linear detrend and at 0.966% and 39.2 under the cubic, which passes the rule at a frequency absent by construction; the base-5 control's nearest maximum to 2 pi / ln 3 sits at 1.060% and 2.21 under the linear detrend and at 0.963% and 1.95 under the cubic, inside the tolerance and held out by the power gate alone. Both control windows are longer than the cell's and give a prediction more room to be met by position alone: span 38.05 with 11 maxima above 10x the median covering 0.113 of the band, and span 38.49 with 4 covering 0.048, against the cell's span 27.41, 4 maxima and 0.020. Witness lab/py/two-base-instrument verb cell 44.
2026-09-14 Verified The multiplicatively dependent pair is a one-base control and not a two-base count. Base 3 digits {0, 1} against base 9 digits {0, 1, 3} satisfies the zero-digit hypothesis of the collapse theorem of bases.md, 0 lying in both digit sets, so the joint set is exactly the one-base design F(9, {0, 1, 3}) of exponent log_9 3 = 1/2; at the aligned height 3^28 = 9^14 its count is exactly 3^14 - 1 = 4782968, the fitted exponent is 0.497809, and its spectrum shows 2 pi / ln 9 = 2.859601 at 2.859889, error 0.010%, at 886 times the median, with the first harmonic 2 pi / ln 3 at error 0.003% and 2.5e3 times the median, on a window of span 17.47 carrying 4 maxima above 10x the median covering 0.042 of the band. Witness lab/py/two-base-instrument verb collapse 28.
2026-09-14 Verified The two-frequency prediction is what the two band structures alone predict, and the cell does not meet it at their scale. A member of the base-3 design with k+1 digits lies in [3^k, (3^(k+1)-1)/2] and a member of the base-5 design with i+1 digits in [5^i, (5^(i+1)-1)/2], so the designs occupy ln(3/2)/ln 3 = 0.369070 and ln(5/2)/ln 5 = 0.569323 of their own decades and the cell's count is exactly constant wherever the two bands miss: of the 47 decades [3^j, 3^(j+1)) below 3^47, the 10 with j = 1, 4, 17, 20, 23, 26, 36, 39, 42, 45 carry no member at all, proved by the bands and witnessed by the ladder, whose new hits are 0 at level = 40, 43, 46. The block model C3(N) C5(N) / N, the same two band structures multiplied with no joint arithmetic, shows 2 pi / ln 3 at error 0.001% and 1.33e6 times the median and 2 pi / ln 5 at 0.004% and 9.79e5, on a window of span 31.55 with 4 loud maxima covering 0.043; the cell's own count on a window of span 30.70 with the same 4 loud maxima and 0.020 reaches 22.5 and 15.7. Witness lab/py/two-base-instrument verbs blocks 47 and ladder 38 47.
2026-09-14 Verified The rule's verdict on the cell is not stable in height, so this run settles the two-frequency prediction neither way. On the full window under both detrends the ladder from 3^38 to 3^47 returns both frequencies present at level = 38, 39, 40, 41, 2 pi / ln 3 at errors 0.473%, 0.073%, 0.342%, 0.704% under the linear detrend and 0.325%, 0.036%, 0.484%, 0.862% under the cubic at 19 to 24 times the median, and returns 2 pi / ln 3 absent from level = 42 up, at errors 1.034% to 1.574% and 22 to 25 times the median, while 2 pi / ln 5 is present at every one of the ten heights. The error at 2 pi / ln 3 rises monotonically from level = 39 to level = 46 under both detrends as the window lengthens, which is a nearest maximum drifting away from the prediction rather than an estimate converging on it. On the upper half of the window at 3^44 the nearest maximum to 2 pi / ln 3 sits at error 0.825% under the linear detrend and 0.791% under the cubic, inside the tolerance and held out by the power gate alone at 5.55 and 6.1 times the median, and at 3^47 at 2.430% and 2.293%. No height rule is stated in advance, so no height is entitled to the verdict. Witness lab/py/two-base-instrument verbs ladder 38 47, cell 44 and cell 47.
2026-09-14 Verified The cell's strongest maximum matches no small combination of the two lattice frequencies. It sits at 1.702087 at 3^44 and 1.697925 at 3^47, at 57 and 68 times the median on windows of span 27.41 and 30.70 carrying 4 maxima above 10x the median covering 0.020 of the band, and the nearest m 2 pi / ln 3 + n 2 pi / ln 5 with abs(m), abs(n) <= 8 is (1, -1) = 1.815239 at errors 6.233% and 6.463%; 2 pi / ln 15 misses by 3.750% and 3.457% and the sum frequency by 1.528% and 1.977% at 2.2 and 2.1 times the median, while 2 pi / ln 5 is met at 0.461% and 0.594%. Block structure does not account for it: the block model's nearest maximum to 1.815239 sits at 1.895843 and 0.588 times the median at 3^44 and at 1.697497 and 1.58 at 3^47. The same verbs read the local counting exponent falling through the budget 0.313536, 0.323301 over the full window and 0.295978 over its upper half at 3^44, 0.318728 and 0.292184 at 3^47, which is a fit at finite height and decides nothing about the limit. Witness lab/py/two-base-instrument verbs cell 44, cell 47 and blocks 47.
2026-09-14 Proved The dimension-one lane of coprime, Conjectures O, W and Z with the shelf lane lemma-b-pincer, is a one-base problem whose automata come in families indexed by the object read, and never meets Cobham's hypothesis. Every object of the lane is read in base 3 alone: the gasket G_n is the base-3 digit pairs from {(0,0),(1,0),(0,1)}, the word binary inside the lane means a base-3 expansion with digits in {0, 1} and never base 2, and the only other bases in the section are the general simplex corollary at coprime.md:150 and the base-4 and base-5 simplex probes at coprime.md:253, each a separate design in its own single base and not a second base on one object. The machines are one automaton per primitive ray, per multiplier pair B(s,t), per band direction and per modulus 3^k; the band family, whose states are the integers in [-(z_2-1)/2, (z_1-1)/2] (coprime.md:233), and the digit-congruence family indexed by 3^k (coprime.md:228) carry no uniform state bound, while B(s,t) carries only the upper bound (s+1)(t+1) (coprime.md:214) and a measured 167 live states at (25,52) and (31,40) (coprime.md:197, gasket-ray-machine), and the ray automata grow like ab in the shelf lane's own words. Cobham asks one set recognized by a finite automaton in two multiplicatively independent bases, which the lane never presents, so the wall of cobham, that no transfer matrix over the digits of one base reads the constraint a second independent base imposes, does not touch the lane. Witness: coprime.md:187; the shelf lane README reading "counting the multiples of a ray inside the gasket is a finite automaton on the base-3 digits of the multiplier, with growth rate rho(a,b). There are infinitely many rays and the automata grow like ab, so no computation settles them"; cobham.md:41.
2026-09-14 Conjecture Block designs are the only semilinear designs, at every base and every dimension. The count alone is not enough, which is what makes the conjecture nontrivial: the base-3 gasket has card F = 3 = 3^1 and is not semilinear. Witness: cobham.md:35.
2026-09-14 Conjecture The true counting exponent of the two-gasket intersection is unknown and twenty-five levels decide nothing. log_3 C(3^m) / m reads 0.754141, 0.749634, 0.746312, 0.743739, 0.738025, 0.732546, 0.729032, 0.724371, 0.721151, 0.717651, 0.714298 at m = 14..24, falling by about 0.004 a level on the mean of those ten steps and still 0.129 above the budget at the last level, consistent with any limit from 0.630929 upward. The cheap falsifier is a linear recurrence for 1, 3, 7, 19, 45, 111, 241, 467, ..., which would make the growth rate algebraic and put the object back inside this tree's machinery; it is not run here. Witness: cobham.md:81 and cobham.md:82, lab/py/two-base-gasket verb terms.
2026-09-14 Conjecture The Schanuel wall. Every growth exponent this tree prints has the shape log(algebraic)/log(base), the Perron root of a nonnegative integer matrix read in its own base, and every two-base budget has the shape sum_i log(k_i)/log(p_i) - (m-1) dim, a Q-linear combination of 1 and the ratios log k_i / log p_i, the wall assuming that a realized two-base exponent has that shape too, which nothing here proves; those two families meet only where one side degenerates, and under Schanuel's conjecture they meet nowhere nontrivial, so no instrument of this tree ever outputs a two-base exponent. This is strictly stronger than Cobham, which forbids the set from being automatic while the wall forbids the number from being a Perron root, and neither implies the other. Witness: cobham.md:108, Burrell and Yu Theorem 1.6 and Theorem 1.11 read at source.
2026-09-14 Conjecture The three-base thin set is exactly {0, 1, 3186, 3187, 20007}. Finiteness is open and no theorem on the page gives it: the dimension bound is 0 and the page already proves that a budget of zero says nothing about finiteness, while the finiteness results in print, Senge and Straus 1973 and Stewart 1980, bound digit sums rather than digits, so applying either to the thin set would need a bound on the base-3 digit sum of a member, which is the finiteness in question. The falsifier is a sixth member and the census is where it would have shown. Witness: cobham.md:101 and cobham.md:102, lab/rs/three-base-thin verb reach 80000.
2026-09-14 Conjecture C(3^m) satisfies no linear recurrence with constant coefficients of any order, so the counting exponent of Object Y is not log_3 of a Perron root and the Schanuel wall holds in data for Object Y. What is proved is the order at most 12 case above; the step to every order is the conjecture, and nothing here rules out a recurrence of order 13 or more. Falsification: order r is testable once the terms leave its system a spare equation, which wants 2r+1 of them, so order 13 wants 27 terms, the two levels m = 25 and m = 26 beyond what is run, about 11 min by the measured per-level factor; the same verb tests any order the census reaches. Witness: lab/py/two-base-gasket verbs hankel and selftest, extrapolated from the r <= 12 negative.
2026-09-14 Conjecture The joint digit constraint destroys the oscillation that either design carries alone. The block model of the same two designs carries both lattice frequencies at 10^5 to 10^6 times the median while at 3^47 the cell's own count carries 2 pi / ln 5 at 15.7 times the median and puts nothing nearer to 2 pi / ln 3 than a maximum 1.488% away, so the intersection is not the product of its two band structures at the level the spectrum reads. A nonlattice Moran system has its complex dimensions off any arithmetic progression and its detrended count carries no sharp frequency, which is consistent with the cell's flat reading, and nothing in this run separates that reading from the height instability the ladder prints. A positive test has to read the spread of the complex dimensions rather than a comb, and that needs a zeta function for the joint object, which needs a gap structure, which is what Cobham denies. Witness lab/py/two-base-instrument verbs cell 44, cell 47, ladder 38 47 and blocks 47.
2026-09-14 Refuted The sentence "no proper design is recognizable in two independent bases, in any dim" is false. At dim = 2 and base = 2 the design F = {(0,0), (1,1)} has S_F = {(n, n)}, the diagonal, which is definable in <N; =, +> and so recognizable in every base by the easy half of Cobham-Semenov. Proper designs recognizable in two independent bases exist as soon as dim >= 2, so the dim = 1 lock does not generalize by itself and the theorem must be stated as Cobham-Semenov states it, with semilinear in place of ultimately periodic. Witness: cobham.md:31, Bes Theorem 25 read at source.
2026-09-14 Refuted The global planar budget dim(A cap B) <= max(0, dim A + dim B - dim) is false at dim = 2, in one line and with product designs. The base-2 design F_A = {(0,0), (0,1)} gives A = {0} x T of dimension 1, the base-3 design F_B = {(0,0), (0,1)} gives B = {0} x C of dimension log_3 2, B sits inside A, so the intersection has dimension log_3 2 while the budget reads max(0, 1 + log_3 2 - 2) = 0. Both sets lie in the line {0} x T, invariant under both maps, and that is exactly where the two codimensions refuse to add. Witness: cobham.md:60, lab/py/two-base-gasket verb budget.
2026-09-14 Refuted The global budget is not a corollary of the per-axis bound, and the step that fails is sum_j max(0, x_j) >= max(0, sum_j x_j), which runs the wrong way. The one-line witness above is where it runs strictly wrong: the per-axis bound reads 0 + log_3 2 and is sharp there, while the global budget reads 0. So in dim >= 2 the two-base budget is a theorem per axis for product designs and open for compounds, and core.md proves almost every design is a compound as dim grows. Witness: cobham.md:62.

Complex dimensions

2026-08-28 Conjecture Every one-base design is lattice, so the Lapidus-Maier machinery is not out of reach but empty: zeta_level(s) = 1/(1 - fill base^(-s)) puts the complex dimensions on one vertical line of period 2 pi/ln(base), and (ISP)_dim, the Riemann hypothesis in the language of fractal strings, has no content on the degenerate case, several incommensurable ratios being the change that gives it content; the lattice half is checked on this tree's own poles and folding tables, the (ISP)_dim half is a literature reading not yet checked at source, and the dichotomy is a theorem for strings only, so dimension two and above is open outside the pluriphase class, inside which the carpet and the interior-hole designs are settled. Witness: lab/py/complex-dimensions.
2026-09-06 Proved The Sierpinski carpet is not Minkowski measurable: its complement in the open square is the disjoint union of 8^(m-1) open squares of side 3^(-m) whose boundaries lie in the carpet, the tube is the exact hole sum, and eps^(log 8/log 3 - 2) V(eps) -> G(t) with G = t^(log 8/log 3 - 2)(1 + 4t/5 - 4t^2/7) on [1/3, 1/2) and t^(log 8/log 3 - 2)(9/8 + 3t/10 - t^2/14) on [1/2, 1), C^1 at the seam, 379/280 at the ends, maximum 1.35561708227 at t = 0.429638, minimum 1.3506702097 at t = 0.692137, swing 0.3662%; a corollary of Kombrink, Pearse and Winter 2016 Theorem 1.1(ii), whose hypotheses are verified for the carpet with the open square, the profile and the elementary proof being the addition. Witness: lab/py/complex-dimensions carpet_tube.py, dimensions.md measurability with its hypotheses.
2026-09-06 Proved Every one-base design at base >= 3 in dim >= 2 removing at least one digit vector, all removed vectors interior and pairwise differing by at least 2 in a coordinate, is not Minkowski measurable: base^(dim-1) < fill < base^dim so log(fill)/log(base) is never an integer, G(t) = t^(log(fill)/log(base) - dim) sum_j fill^(j-1) base^(-j dim) h(t base^j) > 0, and t^(dim - log(fill)/log(base)) G is a polynomial on [1/base, 1/2); includes the parity carpets at every odd base and dimension. Witness: dimensions.md measurability with its hypotheses (proof), lab/py/complex-dimensions at base = 3, dim = 2 only.
2026-09-06 Refuted The hole-sum route for the sponge: the level-1 plus hole's boundary is not in the sponge ((1/2, 1/2, 1) at distance 1/6, (2/3, 1/2, 5/6) at distance 1/18), the tube inside the hole is not its parallel volume, and the pluriphase theorem does not reach the sponge with the open cube; the sponge stays Conjecture. Witness: dimensions.md measurability with its hypotheses, qualification 2, read off the digit rule.

Component counts

2026-08-28 Refuted The boundary state of a Kronecker word grows with level, so the component count is not a finite-state function of the code sequence - one failed trial state (the four-corner partition of the running product, exact at length 2 and wrong on 20 of 216 words at length 3) bounds nothing; a linear representation of rank 4 exists and is exact on all 54240 words of length at most 4; what is unbounded is the naive geometric state kappa, which reaches 2^(level-1). Witness: order-sensitivity-of-kronecker-words.

Conjecture S: even half

2026-08-28 Proved The reduction chain for Conjecture S runs at every odd base, middle-digit design, dim >= 2: the digit polynomial is palindromic with strictly positive support; the transfer recursion hat u_(level+1)(psi) = (1/base) sum_r Phi(y_r) hat u_level(y_r) holds with exact phase cancellation (the palindromic centre is the carry offset dim m); the step identity is W_k = (-1)^(dim-1)(dim-1) base^(k-1) V(k-1) with V(level) = base m0(level) - b(level); the carry core has reachable set exactly {|c| <= floor((dim-1)/2)}, is irreducible and aperiodic, and its Perron root is rho_dim; eventual contraction (V(level) >= 0 for all level >= level_0, even dim) implies rho_dim <= fill/base; and det(fill I - base M_even) == fill^n mod p for any p | base, primality unused, gives strictness whenever p nmid fill; the Perron-Frobenius asymptotic is unnecessary (b(level) >= (M^level)[0,0] suffices) and the base-3 contraction hypothesis weakens to its eventual form since the odd-level dip is a finite transient. Witness: slice-sign-even-half.
2026-08-28 Proved The exact two-step reduction 9 b(2j+2) - fill^2 b(2j) = -(dim-1)[fill V(2j) + 3 V(2j+1)], the two-step weight identity C_2(y) = g_5(2y) (the base-3 two-step symbol is the base-5 symbol, the comb constant having minimal polynomial x^3 - 9x - 9), and the orbit identities G(3t) = cos(t) G(t) and Ntilde(3^a pi) = ((dim-2)/(dim+2))^a Ntilde(pi). Witness: slice-sign-even-half.
2026-08-28 Proved The Collatz-Wielandt certificate closes the even half per dimension at any odd base, strictness included: B >= 0, x > 0, Bx < theta x componentwise imply rho(B) < theta with no irreducibility needed; with beta_K = (M^T)^K 1 > 0 (positive by column-sum positivity alone, colsum(c) = (fill + (-1)^(dim-1)(dim-1)(base[base|c]-1))/base > 0), if base beta_(K+1)(c) < fill beta_K(c) for every |c| <= (dim-1)//2 then rho_dim < fill/base strictly, bypassing the mod-p determinant lemma and every exceptional class; the base-5 mass identity on the right Perron vector of the full core is 5 rho_dim = fill - (dim-1)(5 p_dim - 1), so the even half is p_dim > 1/5, and the left-vector reading is false at dim = 8 (p_LEFT = 0.1428 < 1/5); K = 2 certificates are exact at dim = 16, 30, 44, 60. Witness: slice-sign-even-half.
2026-08-28 Verified The row certificate v^T M^t >= 0 is sound - V(level) = sum_j (v^T M^t)_j u_(level-t)(j) with both factors nonnegative, so one integer t with v^T M^t >= 0 entrywise plus the exact prefix V(0..t-1) >= 0 proves V(level) >= 0 for all level, monotone in t - and at base 5 its minimal depth is t(dim) = max(1, ceil(log_5(2 dim - 3)) - 1), breakpoints exactly at R = (5^(k+1)-1)/4, checked to even dim = 400 with fresh rows at 150, 250 and the boundary 314|316; the 782 at k = 4 is an extrapolation unobservable below dim = 1566. Witness: slice-sign-even-half.
2026-08-28 Proved The 2-adic strictness lemma, complementary to the mod-p lemma: det(fill I - base M_even) == (-base)^n det(M_even) (mod fill) by principal-minor expansion (every k < n term killed by fill^(n-k)), so a prime p | fill, p nmid base with v_p(det M_even) < v_p(fill) forces det(fill I - base M_even) != 0, that is rho_dim != fill/base, which upgrades a certificate's <= to <; the two lemmas' silent classes (p | base against p | fill) are complementary; at base 5 v_2(fill) = 2(dim-1) + v_2(dim+4) while v_2(det M_even) <= 26 out to dim = 156, so the test holds everywhere including the exceptional class dim == 6 (mod 10) to dim = 156, and at base 3 the class dim == 4 (mod 6) to dim = 118; the 5-adic side is large and erratic and the mod-25 angle is dead; open: a uniform bound on v_2(det M_even), <= n sufficing for all even dim >= 4. Witness: slice-sign-even-half.
2026-08-28 Proved The even half of Conjecture S at base 5 is a theorem entire: on the base-5 middle-digit solid rho_dim < fill/5 for every even dim >= 2, hence slice dimension < solid dimension - 1 at every even dim - the Fourier form beta_K(c) = 5^(-K) sum_n F_K(n) e^(2 pi i n c/5^K) (the product-formula phase cancellation iterated), the exact telescoping fill beta_K - 5 beta_(K+1) = 5^(-K)(dim-1) Sigma_K(c) with the fill-power leading terms cancelling identically, the frequency-separation lemma Q_K(n)/Q_K(1) <= 0.768 for every n != +-1 (three-branch residue analysis in rigorous intervals, maximum 0.7679580, read 0.7679541 in an earlier check, the j >= 3 branch exhaustive at j = 3, 4, 5), and the criterion at depth K(dim) = Theta(log dim), analytic for even dim >= 18 (K(dim) >= 2 for dim >= 34 since 4(dim-2) >= 128 > 25) with exact integer certificates below; the log depth is necessary, every fixed K dying at dim = 16, 66, 316 for K = 1, 2, 3; the criterion holds directly at every even dim = 34..600 and at every depth transition to dim = 10^6; the death law dim = 2 ceil(5^(K+1)/4) + 2 is known at three depths only; with the odd-dim theorem, Conjecture S at base 5 is settled everywhere except strictness at odd dim == 1 mod 5, exact through dim = 80. Witness: slice-sign-even-half.
2026-08-28 Verified The even half at base 3 holds per dimension for every even dim = 2..102 and on the grid 106, 110, ..., 178 - rho_dim < fill/3 with strictness at each, by the exact-integer Collatz-Wielandt certificate 3 beta_(K+1)(c) < fill beta_K(c), no determinant lemma and no exceptional class dim == 4 mod 6 needed; beta_K(0) = b(K) exactly, so K_min >= level*(dim) + 1 with level* the last level with V(level) < 0, and K_min = level* + 1 or + 2 at every tested dim; spot depths K_min = 8, 50, 140, 291 at dim = 12, 30, 50, 72; two implementations sharing no code reproduce all 36 rows to every digit, including the two non-monotone slack rows. Witness: slice-sign-even-half.
2026-08-28 Verified The odd half's last gap narrows to the same 2-adic bound: at base 3 and odd dim == 1 mod 3, where the mod-3 strictness lemma is silent, v_2(det M_even) < v_2(fill) = (dim-1) + v_2(dim+2) at every dim = 13, 19, ..., 241, silent only at dim = 7 (v_2(det) = 7 >= 6, closed by the direct computation det(fill I - 3M) != 0), so rho_dim != fill/3 on 13 <= dim <= 241 and, with rho_dim >= fill/3 at every odd dim, rho_dim > fill/3 strictly at every odd dim <= 241; the reference v_2 rows on the two other classes read 1, 2, 3, 1, 4 (base 3, dim == 4 mod 6) and 2, 1, 2, 3, 2 (base 5, even); both exceptional classes of Conjecture S reduce to one uniform statement, an upper bound on v_2(det M_even), and v_2(det M_even) <= n = (dim+1)/2 at every dim = 13..241 in the class, failing only at dim = 7, is exactly strong enough. Witness: slice-sign-even-half.
2026-08-28 Proved The even half of Conjecture S at base 3 is a theorem entire: rho_dim < fill/3 for every even dim >= 2, hence slice dimension < solid dimension - 1 at every even dim, base 3, middle-digit design, strictness included - by the Fourier/telescoping port to base = 3 (the exact phase cancellation load-bearing, off-centre variants failing with integer witnesses), four nested frequency tracks (+-1 at angle 0, the half-points +-(3^(K+1)-1)/2 at the tripling fixed point pi; on-track prefixes nest, exits never return), exit-cost and window/subtree lemmas giving E_K(dim) <= [4(K-1)(0.7528157^(dim-1) + 0.7052518^(dim-1)) + 2 * 0.2266816^(dim-1)] exp(2(K+1) 0.8900159^(dim-1)) for everything off the two leader pairs, and the criterion closing at K_1(dim) = K*(dim) + O(1), analytic for even dim >= 38 (182 interval-certified inequalities to dim = 400, monotone domination beyond, the margin term the true cosine deficit delta(dim) = O(3^(-2 K_1)), < 4.1e-76 at dim = 38, since an absolute 1e-8 term fails at dim = 399999998), exact certificates below; checks: Fourier form to 9.4e-61, telescoping to 2.6e-59, tracks exhaustive over all 3 !| n < 3^10, the subtree bound never exceeded (worst sigma_5 = 1.023 against 8.97), the E-bound dominating exact enumeration at all 35 (dim, K) points and at 28 fresh ones (dim in {10, 14, 22, 26} x K in {3..9}, worst ratio 19.74), class counts exact at K = 9, an exit-level sweep to j = 60, dim = 4000 finding the caps asymptotically exact (worst attainment 0.999121) but never breached, seven certified constants re-derived to 22 digits by exact interval arithmetic on a 10^-90 grid with outward rounding (a hand-rounded 0.6696 reads 0.66966), and the theorem machine-checked in exact integers at dim = 38, 40, 42 (the certificate holds at exactly K_1, fails at 0.8 K*, K_1 = K_min + 1 at all three); the chain's single global safety factor is 2 and the h-exit(2) attainment (two of four residues reach C_H) is load-bearing. Witness: slice-sign-even-half.
2026-08-28 Proved The base-3 transient is identified in closed form: level*(dim) is the greatest odd integer <= K*(dim), K*(dim) = [(dim-1) ln R + s_dim]/ln((dim+2)/(dim-2)), R = prod_(i>=2) cos(pi/3^i)/cos(2 pi/3^i) = 1.2553249438... - the half-point frequency rides the pi fixed point with per-level magnitude advantage cos(pi/3^i)/cos(2 pi/3^i) > 1 against per-level amplitude cost about (dim-2)/(dim+2), its sign alternates as (-1)^K (that is the odd-level dip), and the crossing is the transient - so the certificate depth constant is ln(R)/4 = 0.0568486146... and K_min in [level* + 1, ceil(K*) + 2] for even dim >= 38; exact on 58 of 58 level* rows, every even dim = 6..120, each one exhausted by proof and not by margin - M >= 0 and u_level = M^level e_0 >= 0, so a single t with (M^T)^t v >= 0 entrywise forces V(level) >= 0 at every level >= t and no census window can truncate the answer - with towers level* = 79, 97, 107 at dim = 38, 42, 44 and level* = 811 at dim = 120; a census carried only to a 4 dim + c window is unsound past dim about 70 since level* is quadratic, but no row of dim = 6..120 is in fact false; the column-sum identity fill - 3 colsum(c) = (dim-1) v_c, which is prop:mass by root-of-unity filtering and needs no per-row check, makes that row certificate the even-half Collatz-Wielandt test itself, so the stopping level is K_min exactly, level* + 1 on 36 rows and level* + 2 on 22; scoped to dim >= 6 since dim = 4 has no dip; at dim = 10, 20 the half-pair carries the largest magnitude in the spectrum, the dominant pair only third. Witness: slice-sign-even-half, lab/py/base3-transient-exhaustion.
2026-08-28 Proved Base 3 is the unique hard base: the half-point track exists at strength |A_base(-1)|/A_base(1) per level with A_base(-1) = 1 - (-1)^((base-1)/2), so the ratio is 1 exactly at base = 3, 0 at every base == 1 mod 4 (the symbol dies at pi, the base-5 case) and 2/(base-1) < 1 at every base == 3 mod 4, base >= 7. Witness: slice-sign-even-half.
2026-08-28 Verified Exact Collatz-Wielandt certificates give rho_dim < fill/9 at every even dim = 2..56 and rho_dim < fill/11 at every even dim = 2..74 (machine-pinned to dim <= 42 and dim <= 60), K_min <= 2, V(level) > 0 everywhere, no transient, and base 9 = 3^2 inherits nothing from base 3; at base = 7 the K = 2 -> 3 step lands at exactly dim = 174 as the frontier-race law predicts, the frontier f_2 = 85 converged from dim = 160, the asymptotic death law landing there too; earliness (asymptotic death minus true death) is monotone down in K and up in base - the depth-0 death is dim = 4 at every base, so base 5 is one even step early at K = 0 (4 against 6) and exact at K = 1, 2, K_0(5) = 1; base = 7: 1, 1, 0 steps; base = 9: 2, 1; base = 11: 2, 2 - the asymptotic law being exact for all K >= K_0(base); at base = 9 the window edge is immune when h == 0 mod base (dim = 20, 38), so tightness must be stated mod base; the 12 printed constants of the base-3 chain are asserted against interval endpoints and printed by ceiling. Witness: slice-sign-even-half.
2026-08-28 Conjecture The row certificate's sign law at base 5 is periodic, not one-sided: (v^T M^k)_j >= 0 iff dist(j, 5^(k+1) Z) <= (5^(k+1)-1)/4 (witness dim = 40, k = 1, j = 19 positive), the threshold being the carry-drift radius around every multiple of 5^(k+1), not just around 0. Witness: slice-sign-even-half.
2026-08-28 Conjecture Three negatives on the base-5 even half: quintupling resummation is structurally empty, the two orbit relations of the base-5 tower summed over complete residue systems returning exactly the one-step identity 5 b(level+1) = fill b(level) - (dim-1) V(level), the nontrivial comb mapping into the trivial comb whose orbit factor is 1 and the correctly normalised scaling limit of V being 0 = 0; any envelope bounding numerator and denominator independently dies at psi = 0, the cone having zero width at both census points 2 pi/5 and 4 pi/5 (both slack summands nonnegative with nonpositive sum), so only curvature-coupled envelopes remain; and the neutral Gaussian width of the transfer at psi = 0 is exactly a* = m2/(24 fill) = Var(digit sum)/24 = 5 dim (dim+3)/(48(dim+4)), not 5 dim/48, which explains the measured upward drift of sig2/dim toward 5/48.
2026-08-28 Conjecture At every odd base base >= 5 the even half falls to the base-5 template at depth O(log dim) with no transient: V-towers at base = 5, 7, 9, 11, 13, even dim = 8, 12, level <= 25 show no dip anywhere off base 3. Witness: slice-sign-even-half.
2026-08-28 Refuted The base-3 resummation mechanism built on those identities - the scaling limit 3^n f_n(psi)/fill^n -> Sigma(psi) is false, the exact tower falling geometrically to 0 as it must since a nonzero limit would contradict rho_dim < fill/3; the claimed absolute convergence is false, the per-decade absolute mass of |G(m pi)|^(dim-1) Ntilde(m pi) growing at dim = 4 (block ratios 1.081 to 1.115 out to m = 2e7) and rising through 1 at dim = 6, 8; the tail-to-lead figures -0.1812/-0.0464/-0.0121 are artifacts of the m <= 199 cutoff, still moving at m <= 2e5; Sigma(pi) > 0 is unproved at every dim; the leader bound max_(m>1) |G(m pi)| = |G(7 pi)| = 0.2520527 holds to m <= 20001. Witness: slice-sign-even-half.
2026-08-28 Refuted V_(2j+1) < 0 for every even dim >= 4 - at dim = 4, V_level > 0 for every level <= 40 (V_1 = +4 exactly) and at dim = 6, V_3 = +135092 > 0; the odd-level dip is a transient of length about 0.055 dim^2, and dim = 2, 4 never dip. Witness: slice-sign-even-half.
2026-08-28 Refuted Certificate depth K = 2 closes every even dim at base 5 - 5 beta_3 < fill beta_2 holds for even 16 <= dim <= 64 and fails at every even dim = 66..320, first at dim = 66 at the edge carry |c| = 32 = (dim-2)/2, relative deficit -2.19e-43; beta_K has Fourier support 5^(-K) Z, the dominant frequency n = +-1 gives Sigma_K(c) ~ 2 T_K(1) cos(2 pi c/5^(K+1)), so depth K sees only carries inside the quarter-period |c| < 5^(K+1)/4 and dies at dim = 2 ceil(5^(K+1)/4) + 2 - predicted deaths 16, 66, 316, 1566 at K = 1, 2, 3, 4, the first three exact - every fixed depth is finite, Theta(log dim) growth is necessary, and the minimal K equals the row certificate's t(dim) at every breakpoint tested (14|16, 64|66, 314|316). Witness: slice-sign-even-half.
2026-08-28 Refuted The base-3 certificate depth is exactly (9/160) dim^2 - exact lower bounds put the residual at +2.00 by dim = 120 and +9.78 at dim = 178; 9/160 = 0.05625 is the first two digits of the true constant ln(R)/4 = 0.0568486146.... Witness: slice-sign-even-half.

Conjecture S: odd half

2026-08-28 Proved The slice census has the trigonometric product formula P(e^(i psi)) = e^(i dim psi) (2 cos psi)^(dim-1)(dim + 2 cos psi), and the sheaf census b(level) (coordinate sum == dim(3^level-1)/2 mod 3^level, equally the free-end carry count) is b(level) = 3^(-level) sum_(m<3^level) prod_(j<level) Phi(2 pi m 3^j/3^level) because the extraction phase is the accumulated palindromic phase; every unit tower ends at 2 pi u/3 with factor (-1)^(dim-1)(dim-1), so at odd dim the integrand is pointwise nonnegative, b(level) >= (fill/3)^level and rho_dim >= fill/3, and det(fill I - 3 M_even) == fill^n mod 3 gives rho_dim > fill/3 strictly at every odd dim = 0, 2 mod 3 and through dim = 80 in the class 1 mod 3 by exact determinants (Bareiss, three 61-bit primes and Berkowitz agreeing); the bijection is brute-forced at dim = 2..8, level <= 4 and the phase cancellation matched to 50 digits at dim = 2..12. Witness: slice-recurrence-order.
2026-08-28 Proved The pinning |rho_dim - fill/3| <= 2(dim-1)/3 holds unconditionally (even dim in [fill/3 - 2(dim-1)/3, fill/3 + (dim-1)/3], odd dim mirrored) because the core's column sums take exactly the values fill/3 + 2 eps and fill/3 - eps; exactly 3 rho_dim = fill + (-1)^(dim-1)(dim-1)(3 p_dim - 1) with p_dim the Perron carry vector's mass on carries divisible by 3, well defined since the core is irreducible for all dim; so slice dimension - (solid dimension - 1) -> 0 like dim^2 2^(-dim) regardless of sign, and Conjecture S entire is the parity-free inequality p_dim > 1/3; checked by power iteration at dim = 2..20 and entrywise column sums at dim = 2..80. Witness: slice-recurrence-order.
2026-08-28 Verified The sign-law mechanism is universal: at every base >= 3 and u != 0 mod base the design symbol has g_base(2 pi u/base) = -1, the full digit sum vanishing at a nontrivial base-th root of unity and the middle digit contributing 1, so the innermost tower factor is (-1)^(dim-1)(dim-1) at every odd base and the mechanism is base-th-root evaluation, never P(-1); the odd-dim inequality rho >= fill/base travels with scope dim >= -min g_base (9/4 at base = 5, (34+14 sqrt 7)/27 at base = 7, growing like 0.217 base), strict when dim != 1 mod p for some prime p | base; the sign law is exact by Sturm counts at base 5 dim = 2..26, base 7 dim = 2..18, bases 9, 11 dim = 2..12 and (base,dim) = (21,3), (31,5), (51,5), (101,3). Witness: slice-sign-even-half.

Conjecture S: the even-half transient

2026-08-28 Verified The two-step census contraction 9 b(2j+2) <= fill^2 b(2j) with 3 b(2j+1) < fill b(2j) has zero violations through index 400 at every even dim <= 50, margin peaking near (dim-2)/(dim+2), and in exact integers at even dim = 2..30 to index 40 the margin sits strictly below (dim-2)/(dim+2) and rises toward it, 0.8704914 against 0.875 at dim = 30; with irreducibility and nonvanishing it would close the even half. Witness: slice-recurrence-order.
2026-08-28 Conjecture Every weighted-L2 certificate for the even half fails, the transfer norm being at least sqrt(3) fill for every positive weight, and two Abel pairings fail with it.
2026-08-28 Refuted The even half's early W_k sign alternation at base 3 persists, and a conjecture can be built on it - the alternation is a transient: over all even dim the first break is dim = 6, k = 4, then (8,6), (10,8), the rule k = dim - 2 dying at dim = 16 where the first break is k = 16, then 20, 24, 28, 34, 40 at dim = 18..26 and none through k = 40 for dim = 28..40 (inside the window even 12 <= dim <= 22 the first break is (dim,k) = (12,10)); structurally W_level ~ C 3^(level-1) rho^(level-1)(3 rho - fill) makes the eventual W-sign the even half itself; the pointwise route is dead too, V_2 < 0 for even dim >= 6. Witness: slice-recurrence-order, slice-sign-even-half.

Coprimality at dimension one

2026-08-28 Conjecture No fixed modulus decides mixed-radix coprimality: the smallest moduli labelling coprimality exactly on the n = 12 sets are 27994 and 20736, at which all 4096 values occupy distinct residues, an encoding of the finite set rather than a transfer matrix; a prime not dividing M is invisible modulo M, so no fixed finite state space decides coprimality on an unbounded family, and a finite matrix tracks a fixed finite prime set exactly and nothing beyond, which is why the truncated Euler product through 13 misses by 0.008977 and -0.031676 on the two alternating schedules.
2026-08-28 Refuted The universal pair-prefix transfer matrix is a route to Conjecture W - its Perron root is k^2 = 4 for every dim under coupled digit vectors, or 4^dim under the scalar tensor reading, never 3; the 3 in W belongs to the shift multiplier-pair automata, where lambda(1, 3^r) = 3 exactly and every other coprime pair has lambda <= 2, with 2 attained at (1,4); the octave census that was fitted is the unweighted count, not W's weighted (3/2)^K sum, its exponent on the stabilised octaves j = 0..3 is 9.36, and the all-octave fit alpha = 2.956, CI [2.682, 3.257], leans on right-truncated high octaves with its constant drifting C = 1.042, 1.136, 1.244, 1.356 at n = 13..16, residual Durbin-Watson 0.261. Witness: gasket-ray-machine.
2026-08-28 Refuted Weil's theorem reaches the coprimality window - over F_3[t] the restricted coprime count grows like 4^n, a positive density among ordered pairs, so gamma = log_3(4) = 1.261860 and the window analogue (gamma/2, 1/2] is empty, gamma/2 = 0.630930 already above 1/2; the quantity is positive-density counting with no zeta error term, so the framework is sound over a field where the Riemann hypothesis is a theorem while carrying no zeta content; the like-for-like test is the F_q[t] analogue at a prime power q of the ray-multiplicity second moment, still undone. Witness: lab/py/function-field-density.
2026-09-06 Proved At digit length k = 2t+1 the lift T = [t, 2t-1] has multiplier m_T = 3^(2t) - 3^t + 1 = Phi_6(3^t) dividing the binary 3^(3t) + 1, so K_T carries at least 2^t submasks divisible by m_T against an equidistribution model below 1; no uniform bound of the shape C 2^k / m_T^c survives c > log 2 / (2 log 3) = 0.3154649 while Sum_T m_T^(-c) converges only for c > log 2 / log 3 = 0.6309297, so every exponent that would close the lift-union half is refuted for that shape. Witness: lab/py/ratio-set-saving, ratio.py check 2.5 s and ratio.py lifts --kmax 19 21 s.
2026-09-06 Proved Antipodal lift family: for odd p, 0 <= s <= t, k = (p-1) t + s, the set T = Union_(i odd) [ti, ti + t - 1] has m_T = (3^(pt) + 1)/(3^t + 1) and exactly 2^(((p-1)/2)(t - s) + s) submasks of K_T divisible by m_T, by antipodal pairs, blocks and balanced-ternary uniqueness. Witness: lab/py/ratio-set-saving, ratio.py agg asserts all 74 triples to k = 19, check at k <= 8.
2026-09-06 Proved A binary K with support inside [0, bk - 1], b blocks of k digits, is a multiple of R_k exactly when its column counts satisfy Sum_r c_r 3^r = j R_k, and for b <= 3 that forces the column vector constant, so the binary multiples of R_k below 3^(3k) are exactly 2 * 3^k + 1 lifts, 3^k with one position per column and multiplier 1 + 2 a_(E_1) + 2 (3^k + 1) a_(E_2), 3^k with two, and R_(3k), which yields only submask directions; at b = 4 the column vector branches, 24 non-constant vectors at k = 3. Witness: lab/py/ratio-set-saving, ratio.py tail and ratio.py check.
2026-09-06 Verified Occ_T at the cyclotomic T is exactly the set {(3^t + 1) a_S} with its complements, of size 2(2^(t-1) - 1) whenever R_k is prime and 2, 6, 12, 30, 62, 100, 254, 510 at t = 2..9, and max_T |Occ_T| m_T / 2^k reads 4.562 to 376843.283 at odd k = 5..19 at that T every time; an unconditional statement needs #{S in [1, t-1] : gcd(a_S, R_k) = 1} >= 2^t / poly(t), nowhere proved. Witness: lab/py/ratio-set-saving, ratio.py lifts --kmax 19 21 s.
2026-09-06 Verified The lift union to k = 19 satisfies U_k <= Sum_T |Occ_T| = agg_k L_k Phi_k with L_k = Sum_T 1/m_T < 3/2 Proved; over k = 11..19 agg_k sits inside [1.01748, 1.11457] with no trend, L_k inside [1.41043, 1.41724] and U_k / Phi_k rises monotonically across [1.19611, 1.36517], so U_k = O(2^k) is the boundedness of agg_k alone. Witness: lab/py/ratio-set-saving, ratio.py lifts --kmax 19 --zmax 15 10 min 43 s.
2026-09-06 Verified The absolute-value route on the u != 0 Fourier terms of the lift count is dead: Sum_T (1/m_T) Sum_(u != 0) |F_T(u)| / 2^k reads 1.3839 to 7.9155 at k = 5..11, step ratios all above 1.26. Witness: lab/py/ratio-set-saving, ratio.py agg, the Abs column.
2026-09-06 Verified The cut-free aggregate agg'_k = Sum_T (N_T - 2)/(2^k L_k) sits inside [1.03919, 1.3403] and M_k / 2^k inside [2.47278, 2.89356] over k = 11..19, no upward trend. Witness: lab/py/ratio-set-saving, ratio.py agg, 20 s.
2026-09-06 Verified Writing b(z) for the number of k-blocks the minimal witness lift m(z) R_k fills, U_k = #{b(z) <= 2} (Proved) and the depth-3 census V_k = #{b(z) <= 3} gives (U_k, V_k, Z(R_k)) = (2342, 2350, 2360), (1618, 1624, 1634), (10280, 10310, 10388), (10278, 10310, 10440), (35566, 35630, 36190) at k = 11..15, so depth 3 captures 8, 6, 30, 32, 64 of the deep tail 18, 16, 108, 162, 624, a share falling 0.4444, 0.375, 0.2777, 0.1975, 0.1025, and the one-position lifts add nothing at any k <= 13. Witness: lab/py/ratio-set-saving, ratio.py tail.

Coprimality density

2026-08-28 Proved The base-local coprimality factor is exact at every finite level: #{x in S_level : e | x_i for all i} = fill_e * fill^(level-1) for every squarefree e | rad(base), holding on all 763 census lines; at composite base it does not factor over primes, the base-6 sample giving B(F) = 1/2 for the code's digits F (256 and 240 of 512) against the naive 0.46875; the character contraction c(base,fill) = 1 - (2/fill)(1 - cos(pi/(2 base))) gives 0.804738, 0.966506, 0.946410, 0.986603, 0.991481 at (base,fill) = (2,3), (3,8), (3,5), (3,20), (6,8). Witness: coprime-density-above-dimension-one, lab/rs/design-census.
2026-08-28 Verified Lemma B, the equidistribution estimate the coprimality programme needs, is a uniform equidistribution statement for digit-restricted sets across moduli growing with the level, the subject of Erdos, Mauduit and Sarkozy 1998, Konyagin 2001 and Maynard 2019, with Lemma A supplying the per-character input; it is proved for every design with fill > base by the dimension-above-one theorem and remains open only at fill <= base. Witness: coprime-density-above-dimension-one, REFS.md.
2026-08-28 Proved Every digital design of fractal dimension above one has the classical coprimality density: for dim >= 2, condition (E) and fill > base, A(level)/fill^level -> B(F) prod_{p not dividing base} (1 - p^(-dim)), by the box bound N*_level(m) <= (base+1)^dim fill^level m^(-log(fill)/log(base)) driving a Chebyshev log-gcd sum and a fixed-z sieve; this settles the gasket 16/(3 pi^2), the or-triangle 8/pi^2, the carpet 189/(32 pi^2), the Vicsek plus 27/(4 pi^2) and the sponge (513/520)/zeta(3), and the finite levels approach with oscillating signed error: carpet gap -3.52e-07 at level = 20, sponge -3.45e-05 at level = 18 (lab/rs/dimension-one-ladder), gasket 0.539591 against 0.540380 at level = 16 (lab/rs/oeis-terms, A396934). Witness: coprime-density-above-dimension-one, lab/rs/dimension-one-ladder, lab/rs/oeis-terms, A396934.
2026-08-28 Proved The 36 open dimension-one census lines are one problem: for any full-rank base = 3, fill = 3 design, collecting the corner-choice classes into E_j = sum_{c_l = j} 3^l maps S_level bijectively through the gasket, and for every m coprime to the design's difference determinant the two divisibility conditions become one shifted-target congruence on the gasket pair, so Lemma B stands or falls for the whole family at once, with T* vanishing at m >= 3^level; the same argument reduces every full-rank fill = dim + 1 design at any base to the simplex at that base. Witness: lab/rs/design-census.
2026-08-28 Verified Not every census collision is a shear: base-3 codes 11 and 161 have identical A(level) at every level (2, 4, 12, 34, 108, 322, 992, 3006, 8924, ... through level = 12, and equal T_e for every e <= 40 at level <= 7) though 161 is alone in its GL_2(Z) orbit over the exhaustive entry range [-8, 8]: both have zero corner v_0 = 0 hence gasket target 0, determinants with prime support {3}, and matching base-3 peel, so the finite Mobius sums agree termwise. Witness: lab/rs/design-census.
2026-08-28 Proved A(level) is not C-finite for any design meeting the dimension-above-one theorem with B(F) > 0 and dim even or dim = 3: a rational C-finite sequence with A(level)/fill^level convergent has a rational limit (roots above fill have zero coefficient, oscillatory roots on |z| = fill die by mean-square averaging, the remaining constant is fixed by every Galois automorphism), while delta is an irrational multiple of 1/zeta(dim); all five eligible base-2 plane designs through level = 12 admit no rational constant-coefficient recurrence of order at most 6 and approach their irrational limits (the gasket-type designs read 0.5378546 at level 12 against 0.5403796); the theorem says nothing at odd dim >= 5, at B(F) = 0 or fill <= base, or about polynomial-coefficient recurrences, which 2729 exact P-recursive fits with held-out terms exclude only empirically. Witness: coprime-density-above-dimension-one.
2026-08-28 Proved The Menger sponge's pairwise coprimality density is (13/20) prod_{p != 3} (1 - 3/p^2 + 2/p^3) = 0.251620868451255 = (351/400) C_3 with C_3 = 0.286747428434479, so the sponge rule lowers the full-lattice benchmark by exactly 12.25%: the three-modulus Mobius inversion over the three coordinate pairs does not collapse to one modulus, the base factor 13/20 is exact at every level (13 of the 20 legal digit vectors have at most one zero), each foreign prime contributes (1 - 1/p)^2 (1 + 2/p), and the tail closes on the pair-fibred box bound with kappa_I = 3 and alpha = log_3(20/3) = 1.726833 > 1; the exact census 0, 60, 1434, 32268, 721524, 15141288 at level = 1..6 gives 0, 0.150000, 0.179250, 0.201675, 0.225476, 0.236583; the local factor is not 1 - p^(-s), so no reciprocal zeta value is claimed and the coefficient in 0.4138997384/zeta(2) carries no rationality claim. Witness: menger-pairwise-coprimality.
2026-08-28 Proved The density theorem's spanning hypothesis retires to condition (E): a finite abelian quotient of order m is killed by m, so m Z^3 sits inside the difference lattice, and a character mod d vanishing on F - F forces m t = 0 mod d with gcd(m, d) = 1, hence t = 0, the only step of Lemma A that used spanning; non-parity index-4 and index-8 designs at bases 4 and 6 measure 0.105072, 0.035346, 0.936067 against the widened predictions 0.105639, 0.035261, 0.950751, converging; those three non-parity measurements have no generator in lab/, which computes parity codes only. Witness: coprime-density-above-dimension-one.
2026-08-28 Proved Every parity code at every even base >= 4 has delta * zeta(3) = (8/7)(1 - W_0/|P|), nine values only and independent of the base, because multiples of an odd m | base split evenly by parity while multiples of 2m are all even, so every odd base prime cancels its own Euler correction exactly; all 255 nonempty codes at every even base <= 40 give exactly nine band values, every lattice index at base = 4, 6, 8, 10 lies in {1, 2, 4, 8}, enumerations reproduce 0.712853, 0.951771, 0.709137, and in dim = 2 the parity carpet's band value 8/9 gives 8/9 / zeta(2) = 0.5403796460924681 = 16/(3 pi^2), an even-base band constant rather than the gasket's own. Witness: lab/py/mrlybang-density-classes.
2026-08-28 Proved Odd bases are self-similar across bases and the density trichotomy is exhaustive for odd base >= 5: fill_e(base) = fill_P(base/e) for squarefree e | base, so the bracket is a Mobius convolution of the code's corner-count cubic; the 149 spanning codes converge to 1/zeta(3) along the odds while frozen on their even band, the 43 codes inside their difference span take the corrected factor 1 - 2^(-s2) at 2, and the 63 codes whose affine span avoids the origin have no density at all, with zero all-even points at every odd level and two subsequential limits in ratio 1 - 2^(-s2); the even value equals the odd limit exactly on the 16 subgroup codes; checked for all codes at all odd base <= 75, twelve measured cases to the printed digit including exact zeros at the even levels of {111} at base = 5 and the axes pair 0.987338 / 0.739563 against 0.987319 / 0.740489. Witness: lab/py/mrlybang-density-classes.
2026-08-28 Proved Slice coprimality is finite arithmetic of the height: on x + y + z = s the gcd divides s, so A_s = sum_{d | s} mu(d) N_s^(d) exactly with no tail and no Lemma B; prime slices are fully visible up to the three axis points (coordinates forced into {0, p}), the base prime peels the slice to the previous level one step off-centre, a code without the origin corner owes nothing at its base on any slice, and each foreign prime costs the slice 1/p^2 where it costs the solid 1/p^3 (aggregated locals 0.040902 against 1/25, 0.020446 against 1/49); the parity-walk factor at 2 is 9121792/32002048 on the integer, the net's immunity at 3 holds on all 7^7 points, and the tree dichotomy holds: even slices hold zero visible points, odd slices zero even gcds. Witness: lab/py/slice-coprimality.
2026-08-28 Proved The central slice never converges and its bill is the repunit: s* = (3(base-1)/2) R_level(base) owes 3 always, 2 exactly when base = 1 mod 4 or level is even, and a foreign odd prime exactly when ord_p(base) | level, so the centre's visible density is a quasiperiodic function of the divisors of level; at base = 3 the two streams read 0.892, 0.898 against 0.571, 0.611, 0.652, at level = 7 the whole foreign bill is the Wieferich prime 1093 (2^1092 = 1 mod 1093^2), the central count is A299916(level) on the (9, -12) recurrence exact to level = 14, and the sixth peeled term is 83835 by a meet-in-the-middle count over all 20^7 level-7 points. Witness: lab/py/slice-coprimality, A299916.
2026-08-28 Conjecture The visible density inside a design's base-periodic pattern is exactly delta: 4/pi^2 = 0.405284734569 for the base-2 gasket pattern, 21/(4 pi^2) = 0.531936214122 for the carpet, 19/(26 zeta(3)) = 0.607932310732 for the sponge, by coprime tuples splitting evenly over the nonzero residue classes ((2/3)(6/pi^2), (7/8)(6/pi^2), 19 of 26 classes over 1/zeta(3)), with worst Mobius-count error 6.05e-07 at N = 10^6; designs differing only in the all-zero corner have identical visible density on 8579 pairs, since that class holds no visible points.
2026-08-28 Conjecture Exhaustive endpoints at bases 2..6 and dim = 2, 3 agree with the predicted delta with no inferable rate: base-6 code 34376528265 reads 0.454413 against 0.455945 at level = 8, and the base-5 dim = 3 pair is the worst case at 8.3e-03 and 1.5e-02.
2026-08-28 Conjecture The central-slice peel ratio tends to (sqrt(33) - 5)/8 = 0.0930703308, measured 0.093070331, forced from the shared (9, -12) recurrence of the peeled streams, which itself stays Conjecture. Witness: lab/py/slice-coprimality.

Crop census

2026-09-01 Proved Crop partition and anti-crop complement: classify puts every cell in exactly one of Out, Cut, In, so the keep-cut and strict crops bracket the boundary, and Shape::Anti flips In with Out fixing Cut, so the crop and the anti-crop under the complementary cut rule partition the filled set exactly; read off the definition and asserted both ways on all 118 printed configurations. Witness: mrlymath::shape, lab/rs/crop-counts.
2026-09-01 Verified The inscribed sphere never enters the level-1 sponge: census reads cells [0, 26, 1], the one In cell the empty centre and all 20 filled cells Cut, so the keep-cut crop keeps everything and the strict crop nothing; the inscribed octahedron holds no filled sponge cell fully inside through level 2. Witness: mrlymath::shape, lab/rs/crop-counts.
2026-09-01 Proved Exact dead zones of the inscribed crops: the carpet crop is empty for r < 1/6 under ball and diamond alike and the sponge diamond crop for r < 1/3, the central holes' inradii; the sponge ball's exact contact radius is sqrt(2)/6 = 0.2357, witnessed by the filled level-3 cell [13/27, 14/27] x [8/27, 9/27] x [8/27, 9/27] whose nearest point to the centre is (1/2, 1/3, 1/3), so the 1/24 sweep reads empty through r = 5/24 and first cuts at r = 6/24. Witness: crop.md, lab/rs/crop-counts.
2026-09-01 Verified Saturation and its one failure: the carpet ball crop holds all 4096 filled cells at level 4 from r = 17/24, the first sweep radius past the circumradius sqrt(2)/2, the sponge ball all 8000 at level 3 from r = 7/8, past sqrt(3)/2, and the sponge diamond never saturates in the sweep, reading in = 5356, cut = 1332 of 8000 at r = 1 since the cube's corners sit at L1 distance 3/2. Witness: lab/rs/crop-counts.
2026-09-01 Verified The strict inscribed diamond crop of the full side-2m grid holds exactly 2m(m-1) cells. Witness: mrlymath::shape.
2026-09-01 Proved A grid-aligned polytope crop is digit counting: walls on multiples of 3^-k keep exactly the cells with coordinates in integer intervals at level k, the count factors along digit positions as in mrlylab::press, and the crop adds nothing. Witness: crop.md.
2026-09-01 Conjecture The curved-slice dimension: the log_3 cut-ratio exponents of the inscribed circle on the carpet read 1.140, 0.909, 0.899, 0.951 and of the sphere on the sponge 2.166, 1.579, 1.705, hovering near the straight-slice yardsticks, the dimension minus one, 0.8928 and 1.7268 - yardsticks by analogy only, since Shmerkin 2019 and Wu 2019 cover intersections of xp- and xq-invariant line sets with p, q multiplicatively independent, not same-base carpet slices, straight or curved; five levels decide nothing. Witness: lab/rs/crop-counts, crop.md.
2026-09-06 Proved Over any triadic window r in [R, 3R) the mean of the crossing count C(r) is Theta(R^(log(fill)/log(base) - 1)), from C(r) = A(r) - B(r), the step bound 1 <= |x+1| - |x| <= sqrt(dim), the exact sandwich B(3R) - A(R) <= W(R) <= 2 (A(3R) - B(R)) and the bracket lemma B <= M <= A, with constants ((m-1)/2) G_min and (m-1) G_max. Witness: lab/rs/circle-crop mean lines, carpet r = 2187..6560 sum 13758140 inside [11019880, 22055720].
2026-09-06 Proved C_full(r) = Theta(r^(dim-1)): the shell bound above, and below C_full(r) >= (r/sqrt(dim-1))^(dim-1) from one crossing cell per orthant lattice point of the first dim-1 coordinates. Witness: lab/rs/circle-crop corner assert at every radius, band [2.000152, 2.037038] on the carpet at r = 27..6560.
2026-09-06 Proved Pointwise C(r) = Theta(r^(log(fill)/log(base) - 1)) holds if and only if Phi(r) = C(r) (3^dim/m)^level / C_full(r) is bounded above and below, level the least level with r < 3^level. Witness: lab/rs/circle-crop factor lines.
2026-09-06 Proved The digit transform route's budget is sum_(a != 0) |phi_level(a)| |S_r(a/3^level)| = O(r^(dim-1)), met term by term only if the l^1 mass grows by at most sqrt(3) = 1.7320508076 per triadic step, and the transfer step at the lattice is h(u) = (m + 3^dim - 1)/m = 2 at dim = 2. Witness: crop.md the transform route, and where it stops.
2026-09-06 Proved At dim = 2 the crossing shell is exactly 2r + 1 cells at every integer r >= 1, by the telescoping lo_i = hi_(i+1) of the column intervals with hi_(r+1) := 0; the same count at real radius is 2 floor(R) + 1, so the level-j boxes meeting the shell number at most 2 floor(r/3^j) + 1 and one holds at most 2 * 3^j crossing cells. Witness: lab/rs/circle-crop, asserted at every radius of every carpet level to r = 19682.
2026-09-06 Proved The fraction p_j(r) of crossing cells whose base-3 digit vector at position j is one the design omits obeys p_j(r) <= 2 * 3^j (2 floor(r/3^(j+1)) + 1)/(2r + 1) < 2/3 + 3^j/r at dim = 2, capping every position with 3^j <= r/30 at 0.7 uniformly in r; it does not transfer to Phi, since the sharpest bound the marginals alone support is Frechet-Hoeffding, C >= C_full (1 - sum_j p_j), and sum_j p_j reaches 1.349974 at level = 8 on the carpet and 1.627693 at level = 5 on the sponge. Witness: lab/rs/circle-crop digits and digitrate lines, asserted in exact integers.
2026-09-06 Verified The crossing shell's digits are equidistributed away from the top of the scale: carpet window means 0.111086, 0.111086, 0.111068, 0.111063, 0.111141, 0.109478, 0.109295, 0.166786 at r = 2187..6560 against 1/9, the whole departure in the top three positions and locked to level - j, fine positions inside [0.108363, 0.111141] and the scaled drift (p_j - 1/9) 3^k/3^j inside [-0.111806, 0.063806]; sponge 0.259211, 0.259237, 0.259663, 0.256864, 0.286061 against 7/27 with fine positions inside [0.259103, 0.259237], wholly below the null on three readings. Witness: lab/rs/circle-crop digitrate and digittotal lines.
2026-09-06 Verified Pairwise digit dependence in the crossing shell is bounded per pair and falls off with the gap: consecutive ratios 1.000165, 1.000219, 1.000521, 0.998543, 0.992613, 1.076698, 0.956158 and gap-two 1.000298, 1.000022, 1.000203, 1.000221, 1.003261, 1.005671 on the carpet at level = 8, over every window [0.939130, 1.714286] and [0.988460, 1.126957], sponge [0.954573, 1.151415] and [0.997413, 1.019127]. Witness: lab/rs/circle-crop digitpair and digittotal lines.
2026-09-06 Verified The pointwise factor widens with decelerating drift: carpet maxima rise 1.125000 to 1.518945 by increments 0.140625 down to 0.000808 over eight windows r = 1..6560, minima in [0.588115, 0.900000]; sponge maxima 1.350000 to 1.673315 over five windows r = 1..242. Witness: lab/rs/circle-crop factor lines.
2026-09-06 Verified The window multiplicity kappa = W(R)/((m-1) M(R)) brackets to [1.247746, 1.248322] on the carpet at r = 2187..6560 and to [1.417534, 1.445977] on the sponge at r = 81..242, still climbing there; the [1, 2] bound is asymptotic, the exact slack being (C(3R) + C(R))/((m-1) M(R)). Witness: lab/rs/circle-crop mean lines, form_low = -4.744629 at r = 1..2.
2026-09-06 Verified The l^1 mass Lambda_level reads 1.000000, 3.585973, 9.637999, 23.736907, 56.547512, 132.884543 at level = 1..6 with log_3 step 0.777708, forcing a term-by-term cost r^1.277708 against a budget r^1 and an error no better than r^1.170497, worse than the exact floor Lambda_level >= 2^level - 1 gives; closing it needs |S_r| = O(r^0.222292), below the square-root floor. Witness: lab/rs/circle-crop transform mass lines.
2026-09-06 Conjecture Phi = ind * Psi exactly with ind = prod_j (1 - p_j) (3^dim/m)^level and Psi the dependence correction; carpet means settle at 0.942104 and 1.005714 while the global brackets [0.542697, 1.515753] and [0.793296, 1.374208] still widen with decelerating drift, so ind and Psi bounded is sufficient for the pointwise C(r) = Theta(r^(d-1)) and is the whole of what is left. Witness: lab/rs/circle-crop digits and digittotal lines over eight carpet and five sponge windows.
2026-09-06 Refuted That two is the minimum of the transfer step h: h at the level-5 triadic point (155/243, 155/243) is 1.951261. Witness: lab/rs/circle-crop transform step line.

Density theorem boundary

2026-08-28 Verified The base-2 dim = 4 design census is complete to level 4: 65536 designs close into 402 orbits under the 384 signed coordinate permutations, 400 with fill >= 2, 336 both spanning Z^4 and carrying fill > 2 so the dimension-above-one density theorem applies to them on its stated sufficient condition, and the 400 eligible canonical representatives realize 189 distinct coprimality sequences (A(1), A(2), A(3), A(4)) with 87 collisions covering 298 classes, the largest being 10 classes on (4, 16, 88, 436) and 10 on (5, 25, 165, 985), by exhaustive enumeration with exact integer minors for spanning and an exact four-coordinate gcd at every point; four terms cannot separate an infinite collision from a short coincidence, and the distribution is over minimum-bitmask representatives rather than unoriented orbits, since coordinate complement moves the arithmetic origin and preserves neither B(F) nor delta nor A(level). Witness: coprime-density-above-dimension-one.
2026-08-28 Verified Exact sponge visible census without enumeration, three levels past the feasible: the hybrid A(level) = Sum_(d <= G) mu(d) T*_d(level) - Sum_(gcd > G) S(gcd) with transfer matrix T_d and the big-gcd tail enumerated as multiples in a base^level/g box costs about base^(level(dim+1)/2) against enumeration's fill^level and gives A(7) = 1038074187, A(8) = 20860210527, A(9) = 418429711224 (22.6 seconds against half a trillion points, the whole ladder in 84 seconds), anchored by the four census terms, direct enumeration at level = 5, 6, and cutoff independence (G = 100 and G = 150 split the work differently and agree on A(9) to the integer). Witness: lab/py/sponge-visible-census.
2026-08-28 Conjecture Spanning is the wrong hypothesis for the density theorem and the sharp condition (E) is that F - F has full rank and rad(m(F)) | rad(base): six census degenerate lines satisfying it obey the formula unchanged, a prime p | m(F) not dividing base replaces the Euler factor at p by a coset-corrected one, and A(level)/fill^level can fail to converge at all (base 7, F = {v : v_1+v_2 = 1 mod 3}, fill = 16, index 3, period-3 subsequential limits matched to 3e-04), numerics at 1e-04 on five corrected designs plus seven null cases, provable-looking by the base-peel argument; the same failure shows at base 3, F = {0,2}^2, fill = 4 > 3 with A(level) = 0 at every level against a positive predicted density. Witness: coprime-density-above-dimension-one; A396934.
2026-08-28 Conjecture The b-visible local factor of a design at a base prime is exact at every finite level: on the gasket #{x in S_n : 2 | x_1 and 2^b | x_2} = (2^(b-1)/3^b) 3^n for n >= b, by the last-digit argument, exact to the integer at b = 1, 2, 3 on every level 3..12, thirty matches with ratios exactly 1/3, 2/9 and 4/27; the Euler-product assembly it feeds is unproved for b >= 2.
2026-08-28 Conjecture Directional coprime profiles separate designs that the scalar density cannot: two base-3, dim = 2, fill = 5 designs with identical B(F) = 4/5 and identical predicted density 0.5471344 differ by up to 0.0327 in an eight-bin angular coprime profile at level 7 while their scalar pairwise densities differ by 0.0001001809, by exact enumeration at levels 2 to 7 with displacement multiplicities validated against N(N-1); the bin gap decays by a factor near 0.6 a level (0.375, 0.1461, 0.0946, 0.0492, 0.0327 at levels 3 to 7), so a nonzero limit is open.
2026-08-28 Conjecture The digit-restricted coprime density over F_3[t] with S = {0,1} is 9/16: measured 0.564176, 0.563471, 0.562833 at level = 10, 12, 14 by exact enumeration against (2/3)(3/4)/(8/9) = 0.5625, the correction sitting entirely at the exceptional prime t where pi(t) = 1/2 exactly against the unrestricted 1/3, the other two linear primes measuring 0.333984 and 0.333008; the finite Euler product is neither exact nor monotone, crossing 9/16 between degrees 3 and 4 and landing at 0.560193 through degree 5 while the exact no-shared-prime-below-degree-6 probability is 592189/1048576 = 0.564755, a dependence gap of -0.004563, so the density is a theorem only under three hypotheses: existence of pi_S(p), asymptotic independence over finite prime sets, and a vanishing high-degree tail. Witness: lab/py/function-field-density.
2026-08-28 Conjecture Mixed-radix coprime density depends on the schedule through a mod-3 effect rather than parity: at level = 12 over 8386560 ordered pairs per schedule (4096 points each), alternating base-2/base-3 gives 0.511135 with digits {0,1}/{0,1} and 0.672615 with {0,1}/{0,2}, against pure base-2 0.607874 (near 1/zeta(2) = 0.607927) and pure base-3 0.514692; both alternating schedules are exactly half even, and the 0.161480 gap comes from the divisible-by-3 fraction falling from 1/2 to 1/4, local factor 3/4 to 15/16, log advantage 0.223144, with prime 5 opposing at -0.014253 and primes 2, 7, 11, 13 identical between them; the residue mechanism for {0,1}/{0,1} is exact (every place value from position 2 on is a multiple of 6, so a mod 6 = d_0 + 2 d_1 and residues 0..5 are hit 1024, 1024, 1024, 1024, 0, 0), but no limit is established and the aperiodic staircase is untouched.
2026-08-28 Conjecture The sponge census gaps delta 20^level - A(level) measure 0.347, 0.349, 0.344 in units of 12^level, the subdominant parity-walk scale, so A(level) = delta 20^level - c 12^level + smaller. Witness: lab/py/sponge-visible-census.
2026-09-02 Verified The sponge visible census to level = 18 by two engines: admissibility is pairwise disjointness of the digit-1 masks, so A(level) = W(level) - W(level-1) with W(level) = Sum_(m < 3^level, gcd(m,3) = 1) mu(m) (N_level(m) - 1) and N_level(m) a disjoint-triple count over at most 2^level masks of the multiples of m; the engine splits moduli by their multiples count into a closed-form tail, bitset rows, u16 zeta rows and a rank-truncated ranked cube, about 3^level (level 2^level)^(2/3) work, 122.3 s at level = 18 on eight threads with level ratio 4.32, 3.8x its previous form, which it reproduces term for term from A(10) = 8382927031902 to A(18) = 215134797774716879278017, both matching the hybrid census through A(9) and enumeration through A(6); nine counters agree on every modulus to level 8, pinned counters with auto to level 8-11, 38 probed moduli at level 14 and two pinned probes at level 19 cover the u64 cube gate and the u32 rows branch; the new engine alone gives A(19) = 4302768326366633733102921 in 515 s, unwitnessed. Witness: lab/rs/coprime-terms.
2026-09-02 Proved The tail of the sponge Mobius sum is closed and admits no hyperbola grouping: for 3^level/2 < m < 3^level, 3 not dividing m, N_level(m) - 1 = 3 + 4 [m has no base-3 digit 1], so the top band of W(level) is three times the Mertens sum over the band's moduli coprime to 3 plus four times a Mertens sum over the base-3 Cantor set; the Y = 3 band is 6 + 7 [mask(m) = 0] + 7 [mask(2m) = 0] + 6 [mask(m), mask(2m) disjoint], every band a Mobius sum over a digit-automatic condition on m, 2m, ..., (Y-1) m; N_level(m) depends on the digits of m, not on floor(3^level/m) (level = 2: m = 5, 8 share the floor with N - 1 = 3, 7; at level = 6 every floor band holding two admissible moduli is non-constant), verified exhaustively at level = 6, 7 against a brute triple loop. Witness: lab/rs/coprime-terms.

Dependent bases

2026-09-14 Proved The collapse theorem: for one root r >= 2, bases base_i = r^(e_i) with i = 1..m, any dimension dim >= 1 and any digit sets A_i inside {0,...,base_i - 1}^dim each containing 0, put M = lcm(e_1,...,e_m) and B = r^M; then cap_i F(base_i, A_i) = F(B, A) exactly, where A = (cap_i F(base_i, A_i)) cap [0, B)^dim is the joint set's own bottom block, the count law is exact at card {n in [0, B^level)^dim : n in F(B, A)} = (card A)^level for every level >= 0, and the attractor of {x -> (x + a)/B : a in A} has Hausdorff and box dimension log card A / (M log r). Proof in four steps: e_i dividing M makes every e_i-group of base-r positions sit inside one M-group, so no group straddles and lcm is forced rather than chosen; each digit at base base_i is then a function of one base-B digit, so the constraint is per base-B digit with nothing carried between them; the top base-B digit is read with its full M / e_i base-base_i digits while the base-base_i expansion of n stops at its top nonzero digit, so the two readings agree exactly when 0 in A_i; and membership one digit at a time gives the count with no error term while the maps send the unit cube to boxes with disjoint interiors, so the open set condition gives the dimension. No step uses dim = 1 and no step asks A_i to be a product across the dim axes, so dim >= 2 is covered with no extra hypothesis, compound designs included. Across two dependence classes the theorem says nothing. Witness: lab/py/base-collapse verbs blocks, dim, check; bases.md:92, bases.md:101-107.
2026-09-14 Proved Multiplicative dependence is an equivalence relation on bases >= 2 and each class is the set of integer powers of its least member: reading a base by its vector of prime exponents turns p^a = q^base into parallel vectors, so a class is the set of integer points on one ray through the origin, its least member r is the primitive vector on that ray and every member is r^e for one integer e >= 1; transitivity is one line, p^a = q^base and q^c = s^d give p^(ac) = s^(bd). The half of the rule that follows is collapse first: partition the bases into dependence classes and replace each class by the single design the collapse theorem gives it, with its own base r^M, its own digit set and its exact dimension; a transversality budget inside a class is not allowed, by the Refuted row below. Witness: lab/py/base-collapse verb blocks; bases.md:88, bases.md:126.
2026-09-14 Verified Four dependent cells rebuilt from their digit sets alone, root, exponents, M, block set and exact dimension: base 4 on {0,1} with base 8 on {0,1,2,3} gives r = 2, M = 6, A = {0,1,16,17} and dimension exactly 1/3; base 4 on {0,1} with base 16 on {0,1,4,5} gives M = 4, A = {0,1,4,5} and exactly 1/2; base 9 on {0,1,2} with base 27 on {0,...,8} gives r = 3, M = 6, nine blocks and exactly 1/3; bases 4, 8, 16 on {0,1}, {0,1,2,3}, {0,...,7} give M = 12, sixteen blocks and exactly 1/3. The block set is built twice per cell and asserted equal, once by sieving all r^M base-r words against the per-group constraint and once by testing every integer below r^M for membership in each original design. Witness: lab/py/base-collapse verbs blocks, dim; bases.md:115-120.
2026-09-14 Verified The exact count law survives brute force to 10^13 on all four cells, three checks each: every element of F(B, A) below 10^13 passes a digit test in each original base; the joint count from enumerating the lowest-dimension original design (2^22 - 1 elements at three cells, 3^14 - 1 at the base-9 cell) and filtering it by the others equals the count of F(B, A) below 10^13, which with the first check gives set equality, the four counts reading 32767, 4194303, 19682, 32767; and the joint count below B^level is exactly (card A)^level at every level with B^level <= 10^13, reaching level = 7 and 16384, level = 10 and 1048576, level = 4 and 6561, level = 3 and 4096. Witness: lab/py/base-collapse verb check; bases.md:124.
2026-09-14 Verified A dependence class can give a block count that is not a power of its root, so the exact dimension can be irrational: base 4 on {0,1,2} with base 16 on the full digit set gives r = 2, exponents 2, 4, M = 4 and card A = 9, dimension log_2(9) / 4 = log_2(3) / 2. The dimension is log_r(card A) / M always, and for the least base r of the class it is rational exactly when card A is a power of r, since a primitive r is no perfect power. Witness: lab/py/base-collapse verb dim cell I; bases.md:107.
2026-09-14 Verified The hypothesis 0 in A_i is sharp and not decoration: at r = 2 with base_1 = 2 on A_1 = {1} and base_2 = 4 on the full digit set, M = 2 and the bottom block is A = {0, 1, 3}, yet F(4, A) holds 4, 5, 12 and 13, whose base-2 words carry a digit outside A_1, so the collapse strictly over-counts the joint set. Witness: lab/py/base-collapse verb blocks; bases.md:105.
2026-09-14 Conjecture Budgeting across the collapsed classes alone, with m the number of classes and not the number of bases, is the right second step: nothing here shows a collapsed class behaves in a cross-class budget like an ordinary design of the same dimension, so the step is assumed. Witness: bases.md:126; no proof and no lab verb.
2026-09-14 Refuted The naive budget sum_i dim A_i - (m - 1) is not an upper bound on a multiplicatively dependent cell and is low on every one tested: base 4 on {0,1} with base 8 on {0,1,2,3} has exact dimension 1/3 against a budget of 1/2 + 2/3 - 1 = 1/6; base 4 on {0,1} with base 16 on {0,1,4,5} has exact dimension 1/2 against a budget of 1/2 + 1/2 - 1 = 0; base 9 on {0,1,2} with base 27 on {0,...,8} reads 1/3 against 1/6; and the three-base cell 4, 8, 16 reads 1/3 against max(0, 1/2 + 2/3 + 3/4 - 2) = 0, the budget being read at zero because no dimension is negative and the raw sum there is -1/12. The second cell is the clean failure: F(16, {0,1,4,5}) = F(4, {0,1}) as sets, since a base-16 digit lies in {0,1,4,5} exactly when both of its base-4 digits lie in {0,1}, so the budget prices the intersection of a set with itself at dimension 0 while the truth is that set, of dimension 1/2. A budget built on transversality cannot be applied inside a dependence class. Witness: lab/py/base-collapse verbs dim, check; bases.md:113, bases.md:115-120.

Design census

2026-08-28 Proved Designs up to cube symmetry are the NP-equivalence classes of Boolean functions: an equivariant bijection carries B_dim (order 2^dim * dim!) onto the NP group, so the class counts are 3, 6, 22, 402, 1228158, 400507806843728 at dim = 1..6, reproduced by orbit walk on designs, orbit walk on truth tables and Burnside, checked against the entry to dim = 7; the NPN sibling gives 2, 4, 14, 222 at dim = 1..4. Witness: mrlymath::bang::counting::sequence, A000616, A000370.
2026-08-28 Verified The design census at dim = 2 and any base is the toroidal binary array count: Burnside over one dihedral group per residue axis gives 2, 6, 26, 805, 172112, 239123150, 1436120190288, 36028817512382026 at base = 1..8, with brute-force orbit closure agreeing at base = 3, 4. Witness: mrlymath::bang::baseq::distinct_designs, A255016.
2026-08-28 Proved The isotropic-class count is A005418(dim+2) minus one at even dim: a level set's orbit is {F_S xor t} and the image depends on t only through |t|, so the count is subsets of {0..dim} up to reversal with the even-dim merge, 3, 5, 10, 19, 36, 71, 136, 271, 528, 1055, 2080, 4159, 8256, 16511, 32896, 65791 at dim = 1..16; nameable classes grow like 2^dim, half the 2^(dim+1) subsets. Witness: A005418.
2026-08-28 Verified Census multiplicity in the base-2 census is a bounded perfect-power representation count: over the 16 two-dimensional and 256 three-dimensional designs at side 3 and levels 1..5, the 1360 design-level pairs give 119 distinct fill counts, M(N) = sum_{level=1}^5 sum_{fill=0}^27 c_fill [fill^level = N] with sum_fill c_fill x^fill = prod_{w in {1,2,4}} (1 + x^w) + prod_{w in {1,2,4,8}} (1 + x^w), maximum M(4096) = 29 from 4096 = 8^4 = 16^3 (14 designs of base fill 8 plus 15 of base fill 16), support exactly {fill^level : 0 <= fill <= 27, 1 <= level <= 5}, so only 119 integers occur up to 27^5 = 14348907, coverage 8.29e-6, in 91 maximal missing runs, the longest 11881377..14348906; M is not multiplicative (M(2) = M(3) = 5, M(6) = 12), opens 10, 10, 5, 5, 13, 8, 12, 12, 19, 19, 15, 15, 16, 16, 16, 16, and no classical arithmetic function is behind it: the raw Pearson signals against sigma and phi (-0.332, -0.315) are shared size dependence (M against N is -0.338), and partial rank correlations controlling for log N fall in -0.035..0.096 for d, sigma, phi, omega, Omega. Witness: mrlymath::formulas::counting::fill.
2026-08-28 Verified The census covers only a short prefix of the integers: through Kronecker level 6 the 256 three-dimensional designs produce 37 positive fill values in 1..262144, coverage 0.01411%, contiguous only on 1..9, because fill(code, level) = f^level with f the tile popcount in 0..8; the nine named observables reach a union of 368 integers to 1633932 with 10 the first gap, the graph observables core_edges, tips, junctions extend it to 437 integers and the prefix to 1..24; edges is the broadest single observable (73 distinct positive values, 60 exclusive), then faces 70, vertices 67, surface 66; design 23 reads fill 20, voids 7, surface 72, vertices 64, edges 144, faces 96, Euler -4 at level 1 and fill 400, vertices 896, edges 2304, faces 1728, Euler -80 at level 2, and every record obeys surface = 6 fill - 2 core_edges, faces = 6 fill - core_edges, cycle_rank = core_edges - fill + components, euler = vertices - edges + faces - fill; voids means every empty lattice site, 3D edges means cubical-complex unit edges and core_edges the branch count of the face-adjacency graph; the 2D census gives 176 distinct positive integers to 1064340, contiguous on 1..10. Witness: mrlymath::formulas::counting::fill, mrlymath::three::census::census.
2026-08-28 Verified Unbounded census observables cannot be automatic sequences and are at best regular in some base: an integer-valued automatic sequence has finite range, a finite-range regular sequence is automatic, so d(n) and sigma(n) are automatic in no base and can enter only through unbounded regular representations, weighted substitutions, Dirichlet convolutions or a purpose-built geometric model; Cobham's theorem is the one general rigidity constraint, and substitution incidence matrices generate additive recurrences while d and sigma are multiplicative over primes, so self-similar census counts c_m = u^T A^m v are generically sparse in the integers. Witness: REFS.md.

Design counts at every base

2026-08-28 Refuted The per-base, dim = 3 count 2, 22, 111618, 6005363762644688, 7089215977519836239803174210135872 has no OEIS entry - it is A398348, toroidal n x n x n binary arrays up to layer rotation, layer reflection and axis permutation, data verbatim with a b-file to n = 14 and A255016 as the two-dimensional case; the null search ran against a dump older than the entry. Witness: A398348, lab/rs/oeis-terms.

Diagonal cuts of the parity solid

2026-08-28 Proved Every diagonal slice of bang dim 3, code 126 holds exactly 3^level points at every admissible height, by uniqueness of the binary expansion of the height offset; checked at level = 1..8 by two enumerations sharing no code and to level = 14 by a height recursion; the constancy separates it from the digit-scheduled slices of Nakajima and Watanabe, whose non-autonomous IFS uses A_c^(j) = {0} or {1,2,3} by the height digit, so their digit changes the number of maps and this one only the orientation. Witness: mrlymath::three::diagonal.
2026-08-28 Verified The two central cuts of bang dim 3, code 126 decompose into six congruent Sierpinski gaskets of 3^(level-1) points each tiling a hexagon with an order-12 symmetry group, checked at level = 2..8 by rebuilding the pieces from the previous level (union, pairwise disjointness, sizes); the combined totals are 6 * 3^(level-1) = 2 * 3^level: 18, 54, 162, 486, 1458, 4374, 13122; the ambient object is the octahedron flake of dimension log(6)/log(2). Witness: mrlymath::three::diagonal.

Diagonal designs

2026-08-28 Refuted The diagonal designs are codes 98, 140 and 266 with Z_F(level) = 3^level - 2 - the identity fails at level = 0 (sides 0 and -1) and holds for every level >= 1, and the list is short by one: of the C(9,3) = 84 three-corner subsets of the 3 x 3 digit square, four are diagonal through level = 6, {(0,1),(1,0),(2,2)} (266), {(0,1),(1,2),(2,0)} (98), {(0,2),(1,0),(2,1)} (140) and {(0,2),(1,1),(2,0)}, code 84 (the value 148 once quoted for it names {(0,2),(1,1),(2,1)}, not a permutation design); the three named codes hold to level = 11 with every pair checked for collinearity at level = 7. Witness: gasket-ray-machine.

Diagonal slice ladder

2026-08-28 Verified The even half at base 7 carries exact Collatz-Wielandt certificates rho_dim < fill/7 at every even dim <= 26 and at dim = 172, 174, anchored by P(1) = 6^(dim-1)(dim+6) and a brute-force digit enumeration of P, with depth K_min = 0, 1, 2 stepping at dim = 4 and dim = 26 (and 3 at dim = 174), V(level) > 0 at every even dim = 2..40, level <= 25 with no dip, and c == 0 mod 7 immune at depth 0, so Sigma_K depends on c mod 7^(K+1). Witness: slice-sign-even-half.
2026-08-28 Verified An off-centre diagonal slice cannot break the alternation: a fixed target offset k changes only the initial vector, the transfer matrix commutes with carry reflection and e_0 is even, so e_k^T M^level e_0 = (1/2)(e_k + e_(-k))^T M^level e_0 and the odd component is annihilated, the off-centre slice seeing only the central even block; the boundary condition matches direct polynomial multiplication in 24 cases at dim = 2..7, level = 1..4, exact counts to level = 80 obey minimal rational recurrences on every term, and the dominant root equals the central one at every dim = 2..12 and offset |k| <= 2, the only movement being transient zero modes at dim = 4, |k| = 2 and dim = 2, |k| = 1, 2 that raise the order without touching the growth rate; offsets scaling with 3^level are untested. Witness: slice-recurrence-order.
2026-08-28 Verified The anti-diagonal slice profile factors across the Kronecker product, P_(A (x) B)(t) = P_A(t^(side_B)) P_B(t), since r + c = side_B (r_A + c_A) + (r_B + c_B), giving the stationary product prod_(j<level) P(t^(base^j)) and its mixed-radix form, so one identity covers fractal slices, mixed-product slices and the dimensional ladder; exact on all words of length 2 and 3 at base 2, dim = 2, zero mismatches. Witness: lab/py/slice-ladder-controls.
2026-08-28 Verified The level-1 central diagonal slice of the base-3 Menger analog is the vertex set of the cube's central cross-section: the hypersimplex vertex counts 2, 6, 6, 30, 20, 140, 70 at dim = 2..8 match C(dim, dim/2) for even dim and C(dim, (dim-1)/2)(dim+1)/2 for odd, so the ladder starts on a polytope rather than on an analogy. Witness: lab/py/slice-ladder-controls.
2026-08-28 Conjecture The 2-adic Smith form of M_even at base 3 and odd dim has elementary-divisor valuations 0^r 1^p 2^e A with exactly one divisor >= 2^3 and sum a_i = v_2(det), so v_2 = nullity + #{a_i >= 2} + max(a_max - 2, 0) reduces the uniform bound v_2 <= n to the tent rank law nullity_2(M_even) = min_(t in T) (|dim - t|/2 + 1), T = {2J(k)+1, 2J(k)+3 : k >= 2}, J(k) = (2^k - (-1)^k)/3 (exact 255/255 at odd dim = 3..511, peaks J(k-1) at dim = 2^k + 1, hence nullity <= ceil(n/3), troughs at the odd dim with 3 dim nearest a power of 2, so the rank deficiency measures the 2-versus-3 carry mixing) plus a small-excess bound whose constants are domain-limited: max a_i <= 9 and #{a_i >= 2} <= 5 hold at odd dim = 5..121, but max a_i <= 9 first fails at dim = 127 (12 by 511), #{a_i >= 3} = 1 at dim = 175 (reaches 5) and #{a_i >= 2} <= 5 at dim = 183 (reaches 21); the mod-2 form is P == (1+t)^(2D-3)(1+t^3) with 1 + Dt + t^2 == 1 + t + t^2 irreducible, dim = 7 is a tent trough with the whole valuation in the lone big divisor (a = {0,0,0,7}, max a_i = 7 > n = 4, a size effect), dim = 5 (a = {0,0,4}) is the only other v_2 > n at odd dim = 5..121, equality holds at dim = 9, 15, and the fold puts the 2-content in the even block because the palindromy row c' = 0 is 2 P[dim+c] entrywise and at dim = 7 the odd block is 2-adically unimodular. Witness: lab/py/smith-cascade; slice-sign-even-half.
2026-08-28 Conjecture The base-7 certificate extends with logarithmic depth and no transient to every even dim = 2..40, and the depth-death law is asymptotic rather than exact: both measured breakpoints land one even step early of dim = 2 ceil(7^(K+1)/4) + 2 because the depth-K positivity frontier f_K(dim) is still climbing when the window edge reaches it, so the corrected law reads K_min = min{K : (dim-2)/2 <= f_K(dim)}.
2026-08-28 Conjecture The central diagonal slice census of the base-3 dim-dimensional Menger analog obeys a linear recurrence of order exactly ceil(dim/2): the digit polynomial factors as P(t) = (1 + t^2)^(dim-1)(1 + dim t + t^2), the carry map c' = (c + dim - s)/3 contracts to {|c| <= floor((dim-1)/2)}, the symmetry v -> (2,...,2) - v gives P[s] = P[2 dim - s], so the Krylov subspace from e_0 sits in the reflection's +1 eigenspace of dimension floor((dim-1)/2) + 1 = ceil(dim/2) and the order bound holds at every dim; exactness is checked at dim = 2..14 by distinct eigenvalues of M_even with minimum gap above 6.9 and at dim = 2..24 by nonzero Hankel determinants, and is open for general dim (a square-free characteristic polynomial); controls: dim = 2 gives 2^level at order 1, dim = 3 gives 6, 42, 306, 2250, 16578, 122202 and A299916's 9a(n-1) - 12a(n-2) at order 2 by a route that never mentions a hexagon, dim = 4 gives 6, 132, 1848, 29040, 441408, 6772128 at order 2 with dominant root (11 + sqrt(385))/2, dim = 5 gives 30, 1000, 35700, 1321600, 49786200, dim = 6 gives 20, 4030, 242300, 24642700, and rational Hankel elimination on nine terms reads orders 1, 2, 2, 3, 3 at dim = 2..6. Witness: slice-recurrence-order; A299916.
2026-08-28 Conjecture Conjecture S, the sign law sgn(slice dimension - (solid dimension - 1)) = (-1)^(dim+1) at every dim: the even half at bases 3 and 5 and the odd classes dim != 1 mod 3 are settled on the shelf, the odd class dim == 1 mod 3 beyond dim = 80 is open, and the route through a named lemma, an explicit positive vector x_dim with sgn((M_even x)_i - (fill/3) x_i) = (-1)^(dim+1) at every index, has only the Perron vector, which supplies it numerically at every dim <= 50 with worst componentwise discrepancy 9.15e-46 at 90 digits, peaked at index 0 and non-increasing, with no closed form; its two silent hypotheses, a real spectrum (exact at dim <= 20, numerical at dim = 2..60) and no non-Perron eigenvalue crossing fill/3 (checked at dim = 2..60 against a 180-digit reference), are themselves unproved. Witness: slice-recurrence-order; slice-sign-even-half.
2026-08-28 Conjecture The second eigenvalue of the even carry block tracks the digit polynomial at -1: lambda_2 -> (-1)^(dim+1) 2^(dim-1)(dim-2)/3 = (-1)^dim P(-1)/3 with exponential convergence but never exactly (the characteristic polynomial is nonzero at that value in exact arithmetic at every dim = 2..40, so lambda_2 = -9007199254740992 = -2^53 to 22 digits at dim = 50, which is 2^49 * 48 / 3, is display rounding), hence rho/|lambda_2| -> (dim+2)/(dim-2) -> 1, measured 13/12 to 2.58e-22 at dim = 50 and 1.04081632653 at dim = 100, with |lambda_2|/rho = (dim-2)/(dim+2) to nine digits by dim = 36, so the spectral gap closes and no argument may assume a fixed one; the asymptote must not be quoted at small dim, where dim = 4 gives a true lambda_2 = -4.310708 against -16/3, a 19% gap consistent with an O(2^(-dim)) approach; measured at 420 to 650 digits. Witness: slice-recurrence-order.
2026-08-28 Conjecture The base-5 middle-digit analog (keep a cell when at most one coordinate is the middle digit 2) has P_5(t) = A(t)^(dim-1)(A(t) + dim t^2) with A(t) = 1 + t + t^3 + t^4, fill = 4^(dim-1)(dim+4) and carry rule c' = (c + 2 dim - s)/5, and sgn(slice dimension - (solid dimension - 1)) = (-1)^(dim+1) holds at dim = 2..15 down to a smallest excess of 1.055e-9 at dim = 15 (exact-integer sign sweep with 80-digit root refinement, agreeing with substitution-product convolution in all 16 cases at dim = 2..5, level = 1..4, the dim = 3 fill 112 of 125 matching the middle-digit count), yet A(-1) = 0 makes P_5(-1) = 0 for every dim, so the alternating mass that carries the base-3 explanation is absent while the alternation survives, and no mechanism yet survives that. Witness: slice-sign-even-half.
2026-08-28 Conjecture The alternation is universal across base-3 designs and its phase is not: over four families at dim = 2..7, Menger with at most one middle digit and P(-1) < 0 for dim > 2 gives -+-+-+-, at most two middle digits with P(-1) > 0 gives --+-+-+, Cantor with no middle digit and P(-1) = P(1) gives +-+-+-+, exactly one middle digit with P(-1) < 0 gives +-+-+ from dim = 3, so the phase tracks the sign of P(-1); the range stops at dim = 7 and the four sign patterns rest on a prose table alone.
2026-08-28 Conjecture The excess rho_dim - fill/3 decays at the rate r_inf = 1/prod_(k>=2) cos(2pi/3^k) = 1.3461220067642173 per dimension on the eigenvalue scale, 2 r_inf = 2.6922450 on the dimension scale, with a linear prefactor |delta_dim| ~ A (dim-1) r_inf^(-dim), A -> 2/(3 prod cos) = 0.89741, from the 3-adic angle-tower product formula, matched within 1e-8 by exact rational bisection at dim = 61; a third-order Richardson fit in 1/dim over dim >= 60 at 320 digits gave the one-step ratio 0.742874554813847413, r_inf = 1.34612251727283689 (seven true digits, the rest fit residue), even and odd extrapolations 4.5643e-8 apart and A ~ 0.897520192686, and the shape check (6A/ln 3)(dim/(dim+2))(2 r_inf)^(-dim) = 1.47e-21 at dim = 50 against the measured 1.42672e-21 tests the form and not the constant. Witness: slice-recurrence-order.
2026-08-28 Conjecture Conjecture S reduces to one separation lemma along an explicit chain: with M_even the reflection-even block of the carry automaton M[c,c'] = P[c + dim - 3c'], P(t) = (1+t^2)^(dim-1)(1 + dim t + t^2) and fill = P(1), if every non-Perron eigenvalue of M_even has modulus below fill/3 then sgn det(fill/3 I - M_even) = sgn(fill/3 - rho), and that determinant sign is (-1)^dim, exact in integer arithmetic at dim = 2..20 and to dim = 40, which is Conjecture S; the separation hypothesis is checked at dim = 2..60 and not proved, the row-sum lemma feeding it holds for the full carry matrix on states c = 0..dim and is false in the recurrent even basis (dim = 3 row sums (12,4) against the formula's (8,6)), and the product formula settles the odd half without separation, so this chain is a route to the even half only. Witness: slice-recurrence-order; slice-sign-even-half.
2026-08-28 Conjecture The balanced-mask homotopy reduces Conjecture S to a one-variable determinant inequality and owes two lemmas: with a_dim = (-1)^(dim-1)(dim-1) and Q_dim = P_dim - a_dim t^dim, the root-of-unity identity forces 1 + t + t^2 | Q_dim, and fill/3 I - M_dim = L_dim + a_dim K_dim exactly with K_dim = I/3 - E_dim, E_dim the dilation 1_(j=3i); then f_dim(z) = det(L_dim + z K_dim) = z h_dim(z) and S becomes h_dim(a_dim) < 0, exact at dim = 2..30; det L_dim = 0 is exact to dim = 30 but does not follow from residue balance, because N_dim lacks constant column sums in the unnormalised even basis after truncation and folding, and coefficient negativity of h_dim, which settles every odd dim since a_dim > 0 there, is useless at even dim where a_dim = -(dim-1) is negative, so a uniform root bound is still missing.

Diagonal slice stack

2026-08-28 Verified The level-1 slice is exactly a lattice-plane object: carpet_cut(n, 1) at odd scale n equals the set x + y + z = 6n - 2, z even, in [0, 4n)^3, filled iff at most one of floor(x/4), floor(y/4), floor(z/4) is odd, with |slice| = 6n^2; two cell-for-cell reconstructions (n = 1..63 and n = 1..13) show zero mismatches. Witness: walsh-spectrometer.
2026-08-28 Proved The 1:6:1 three-plane law: every micro point of the slice lies in a macro cell with i + j + l in {K, K-1, K-2} at K = (3n - 1)/2 with multiplicities 1, 6, 1, so fill(n) = 6 F(K-1) + 2 F(K), a two-line derivation of the cut fill closed forms, exact for all odd n <= 21 and holding at n = 1 where the outer planes are empty. Witness: walsh-spectrometer.
2026-08-28 Proved The slice's two mod-4 families are the Dirichlet character chi_4: per-gram ink is exactly 3/8 + 1/(2n) + 1/(8n^2) at n = 1 mod 4 and 5/8 + 1/(2n) - 1/(8n^2) at n = 3 mod 4, because the plane constraint pins the triple parity product to (-1)^K; 14 + 14 layers cancel it, so the stacked snowflake sits at background 1/2 while the flat carpet stack sits at 3/4 ink; the closed forms reproduce all 28 layers with zero error. Witness: walsh-spectrometer.
2026-08-28 Verified The chi_4 twist kills the pair-resonance ray family: the average of chi_4(n) T(nx) over odd n <= N falls like 1/N at every x, rational or not (x = 0, 1/3, 2/3, 1/5, 1/7, 1/2, 1/4, 1/9 and irrational), so the snowflake stack has no analogue of the carpet stack's bright main diagonal, its A = C excess going -0.0036 at N = 55 to -0.000052 at N = 5555; its visible rays are only the three single-wave crosshair families parallel to the hexagon's edge directions, one per lattice axis, at odd-denominator rational coordinates. Witness: lab/rs/hexagon-moire.
2026-08-28 Verified The ghost star at the hexagon's centre is a finite-layer artifact: each layer's centre is entirely ink or entirely paper, flipping with n mod 4, so 28 layers give exactly 1/2, which is also the limiting background; the star-minus-background contrast decays as (ln L)/L in the layer count L (-0.094 at 5 layers, -0.031 at 28, -0.018 at 56; excess * L running -0.7779 to -1.2519 in the ideal frame and -1.0212 to -1.3645 in the lattice frame from L = 28 to L = 400), and the exact rate constant is open and frame-dependent (-1/8 per ln L in one frame, -0.18 in another). Witness: lab/rs/hexagon-moire. Superseded in the cell frame: the decay coefficient is a closed form at every band width, see the width family rows under Diagonal slice stack in SETTLED.
2026-08-28 Verified The three 60-degree crosshair families obey a limit law: the line at coordinate a/q carries strength 1/(4q) for odd q and nothing for even q, converging in the arithmetic model (1/3 -> -0.0837 against -0.0833 at N = 5555) and visible in the real render in registration-correct frames (the X + Z = 1.25 line at N = 55: -0.045/+0.029; the X = 1/3 one-sided bands +0.021/-0.058); the per-layer registration drift of 1/(2n) in the slice plane is what a drifted scan raster misreads (1/7 at -0.058 against -0.036, 1/2 at -0.014 in the coarse crosshair model), and a null claiming no rays above 0.013 was about the scan geometry, not the object. Witness: lab/rs/hexagon-moire.
2026-08-28 Proved The Walsh spectrometer: the diagonal-slice ink of every 3D parity design is the exact quasipolynomial ink(n) = Sig0 - (1/2) Sig3 s + [(2/3) Sig1 - (1/3) Sig2 s]/n + [(2/3) Sig2 - ((1/3) Sig1 + (1/2) Sig3) s]/n^2 with s = (-1)^((3n-1)/2) and Sig_j the design's level-j Walsh coefficient sums: the background is the mean Walsh coefficient, the mod-4 blink is minus half the top coefficient, the 1/n orders read the middle levels; exact in rationals on all 256 codes at every odd n <= 55 and at the cold sizes 101, 555, 999, 9991; the attempt to break it recomputed P_n from the definitions for all 28 odd n <= 55 independently of the lane's scripts and of the crate, found |P_n| = 6n^2, weight-only dependence with zero splits and zero law mismatches on all 256 codes. Witness: walsh-spectrometer, mrlydemo::walsh_spectrum.
2026-08-28 Proved Nine of the 22 design classes never blink: |b| takes exactly the values {0, 1/16, 1/8, 3/16, 1/4}, zero iff the top Walsh coefficient vanishes (tree and void among them), carpet and net blink at the middle rung 1/8, the xor pair maximally at 1/4; (a, |b|) is orbit-invariant on all 256 codes, the named codes are carpet 23, net 232, tree 3, void 129, and carpet and net are the same symmetry class (net is carpet with all parities flipped). Witness: walsh-spectrometer.
2026-08-28 Verified The corrected law on the hexagon is dyadic: what breaks coprime independence is a hidden half-cell-shifted overtone at doubled frequency, chi_4(n) s(2nX + 1/2)/8, that the plane constraint forces into every carpet slice, plus the hexagon's non-product tent marginal; together they couple layer m to layers 2m +- 1 and m +- 2 regardless of gcd, and the doubling sign law sign r(m, 2m +- 1) = -chi_4(m) chi_4(2m +- 1) holds on 18 of 18 pairs from (3, 5) to (601, 1201) across all four residue branches. Witness: lab/rs/hexagon-moire.
2026-08-28 Verified The breakage is rule-specific in the limit: the persistent doubling coupling is carpet and net only (void and tree doubling correlations die, +0.001 at (201, 401)), tree keeps a neighbour coupling r(m, m+2) -> -0.0704, void stays essentially independent (adjacent +0.008), and the gcd echo survives in all four (carpet (m, 3m) -> +0.2148, tree +0.1498, void +0.0772 at (67, 201)); on the full hexagon the coprime pairs (5, 9) and (5, 7) read -0.142 and -0.085, so any published number must pin the mask convention. Witness: lab/rs/hexagon-moire.
2026-08-28 Verified Eisenstein is absent from the base-2 slice stack: L(2, chi_-3) = 0.7813024129 appears nowhere, the only character the slice generates is chi_4, and the hexagonal geometry contributes rational tent integrals. Witness: lab/rs/hexagon-moire. Superseded on the character claim: the arm of the ghost star carries chi_8, see the width family rows under Diagonal slice stack in SETTLED.
2026-08-28 Verified The quarter-line law: the strongest interior lines of the stacked hexagram sit at quarter-cell coordinates a/4 (generally a/(4b), b odd), an exact one-sided step of +-1/8 that every layer votes for identically because the overtone's chi_4 sign meets the layer's own chi_4 and squares away, converging 0.1221, 0.1234, 0.1241, 0.1245 at N = 151, 301, 601, 1201, the odd-fraction crosshairs at 1/(4q) following behind; the X, Z, W profiles are numerically identical on the render by the slice's permutation symmetry, so "in five directions but never horizontal" is false and the missing-Z-overtone statement holds only in the rectangle-cell frame. Witness: lab/rs/hexagon-moire.
2026-08-28 Verified The void slice stack keeps its star forever: six central lines at plateau ink 1/2 against background 1/4 (ratio 2, the model-frame Z arm weaker at 3/8), every layer voting on all six, plus a centre dot that is ink at every odd n by a two-line parity proof; the carpet's star fades as log-corrected 1/L, so the two snowflake stacks differ by a theorem. Witness: lab/rs/hexagon-moire.
2026-08-28 Verified Void and carpet have complementary line spectra on the cut: void's lines sit at even-denominator twisted positions X = a/(2b), b odd, where carpet is silent, and void is silent at carpet's odd rationals; tree carries the only untwisted crosshair family plus a permanent ratio-2 line at K = 3/2 and ignores its free axis; net is the exact pixelwise complement of carpet, since "at most one odd" and "at least two odd" exhaust the cases. Witness: lab/rs/hexagon-moire.
2026-08-28 Proved The cut ink laws of all four families are exact closed forms with chi = (-1)^((3n-1)/2): carpet 1/2 + chi/8 + 1/(2n) - chi/(8n^2), net 1 - carpet, tree 1/4 + (1/3 - chi/12)/n + (1 - chi)/(6n^2), void 1/4 - chi/(4n) + 1/(2n^2), exact for all odd n <= 55 in lab/rs/hexagon-moire; the wider range n <= 101 has no generator. Witness: lab/rs/hexagon-moire, walsh-spectrometer.
2026-08-28 Proved The slice stacks' surviving constants are Leibniz, odd Basel and Catalan: the carpet split M (I1 - I3 + 1/4) -> pi/4 + pi^2/32 unconditionally at balanced layer counts (1.08267 at M = 28 against 1.09382), the void background (pi + pi^2)/16 = 0.8131998159 (a fourth-decimal near-collision with the flat-stack pi^2 ln 2/(7 zeta(3)) = 0.8130217042, explicitly separated), the tree background pi/48 + pi^2/48 + G/6 with G Catalan's constant, all character series over the exact ink laws, to 7 digits each. Witness: lab/rs/hexagon-moire.
2026-08-28 Conjecture On rendered cut grams masked to the common hexagon, the 290 coprime layer pairs have Pearson mean -0.037 and range [-0.205, +0.147] with 85 of 290 beyond |0.05|, against a flat-stack coprime maximum of 0.017 on the same raster and exactly 0 in the continuum; the gcd echo survives with the (m, 3m) family topping the table at (17, 51) = +0.248; the two strongest coprime pairs, (5, 9) = -0.205 with the same residue mod 4 and (5, 7) = -0.204 with different residues, show the mod-4 alternation is not the mechanism; these half-mask values are superseded by the full-hexagon -0.142 and -0.085 below.
2026-08-28 Conjecture The xor pair's 14 + 14 stack is the flattest nontrivial field measured, carrying six eternal points at the permutations of (1/4, 1/4, 1), ink at all 28 layers for 105 and paper at all 28 for 150, by a one-line parity proof.
2026-08-28 Conjecture The Catalan statement M (mean ink - 1/2 - eps/2) -> G/8 holds only along N = 3 mod 4 (0.1144757884 at N = 55, 0.11448 at M = 28 against 0.11450); along N = 1 mod 4 the limit is G/8 - 1/8 (-0.0104828892 at N = 53). Witness: lab/rs/hexagon-moire. Superseded: the statement is Proved at both residue classes by the summed ink law, see the Catalan row under Diagonal slice stack in SETTLED.
2026-09-03 Verified The xor pair blinks hardest: code 105 has ink(n) = 1/2 - s/4 - s/(4n^2) and its complement code 150 the reflection 1/2 + s/4 + s/(4n^2), both swinging 1/4 to 3/4; the attempt to break it checked n = 1, where 150 inks 0 of 6 cells against the 1 the shared formula would demand. Witness: walsh-spectrometer.
2026-09-06 Verified The doubling magnitude reads between 0.11711630 and 0.11715991 (Richardson extrapolation on sliding triples of m = 157..601), and the two branch extrapolations in 1/m land on 0.1171270 and 0.1171274, so the exact rational -19/162 = -0.117284 is dead at 1.57e-4. Witness: lab/rs/hexagon-moire.
2026-09-06 Conjecture The doubling magnitude is 253/2160 = 0.11712963, fitting both branches to 1e-6. Witness: lab/rs/hexagon-moire. Superseded: the constant is exactly 253/2160 by the phase-map integral, see the layer-pair row under Diagonal slice stack in SETTLED.
2026-09-07 Proved The ghost star's decay coefficient is a closed form at every band half-width, not just at the arm. Widen the star to the band |x - y| <= W cells; x - y is even on the cut, so W enters only through K = floor(W/2). With b = 1 when floor(K/2) is even, chi = (-1)^((3n-1)/2) the ink law's character (-1 at n = 1 mod 4), chi_8 the real character mod 8 of Q(sqrt 2), and E(K) = #{|j| <= K : j = 3, 4, 5 mod 8} + floor((K + 2)/4) - K the block tail's chi_8 weight, the band's excess over the hexagon's ink law is exactly kappa chi + (m + q chi_8(n))/n + chi/(8 n^2) at every odd n >= K, with kappa = -(-1)^K/(8(2K + 1)), m = -(K + b)/(2(2K + 1)) and q = (1 - 2E(K))/(2(2K + 1)); below n = K the band is clipped and the identity is false, W = 6 at n = 1 missing by 2/7. The decay coefficient is therefore -(K + b)/(4(2K + 1)), the conjectured -(W + 2b)/(8(W + 1)) at even W and -(W - 1 + 2b)/(8W) at odd W, tending to -1/8. E is 8-periodic because a block of eight adds 6 + 2 - 8 = 0, matching the run 0, -1, -1, 0, 1, 2, 2, 1 at 201 of 201 values K = 0..200, and the identity matches the counted band in exact rationals 1354 of 1354 at 14 distinct half-widths, every odd n from K to 201, with the four classes n = 1, 3, 5, 7 mod 8 counted apart. Witness: lab/rs/hexagon-moire.
2026-09-07 Proved The width family's constant and both its 1/L^2 branches are closed forms at every width. At an even layer count L the ladder is L * excess_L = (m/2) ln L + C_W + O(1/L^2) with C_W = m (ln 2 + gamma/2) + q L(1, chi_8) - G/8 + Delta_W, where Delta_W is the sum over odd n < K of the counted excess less the identity, the exact rational the clipped layers contribute, 0 through W = 5 and 2/7 at W = 6. The chi_4 components cancel at the 1/n order only, so L(1, chi_4) = pi/4 is absent at every width while L(2, chi_4) = G sits in every one: C_W carries gamma, ln 2, L(1, chi_8) and Catalan's G. Three tails give the 1/L^2 coefficient: the chi_8 tail over odd n > 2L is -q/4 at L = 0 mod 4 and +q/4 at L = 2 mod 4, since the sign pattern +--+ on the four odd residues starts at n = 2L + 1; the Catalan tail of the background's chi/(8 n^2) gives +1/64 blind to the residue; and the harmonic remainder of m (H_{2L} - H_L/2) gives +m/48. So the coefficient is -q/4 + 1/64 + m/48 against +q/4 + 1/64 + m/48, which at W = 0 is -23/192 and +25/192. The sliding-window slope the sweep reads cancels the oscillation only at L = 0 mod 4 and converges to m/2 + kappa/ln 2 at L = 2 mod 4: the generator reads -0.24999980 at L = 1600 against m/2 = -1/4, and -0.43078703 at L = 1602 against the limit -0.43033688. Witness: lab/rs/hexagon-moire.
2026-09-07 Proved For a fixed affine phase map n = a m + c with m growing inside one class mod 4, and on the mask this page always uses - the full hexagon of the common cut, area-weighted exactly - the layer-pair correlation limit is an exact rational; a general pair (m, n) has no limit theorem here. The cut cell obeys s_y = s_x + s_z + w mod 2 with w = 1 at exactly the phase cells (p, q) = (0, 0) and (3, 1) when N = 1 mod 4 and its complement when N = 3 mod 4; the map sends alpha = mX mod 2 to (a alpha + c X) mod 2, and (mX mod 2, mZ mod 2) equidistributes on the fixed polygon at O(1/m), leaving a piecewise-constant integral with rational breakpoints. It returns the doubling constant exactly 253/2160, covariance 253/9216 over variance 15/64, at all four branches with the sign law's sign; the adjacent limit exactly -11/135 and the gcd echo exactly 29/135; the tree 0, -61/864, 4/27 and the void 0, +7/864, 2/27, the two doubling zeros exact. 19/162 is refuted. Both residue classes converge: (301, 601) reads -0.11745304 and (601, 1201) -0.11729091 at m = 1 mod 4, (103, 205) reads +0.11914004 and (203, 405) +0.11814528 at m = 3 mod 4, gap times m at -0.097, -0.097, +0.207 and +0.206. Witness: lab/rs/hexagon-moire.
2026-09-07 Proved Every constant of the stack's recentred one-layer cut ink is one character sum, and the layer count's parity is the only residue it reads. The hypothesis carries two limbs: the object is the cut ink of a 3D parity design, whose Walsh quasipolynomial carries only chi_4 and terminates at the 1/n^2 order by the Walsh spectrometer's ink theorem, and the quantity is the recentred M (mean - A - c eps). Writing a family's ink law as I(n) = A + B chi + (c + d chi)/n + (e + f chi)/n^2 with chi = -chi_4(n), the average over the first M odd sides obeys M (mean - A - c eps) = -B S - d s_1 + e s_3 - f s_2 exactly, with eps the mean of 1/n, three chi_4 sums and the zeta tail s_3 = sum 1/n^2 over those layers, so the limit is -B [M odd] - d pi/4 + e pi^2/8 - f G: pi enters only through the 1/n order of the ink law, Catalan only through the 1/n^2 order, the residue class only through B, and inside those two limbs no other constant can appear, so L(2, chi_-3) is absent by a theorem rather than by a search; outside them it is not, the ghost star's width family being a one-layer object of the same stack whose constant carries gamma, ln 2 and L(1, chi_8) because its character is mod 8 and its 1/n limb is not subtracted. The four families read (A, B, c, d, e, f) as (1/2, 1/8, 1/2, 0, 0, -1/8), (1/2, -1/8, -1/2, 0, 0, 1/8), (1/4, 0, 1/3, -1/12, 1/6, -1/6) and (1/4, 0, 0, -1/4, 1/2, 0), every row of the constants table is an instance, and the leading term A + B chi is the pair sections' own phase-map integral taken at the identity map a = 1, c = 0. The generator holds the summed identity against the counted hexagons in exact rational arithmetic at every layer count to N = 55, all four families, the classes n = 1, 3, 5, 7 mod 8 counted apart, 7 of 7 in each. Witness: lab/rs/hexagon-moire.
2026-09-07 Proved The Catalan statement holds at both residue classes and neither one is a fit. The carpet is B = 1/8, d = e = 0, f = -1/8, so M (mean ink - 1/2 - eps/2) -> G/8 = 0.1144956993 along N = 3 mod 4, measured 0.1144757884 at N = 55, and -> G/8 - 1/8 = -0.0105043007 along N = 1 mod 4, measured -0.0104828892 at N = 53. The 1/8 step is the ink law's own chi averaged over an odd number of layers, the same parity term the ghost star's even-L hypothesis carries, and Catalan enters only as L(2, chi_4), one order below the pi the tree and the void collect. This closes the Conjecture of the same name. Witness: lab/rs/hexagon-moire.
2026-09-07 Proved The approach to every one-layer constant is a closed form. With sigma = +1 at even M and -1 at odd M, the three tails past a = 2M + 1 solve T(a) + T(a + 2) = a^-s twisted and T(a) - T(a + 2) = a^-s untwisted in powers of 1/a, giving sigma/(4M), sigma/(8M^2) and 1/(4M) with the 1/M^2 limb of each cancelling, so the gap to the limit is (sigma d - e)/(4M) + sigma f/(8 M^2) + O(1/M^3). Carpet and net read gap times M^2 as -1/64 at even M and +1/64 at odd, the void gap times M as -3/16 and -1/16, the tree as -1/16 - 1/(48M) and -1/48 + 1/(48M), and the carpet split as -5/16 at even M only. The generator's ladders at M = 400, 1600, 3200 print all four classes of M mod 4, so all four of N mod 8, and match to eight decimals at M = 3200. Witness: lab/rs/hexagon-moire.

Digit designs and the Euler product

2026-09-07 Proved The indicator of S_F, the integers whose base digits all lie in the digit set F, is multiplicative exactly at the full digit set. 1 in F is forced by f(1) = 1; if a digit c >= 2 is missing take the least, and R_c R_(c+1) has no carry because its base^m coefficient is min(m+1, c, 2c-m) <= base-1, so its digit set is exactly {1..c} while gcd(R_c, R_(c+1)) = R_1 = 1; if only 0 is missing then odd base gives the coprime pair (2, (base^2+1)/2) with product base^2 + 1 = 101, and even base gives (base^2-1, base^2+1), coprime and odd, whose product base^4 - 1 has every digit base-1 while base^2+1 does not lie in the set. Over all 8177 sets with 2 <= base <= 12 the constructed witness is asserted at each of the 4083 sets that pass f(1) = 1 and are not full, and an independent search finds a minimal witness for every one, hardest base = 12, F = {1}, pair (5, 377). No design outside the full set carries an Euler product over primes; 0 excluded and a single digit both fail. Witness: lab/py/mrly-euler verb wall.
2026-09-07 Proved For every F strictly inside {0..base-1} the design zeta and the design Mobius series obey a disjunction and not a universal: if 1 is outside F the constant coefficient of zeta_F M_F is 0; if a prime p of S_F has p^2 outside S_F the coefficient at p^2 is -1, since (p,p) is the only admissible factorisation; and otherwise zeta_F M_F = 1 forces the least element g > 1 of S_F to be prime with every power g^j in S_F, a necessary condition on an escapee and not a contradiction. At the full digit set the two are inverse, zeta_F = zeta and M_F = 1/zeta. Over 257 sets the least n > 1 with a nonzero coefficient is at most 50, first at n = 4 for base 3 {0,1} and n = 9 for base 10 missing 9, while the eight full sets have none below 4000. Witness: lab/py/mrly-euler verb pair.
2026-09-07 Proved The position product. With G_level(t) = prod_(i<level) sum_(d in F) e(d base^i t) = fill^level hat F_level(t), uniqueness of the digit expansion gives int_0^1 G_level(t) e(-nt) dt = 1_(D_level)(n) for every integer n, hence sum_(n in D_level, n >= 1) a(n) n^(-s) = int_0^1 G_level(t) A(s,t) dt for every absolutely convergent Dirichlet series, with A(s,t) = sum_(n >= 1) a(n) e(-nt) n^(-s); a = 1 is the periodic zeta of DLMF 25.13.1 and a = mu the Lerch-Mobius series, so zeta_F and M_F are pairings of one set-only product against one arithmetic-only kernel. The set enters through the digit positions and never through the primes. Checked to 1.95e-16 and 2.04e-16 at base 10 missing 9, level = 3 and level = 4, and 2.9e-16 at base 3 {0,1}, level = 3, 4, 5. Witness: lab/py/mrly-euler verb position.
2026-09-07 Proved The tree's pair route is Holder on the position identity. When 0 is in F, at x = base^level the identity is finite on both sides, M_F(base^level) = int_0^1 G_level(t) S_level(t) dt with S_level(t) = sum_(n < base^level) mu(n) e(-nt), so abs(M_F(base^level)) <= (int_0^1 abs(G_level)) max_t abs(S_level) is at most fill^level base^(level(alpha_1 - 1)) x^b = x^(alpha + alpha_1 - 1 + b); when 0 is outside F the same upper bound holds after summing the levels, a geometric sum of ratio base^(alpha + alpha_1 - 1 + b) > 1 by the floor alpha + alpha_1 >= 1 of mobius.md. It sits under the trivial x^alpha exactly when alpha_1 < 1 - b, which is the bar of coprime.md and mobius.md derived rather than posited, with b = 3/4 + eps under GRH from Baker and Harman 1991. Witness: lab/py/mrly-euler verb position.
2026-09-07 Proved The fibres of the Lerch-Mobius series are inverse Dirichlet L-functions. Splitting n by g = gcd(n,Q) and expanding on the characters of (Z/(Q/g))^* gives M(s, a/Q) = sum_(g divides Q) mu(g) g^(-s) phi(Q/g)^(-1) sum_(chi mod Q/g) tau_a(chi) L(s,chi)^(-1) prod_(p divides Q not Q/g) (1 - chi(p) p^(-s))^(-1), so M(s, a/Q) continues to C with singularities in Re s > 0 only at zeros of L(s,chi) of modulus dividing Q, and M(s,0) = 1/zeta(s). Since G_level(a/base^j) = fill^(level-j) G_j(a/base^j) are the largest values the position product takes, the design's major arcs are the base-power rationals, and on that family holomorphy in Re s > 1/2 is exactly GRH for base-power modulus. Coefficient identity checked to 2.6e-12 at eleven pairs (Q,a) including Q = 3, 9, 27, 100, the Euler-factor step to 7.4e-16. Witness: lab/py/mrly-euler verb fibre.
2026-09-07 Proved The reflection moves the kernel and not the design. Solving Hurwitz's formula DLMF 25.13.3 at x = t and x = 1-t gives Z(s,t) = ((2 pi)^s Gamma(1-s)/(2 pi i))(e^(pi i s/2) zeta(1-s,t) - e^(-pi i s/2) zeta(1-s,1-t)) for s not a positive integer, the derivation dividing by 2i sin(pi s); this is DLMF 25.13.2 recovered, the gain being the range Re s > 0 in place of Re s > 1. The position identity turns it into a dual integral of the same G_level against Hurwitz zetas at 1-s, never a relation between zeta_F(s) and zeta_F(1-s); the design's own symmetry is the base-adic scaling G_level(t) = g(t) G_(level-1)(qt), whose transfer eigenvalue fill base^(-s) is what makes the vertical pole lattice. Formula checked to 2.1e-30 at s = 3.3, 2.7 + 1.9i and 0.6 + 4.1i. Witness: lab/py/mrly-euler verb dual.
2026-09-07 Proved The design's multiplicative shadow is a Lyndon Euler product with no RH content. On the free monoid over F with norm N(w) = base^(abs(w)), sum_w N(w)^(-s) = 1/(1 - fill base^(-s)) = prod_(level>=1) (1 - base^(-level s))^(-c_fill(level)) with c_fill(level) the Lyndon count, by Chen-Fox-Lyndon: every word factors uniquely as a non-increasing product of Lyndon words, so the free monoid on F is equinumerous by norm with the free abelian monoid on Lyndon words and is not equal to it. The primes are the Lyndon words, the zeta is zero-free, its Mobius is supported on the empty word and the letters so its Mertens is 1 - fill beyond norm 1, and its poles are exactly s = alpha + 2 pi i m / log base, the design pole lattice. All RH content of zeta_F therefore sits in the cofactor zeta_F(s)(1 - fill base^(-s)). Expansion verified through u^16 at fill = 2, 3, 4, 9, 10, c_2(level) being A001037. Witness: lab/py/mrly-euler verb word, A001037.
2026-09-07 Proved The Beurling system of a design with non-unit digit gcd is finitely generated, on two branches. If gcd(F) = a > 1 every element of S_F is a multiple of a; when a is prime the primes of the design are {a}, N_F is the powers of a and M_B(x) = 0 for x >= a, and when a is composite S_F holds no prime at all, N_F = {1} and M_B is identically 1, witness base = 10, F = {0,4,8}. Either way the eight scaled census families of mobius.md are exactly the columns the Beurling route cannot see, while the scaling transfer reads them exactly. Witness: lab/py/mrly-euler verb beurling.
2026-09-07 Verified The Beurling census on the primes of a design, to x = 10^6. Base 3 {0,2} has the single prime 2 and M_B identically zero past 2; base 3 {0,1} has 525 primes, N_F(920483) = 2198, running max abs(M_B) = 98 and exponent 0.3339 against alpha/2 = 0.3155; base 10 missing 9 has 35139 primes, N_F(10^6) = 488864 against x^alpha = 531441, M_B(10^6) = 1860, running max 1866, and exponent log(running max)/log x reading 0.4203, 0.4882, 0.5452 at 10^4, 10^5, 10^6 against alpha/2 = 0.4771, where full base 10 as control reads 0.4084, 0.4241, 0.4276 at the same points against its own alpha/2 = 0.5. What the census reads is the level and not a trend: +0.068 over alpha/2 for the design against -0.072 for the control, a running maximum climbing in both. There is cancellation, 0.545 against the trivial alpha = 0.954, and it is above alpha/2, so the census supports cancellation and does not support the square-root conjecture on N_F; N_F is not S_F. Full base 10 reproduces -23, -48, 212 at 10^4, 10^5, 10^6, A084237. Witness: lab/py/mrly-euler verb beurling, A084237.
2026-09-07 Proved The identity that replaces zeta M = 1 on a design. For every (base,F) with 1 in F the indicator 1_(S_F) has a Dirichlet inverse nu_F, given by nu_F(1) = 1 and nu_F(n) = -sum_(d divides n, d > 1, d in S_F) nu_F(n/d), so zeta_F(s) N_F(s) = 1 with N_F(s) = sum nu_F(n) n^(-s); the support of nu_F lies inside the multiplicative semigroup generated by S_F and strictly inside it, since 9, 27 and 36 lie in the semigroup with nu_F = 0 while 16, 48 and 52 lie in the semigroup and outside S_F, so the semigroup is a third set beside S_F and the Beurling integers on the primes of the design and the support is a fourth, and nu_F is mu exactly at the full digit set, where the classical identity is the special case. If rho is a zero of zeta_F with Re rho > alpha then sigma_c(N_F) >= Re rho, by the identity theorem on the connected pole-free half plane Re s > max(sigma_c(N_F), alpha), so sum_(n <= x) nu_F(n) is not O(x^(Re rho - eps)) for any eps > 0; the converse bound sigma_c(N_F) <= sup Re rho is not claimed. Checked against mu term for term on the full digit set to n = 131072 at base = 2 and n = 177147 at base = 3, and the partial sums of N_F(sigma) meet 1/zeta_F(sigma) to 1.60e-3 at sigma = Re rho + 0.08 = 0.8008 and 1.96e-4 at sigma = Re rho + 0.20 = 0.9208 at base 3 {0,1}, and to 1.72e-2 at sigma = 1.0816 and 2.39e-3 at sigma = 1.2016 at base 10 missing 9, both offsets sitting above Re rho. Witness: lab/py/mrly-pairing verb inverse, lab/py/design-zeta.
2026-09-07 Proved The design's own Mobius has anti-cancellation, and that is what makes the decoupling a blessing. Winding boxes on zeta_F by the argument principle certify one zero each and pin Re rho to the box edges: winding 1 on Re in [0.72074, 0.72084], Im in [28.60563, 28.60573] at base 3 F = {0,1} with contour minimum abs(zeta_F) = 8.298e-4 against the engine bound 6.284e-30, and winding 1 on Re in [1.00150, 1.00168], Im in [2.73915, 2.73925] at base 10 missing 9 with contour minimum 6.865e-4 against 2.798e-23, while the control rectangle Re in [0.99900, 1.00050], Im in [2.73810, 2.74030] there returns winding 0. Both boxes lie strictly right of alpha = 0.6309297536 and 0.9542425094, so sum_(n <= x) nu_F(n) is not O(x^(0.72074 - eps)) and not O(x^(1.00150 - eps)) respectively: the limsup of the design's own Mertens function exceeds the design's own mass A_F(x), and at base 10 missing 9 exceeds x itself, the box lying right of Re s = 1. The square-root conjecture in the alpha/2 shape is therefore false for nu_F and can only be carried by mu restricted to S_F; the sibling's decoupling theorem is what protects it. Pointwise the census is far below both limsups, max/A_F = 0.0738 at base 3 level = 16 and max/x = 0.0847 at base 10 level = 7: the running maximum of sum nu_F(n) grows by 9.4474, 11.5000, 10.2220, 10.0354 per level at base 10 missing 9, level = 4..7, against base^(Re rho) = 10.036661 and the trivial fill = 9, only the last of the four landing on the predicted rate, with max/A_F(base^level) rising 0.1043, 0.1094, 0.1398, 0.1588, 0.1771; at base 3 {0,1} the geometric mean of the four steps level = 12..16 is 2.059 against 2.207512 and 2 while the arithmetic mean of the five printed level ratios is 1.9972, below the trivial 2, a census too short to separate them. Witness: lab/py/mrly-pairing verbs box and inverse, lab/py/design-zeta.
2026-09-07 Verified The pair zeta_F M_F = 1 + D_F gains nothing: D_F has abscissa exactly alpha. Absolute convergence of zeta_F^2 puts sigma_a(D_F) <= alpha and that half is proved; for the other half, if sigma_c(D_F) were below alpha then M_F(sigma) = (1 + D_F(sigma))/zeta_F(sigma) would tend to 0 as sigma -> alpha+, since zeta_F has nonnegative coefficients and is singular at its abscissa by Landau, so zeta_F(sigma) -> +infinity, and there is no circularity in the argument because M_F is dominated termwise by zeta_F and so converges absolutely at every sigma > alpha with no hypothesis on theta(F). That half rests on a measurement, unconditional in shape since sigma_c(D_F) < alpha would force P(x) = o(x^alpha): P(x) = sum_(n <= x) c_F(n) divided by x^alpha is bounded away from 0 and from infinity, reading 0.493767, 0.699235, 0.758519, 0.587055 at four sampling phases at base 3 {0,1}, the four phases being needed because P(x)/x^alpha is log-periodic and sampling only at x = base^level aliases every Fourier mode onto one number. The M_F(sigma) -> 0 limit test is not a witness here: the tail the generator prints beside it is base^(-level alpha/2), which assumes the square-root conjecture, and against the unconditional tail (fill-1) base^(-level eps)/(1 - base^(-eps)) from A_F(base^l) = fill^l no printed M_F value at base 10 missing 9 is distinguishable from 0. Since M_F = (1 + D_F) N_F and sigma_c(N_F) > alpha, the glue is not neutral but lossy. Witness: lab/py/mrly-pairing verb glue.
2026-09-07 Proved The position pairing is exact on the grid and its l^1 mass sits at the top level, which kills the per-denominator split. For 0 in F, M_F(base^level) = base^(-level) sum_(a mod base^level) G_level(a/base^level) S_level(a/base^level) exactly, both factors being trigonometric polynomials of degree below base^level; writing a = base^v a' with base not dividing a' and j = level - v gives G_level(a/base^level) = fill^(level-j) G_j(a'/base^j) and the exact level decomposition C_level = sum_(j=0)^level fill^(level-j) c_j of the l^1 mass, with C_j = fill C_(j-1) + c_j. The l^1 floor C_j >= base C_(j-1) forces the top-level share c_level/C_level >= 1 - fill/base = m/base at every base and digit set, measured 0.485846, 0.602606, 0.687994, 0.510055 against floors 0.333333, 0.500000, 0.600000, 0.100000, with levels j >= level/2 carrying 0.995116, 0.996061, 0.997043, 0.942350. Since the Baker-Harman Proposition beats the uniform x^(3/4) only below j = level/2, weighting the Mobius input per denominator saves exactly log(C_level/c_level)/(level log base), a constant factor capped by base/m: the numerator is 0.657068 at base 3 {0,1}, identical at every level = 6..14. The split exponents are 0.988106, 0.912502, 0.905006, 1.012881 uniform and 0.941173, 0.879287, 0.879188, 0.964150 per denominator against alpha = 0.630930, 0.500000, 0.430677, 0.954243, while the Cauchy-Schwarz split is (alpha+1)/2 exactly since int abs(G_level)^2 = fill^level; the base 2 and base 3 full-set controls return 0.500000, the classical RH exponent. Witness: lab/py/mrly-pairing verb split.
2026-09-07 Proved The principal fibre of the grid pairing has exponent alpha - 1/2 under RH, below the conjectured alpha/2, and that is an asymptotic statement only. The a = 0 term of the grid pairing is base^(-level) fill^level M(base^level), of exponent alpha - 1/2 under RH, and alpha - 1/2 < alpha/2 for every alpha < 1, so in the limit the classical Mertens function cannot carry the conjectured size of the design meter. At finite depth it carries a great deal: the a = 0 term reads -0.31857 of 11, 0.11133 of 6, -0.05924 of 9 and 112.66549 of 276 at base 3 {0,1} level = 14, base 4 {0,1} level = 11, base 5 {0,1} level = 9 and base 10 missing 9 level = 6, shares -0.028961, 0.018555, -0.006583, 0.408208, and exactly all of the meter on the two full-set controls. So at base 10 missing 9 the principal fibre carries 40.8 percent of the meter at the only measured level, which refutes any claim that the square-root conjecture lives entirely off the principal fibre at finite depth: the exponent gap there is 0.454243 against 0.477121, and a factor of 10 between them needs x = 10^44. Witness: lab/py/mrly-pairing verb split.
2026-09-07 Proved The one-step constant of the digit transform never exceeds the triangle-split bound, and is strictly below it at every family measured beyond level = 1. With H(t) = sum_(r mod base) abs(g_F((t+r)/base)) and B_base(F) = sup_t H(t), the identity C_level = sum_(a mod base^(level-1)) abs(G_(level-1)(a/base^(level-1))) H(a/base^level) gives C_level <= B_base(F) C_(level-1), so C_level/C_(level-1) <= B_base(F) at every level and every family with no computation at all; the inequality is not strict in general and equality is attained, C_1/C_0 = 4 = B_base(F) exactly at base 3 {0,1}, so strictness needs level >= 2. Verified there: C_level/C_(level-1) reads 3.889888518, 5.032783116, 6.410132461, 18.369402635 at base 3 {0,1}, base 4 {0,1}, base 5 {0,1} (all level = 9) and base 10 missing 9 (level = 6), against B_base(F) = 4.000000000, 5.226251860, 6.472135955, 19.888543820, and the ratio agrees between the two consecutive level the generator prints to 8.5, 7.4, 10, 5.0 digits by family, so the stability is family by family and two values of level are all that is measured. Witness: lab/py/mrly-pairing verb split.
2026-09-07 Proved The design Mobius of the two-digit design is base-free. Let S* be the nonzero 0/1 polynomials of Z[x], M* the monoid they generate, nu* the Dirichlet inverse of 1_(S*). For F = {0,1} at every base >= 2, nu_F(n) = sum over P in M* with P(base) = n of nu*(P). Evaluation is a bijection S* -> S_F, a monoid homomorphism, and of finite fibres, since an element of M* has nonnegative coefficients so P(base) = n caps every coefficient by n and deg P by log(n)/log(base); the pushforward g therefore exists, g(1) = 1 because 1 is the only element of M* of value 1, and grouping the pairs (D, Q) in S* x M* with D(base) Q(base) = n by P = DQ turns 1_(S*) * nu* = delta into 1_(S_F) * g = delta, where the Dirichlet inverse is unique. Hence sum_(n <= x) nu_F(n) = sum over P in M* with P(base) <= x of nu*(P) at every x: the base enters only as the order in which one base-free function is summed, and at base = 2 the classical mu is that pushforward. Checked term for term with 0 mismatches to n <= 2^15, 3^10, 4^8 and 5^7, where 108978 elements of M* collapse onto 32768 integers at base 2. Witness: lab/rs/carry-free-mobius verb lemma.
2026-09-07 Proved The degree-graded mass of the base-free design Mobius is 1 - 2t exactly. Degree is a monoid homomorphism M* -> N with finite fibres because 1 is the only constant in M*, which holds for F = {0,1} and for no design carrying a digit at least 2, where a constant c >= 2 makes the degree-zero fibre {c^k} infinite; pushing 1_(S*) * nu* = delta along it with 2^d polynomials of degree d gives A(t)/(1 - 2t) = 1, so the graded sums are 1, -2, 0, 0, ... and sum over deg P < level of nu*(P) = -1 for every level >= 2. The design Mertens function is therefore pinned to -1 at every level boundary level >= 2 inside the carry-free window, and reads 0 at level = 1. Witness: lab/rs/carry-free-mobius verb sequence.
2026-09-07 Proved The design zeta has an explicit zero free half plane, and it closes the census right of the abscissa. Let a_min be the least nonzero digit of F, hence the least element of S_F, every element of two digits or more exceeding base. If a real sigma > alpha satisfies a_min^sigma zeta_F(sigma) < 2 then zeta_F has no zero in Re s >= sigma: the coefficients are nonnegative and the series converges for sigma > alpha, so for Re s = sigma' >= sigma one has abs(a_min^s zeta_F(s) - 1) = abs(sum_(n in S_F, n > a_min) (n/a_min)^(-s)) <= sum_(n > a_min) (n/a_min)^(-sigma) = a_min^sigma zeta_F(sigma) - 1 < 1. The hypothesis sigma > alpha is load bearing and the test is one real evaluation carrying the ladder's own error bound. On the grid alpha + 0.05 n the edge sigma_1 reads 0.5 at base 4 {1} to 1.75 at the three full digit sets over twenty-four designs, with a_min^sigma zeta_F(sigma) in [1.8635, 1.9995] and largest sigma_1 - alpha equal to 0.95, so a census of the zeros right of alpha needs no hand chosen right edge and the alpha + 3.02 strip of the locus sweep is three times wider than the zeros need. Witness: lab/py/transport-census verb census.
2026-09-07 Verified The transport census: every proper design censused carries zeros right of its abscissa, the full digit set alone carries none, and each rightmost is certified by a winding box. On the box alpha + 1e-6 < Re s < sigma_1, where the cofactor Z = zeta_F(s)(1 - fill base^(-s)) is analytic and its zeros right of alpha are exactly those of zeta_F, the transfer failing only at residue null poles which sit on the line Re s = alpha, the argument principle counts 157 zeros right of alpha below Im s = 40 over twenty-three designs, all 157 located, plus 2 at base 50 missing one digit below Im s = 4. The count is exact on the box and a lower bound for the half plane, since the sliver alpha < Re s <= alpha + 1e-6, the band 0 < Im s < 0.02, everything above the census height and the conjugate half plane are uncounted. Twenty-one of the twenty-four designs carry such a zero; the three that do not are the base 2, 3 and 4 full digit sets, whose windings read -1.97e-33, 1.73e-33 and 1.53e-33. Every rightmost carries the height it is read below, because the teeth of the level zero comb drift right with the pole index: base 20 missing one digit reads 1.000285484146 at Im s = 2.0988, 1.000549674321 at 4.1971 and 1.002685494779 at 14.6920. Below Im s = 40 the rightmost real parts run 0.441505537191 at base 5 {0,1} to 1.002685494780 at base 20 missing one digit, each certified by a winding 1 box on zeta_F of half width 5e-5 in Re s and in Im s whose sampled contour minimum, 1.2e-4 to 6.1e-3, beats the engine's error bound by at least eight orders of magnitude and whose distance to the pole lattice s_(i,j) = alpha - i + 2 pi i j/log base is at least 0.00517845, four hundred box half widths. The two published boxes of lab/py/mrly-pairing reproduce at their own edges, winding 1 and 1 with contour minima 8.298e-4 and 6.865e-4, and its control rectangle returns winding 0. Witness: lab/py/transport-census verb census.
2026-09-07 Proved A certified zero right of the abscissa refutes every square-root-shaped bound for the design's own Mobius, and the digit 1 is the hypothesis that bites. Let 1 in F, let rho be a zero of zeta_F certified by a winding 1 box with left edge x_0 > alpha containing no pole, and let nu_F be the Dirichlet inverse of 1_(S_F). The transport theorem gives sigma_c(N_F) >= Re rho >= x_0 > alpha, so sum_(n <= x) nu_F(n) is not O(x^(x_0 - eps)) for any eps > 0; since A_F(x) has exponent alpha the square-root exponent is alpha/2 <= alpha < x_0, so the design's own Mobius satisfies no square-root-shaped bound and misses even the trivial O(x^(alpha - eps)), the first inequality failing to be strict only at the two designs with alpha = 0, where A_F(x) grows like log x. Nineteen of the twenty-four designs censused meet all three hypotheses and get a bound, and seventeen of the twenty-two the locus and family sweeps censused, the bounds running theta(nu_F) >= 0.4414555 at base 5 {0,1} to theta(nu_F) >= 1.0026354 at base 20 missing one digit. Four designs have Re rho > 1, so their own Mobius outruns the count of all integers below x: base 10 missing two digits, base 10 missing the digit 9, base 20 missing one digit and base 50 missing one digit, at fill/base = 0.8, 0.9, 0.95, 0.98 and alpha = 0.9030900, 0.9542425, 0.9828779, 0.9948357; only the SIGN of Re rho - 1 is read and never its size, three of the four being censused to Im s = 40 and base 50 to Im s = 4. The fill/base reading dies on its control, base 5 {0,1,2,3} at the same fill/base = 0.8 with rightmost 0.989748105861. Two designs carry a zero right of alpha and no bound: base 4 {2,3} and base 4 {0,2,3} omit the digit 1, so 1 is outside S_F, the indicator vanishes there and nu_F does not exist. Witness: lab/py/transport-census verb law.
2026-09-07 Refuted The gain of a design's rightmost zero over its abscissa is not a function of alpha and fill/base. The refuted functional is new: the locus row already refutes a law for the POSITION of the zeros, this refutes one for the single statistic the transport theorem reads, the rightmost real part less alpha. Four equal key families, one base and one digit count each so alpha and fill/base agree exactly and not to a rounding, read unequal gains: at alpha = 1/2, fill/base = 1/2 the four base 4 two digit designs give 0.0853043873, 0.4400124317, 0.2706238545, 0.3439264581, a spread of 0.35470804; base 5 at alpha = 0.4306766, fill/base = 0.4 spreads 0.37474232; base 3 two digit at alpha = 0.6309298 spreads 0.17605693; base 4 three digit at alpha = 0.7924813 spreads 0.060972003, which is still six hundred box widths. The two columns disagree in direction: the gain is largest at the sparsest designs, 0.5291214025 and 0.4485242462 at alpha = 0, while the rightmost real part itself is smallest there. What rises with alpha is the floor, the least rightmost real part at each alpha reading 0.4485242462, 0.4415055372, 0.5853043873, 0.7207876015, 0.9126562295, 0.9897481059, 1.0015143877, 1.0015892753, 1.0026854948, 1.0000614750 up ten rungs alpha = 0, 0.4307, 0.5, 0.6309, 0.7925, 0.8614, 0.9031, 0.9542, 0.9829, 0.9948, rising at every step but the first and the last, the last being where the census height drops from 40 to 4; one design per rung above alpha = 0.86 against six at alpha = 0.5, and no fit is taken. Witness: lab/py/transport-census verb law.
2026-09-11 Proved nu* vanishes at every polynomial divisible by x^2, and nu*(x b) = -nu*(b) at every b of nonzero constant term. The convolution runs over Z[x] divisors and M* is not divisor-closed, 1 + x^2 + x^4 = (1 + x + x^2)(1 - x + x^2), so the claim lives on A, the nonzero polynomials mod units, where nu* vanishes off M* by induction. A splits as N x R, R the classes of nonzero constant term and divisor-closed; x^k b lies in S* exactly when b lies in S*_odd, so 1_(S*) is the outer product of the all-ones function on N with 1_(S*_odd), inversion factors, and the inverse of the all-ones function on N is 1 - t. Witness: lab/rs/carry-free-mobius verb ladder.
2026-09-11 Proved Every element of M* of degree d has its coefficient of x^i at most binomial(d, i), so the maximum coefficient at degree d is exactly binomial(d, floor(d/2)), A001405, attained by (1+x)^d. A product of 0/1 polynomials of degrees summing to d is dominated coefficientwise by prod_k (1 + x + ... + x^(d_k)), each factor by (1 + x)^(d_k), and domination survives products of nonnegative polynomials, so the product is under (1 + x)^d, itself in M*. This retires the measured clause and the crude cap 2^(L-1) of the carry-bound row. Checked at every degree to 22, maximum 705432. Witness: lab/rs/carry-free-mobius verb ladder.
2026-09-11 Proved On R, the classes of nonzero constant term among the nonzero polynomials up to units, nu* is fixed by the reciprocal b -> x^(deg b) b(1/x). There the reciprocal is degree-preserving, multiplicative and involutive, hence a monoid automorphism, and it carries S*_odd onto itself by reversing the bitmask, so it preserves 1_(S*_odd) and its Dirichlet inverse. It is no invariance on all of Z[x]: the reciprocal drops the x power and nu*(x) = -1 against nu*(1) = 1. Checked with 0 mismatches over the 35121747 classes of degree 1 to 21. Witness: lab/rs/carry-free-mobius verb ladder.
2026-09-11 Proved The carry-free window of a two-digit design is (base+1)^(level-1) < base^level. A 0/1 polynomial of degree d has P(base) <= (1+base)^d and degrees add over a product, so the maximum of P(base) over M* at degree below level is exactly (base+1)^(level-1), attained by (1+x)^(level-1), and the least base holding every such element under base^level is the least base with (base+1)^(level-1) < base^level. Verified by enumeration at level = 3..14, reading 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7. The windows are 4, 7, 9, 12 and 15 at bases 3 to 7, and sum_(n <= base^level) nu_F(n) leaves -1 at level 5, 8 and 10, one level past the window each time. Witness: lab/rs/carry-free-mobius verb lemma.
2026-09-11 Conjecture The base-free Mertens maximum of the two-digit design gains on the design's mass without reaching it. The running maximum of sum nu* over degree below level reads 1, 1, 2, 3, 4, 7, 15, 23, 45, 86, 162, 331, 741, 1665, 3173, 7508, 17753, 36147, 79645, 182432, 427806, 858703, 2026147 at level = 1..23, and the ratio max/2^level bottoms at 0.079102 at level = 11, falls for the last time at level = 15, and rises at every step from there to 0.241536 at level = 23. A turn-down at a deeper level kills the trend and none is seen to 23. Witness: lab/rs/carry-free-mobius verb ladder.
2026-09-11 Conjecture The rate of that maximum exceeds the design's mass rate 2, and its estimate is window-unstable. At depth 23 the geometric mean step reads 2.245836, 2.202419, 2.242075, 2.206405 over the last 4, 6, 8, 10 levels but 2.194975, 2.149760, 2.092480, 2.074545, 1.996559 over the last 12 to 20, a hull of [1.996559, 2.245836] straddling 2, the long windows opening inside the levels where the ratio still fell. Every short-window reading at depths 20 to 23 sits above 2.18 and above its depth-18 value. No constant is claimed; log_2(max)/level reaches 0.910883 at level = 23 unsettled. Witness: lab/rs/carry-free-mobius verb ladder.
2026-09-19 Proved Weil's theorem reaches a design zeta along no evaluation bridge: P -> P(base) on polynomials with coefficients in {0..base-1} is a bijection onto the nonnegative integers and additive only where no carry occurs, a coefficient of the polynomial product P R reaching (base-1)^2 (min(deg P, deg R) + 1) against the digit cap base - 1, so an integer product's digit string is the carry reduction of the polynomial product and R_c R_(c+1) is the shortest coprime pair whose carry-free product shows the missing digit c. Witness: lab/py/mrly-euler verb wall.
2026-09-19 Proved Under the hypothesis alpha < 1 the large sieve does not rescue the per-denominator split on the major arcs: writing a = base^v a' with base not dividing a' and j = level - v, the l^2 mass of the grid pairing over the levels j <= J is exactly fill^(2 level - J) base^J, and the spacing form of the large sieve on those base^(-J) spaced points gives (base^level + base^J) base^level, so the levels below J = u level cost x^(alpha + u(1-alpha)/2), strictly above alpha at every u > 0 and equal to alpha only at u = 0. Witness: lab/py/mrly-pairing verb split, Montgomery and Vaughan 1973.

Digit strings across divisors

2026-09-01 Proved Orthogonality for digit strings against a divisor: N_F(level; d, r) = (1/d) sum_{a mod d} e(-a r/d) prod_{j < level} g_F(a base^j/d) with g_F(t) = sum_{f in F} e(f t), by expanding the divisibility indicator in additive characters mod d, the digits being independent so the character sum factors over positions, the a = 0 term giving fill^level/d; the attempt to break it rebuilds the whole residue vector by dynamic programming against brute-force string enumeration at four bases and checks its total against fill^level at every census cell, with no mismatch. Witness: mobius.md digit strings across divisors, lab/rs/rho-decoupling.
2026-09-01 Proved The uniform geometric equidistribution bound: for fill >= 2, d >= 2, (d, base) = 1 and gcd(d, Delta_F) = 1 with Delta_F the digit-difference gcd, |N_F(level; d, r) - fill^level/d| <= ((d-1)/d) fill^level (1 - 8/(fill^2 d^2))^level <= fill^level exp(-8 level/(fill^2 d^2)) for every r and level >= 1, since |g_F(a/d)|^2 = fill^2 - 4 sum_{f < f'} sin^2(pi a (f' - f)/d) and d | a(f' - f) at every pair would force d/gcd(a, d) | Delta_F hence d | a; the attempt to break it asserts the weaker form as an exact integer inequality at every census cell where the hypotheses hold, five bases and depths to level = 96, with no failure and largest observed-to-bound ratio 0.187 at base = 100, F = {0,1}, level = 16, and the hypothesis edge d = 2, fill = 2 holds at bound factor (1 - 1/2)^level; the d^(-2) in the exponent is sharp in shape, since d | base - 1 with F an arithmetic progression of difference m' and a m' = 1 mod d gives |g_F(a/d)|/fill = sin(pi fill/d)/(fill sin(pi/d)) = 1 - Theta(fill^2/d^2). Witness: mobius.md digit strings across divisors, lab/rs/rho-decoupling.
2026-09-01 Proved The dense-digit bound: for F = {0..base-1} minus E with m = |E|, fill = base - m and (d, base) = 1, gamma_F(d) <= (d/2 + m)/fill because g_F is the full Dirichlet kernel less g_E, |D_base(a/d)| <= 1/(2||a/d||) <= d/2 and |g_E| <= m, so for d/2 + m < fill the error is at most fill^level ((d/2 + m)/fill)^level uniformly in r; the attempt to break it looks for the gain at fixed digit count, where the bound is vacuous and stays vacuous - at d = 7 the per-digit rate falls 0.4869, 0.3312, 0.2484, 0.1104, 0.0167 as fill runs 2, 3, 4, 9, 99 but reads 0.4992 for F = {0,1} at base = 100, against the same ceiling 0.9010 that F = {0,1} carries at base = 3. Witness: mobius.md digit strings across divisors, lab/rs/rho-decoupling.
2026-09-01 Proved A power saving over a whole level for dense digit sets: for eps in (0,1), base >= 4^(1/eps), m <= base^(1-eps)/2 and level >= 4/eps, every 2 <= d <= base^(1-eps) coprime to base has per-digit factor (d/2 + m)/fill <= base^(-eps/2), so sum over those d of |N_F(level; d) - fill^level/d| <= fill^level base^(1 - eps level/2) <= fill^level x^(-eps/4) at x = base^level, a level of distribution base^(1-eps) with no conditional input; the attempt to break it pushes the level past a constant power of the base and fails, since summing the geometric bound alone caps the level at d ~ sqrt(level)/fill, and the census argmax at every family's deepest level is a divisor of base^t - 1 with t <= 8, where no per-factor bound decays. Witness: mobius.md digit strings across divisors, lab/rs/rho-decoupling.
2026-09-01 Proved The exact split across the base's own divisors: for d = d1 d2 with d1 | base^m, m <= level, and (d2, base) = 1, the low m digits fix the value mod d1 and reach the rest only through the invertible multiplier base^m mod d2, so N_F(level; d) = sum over w in F^m with d1 | val(w) of N_F(level - m; d2, r_w) with r_w = -val(w) (base^m)^(-1) mod d2, and the density splits as rho_F(d1 d2) = (N_F(m; d1)/fill^m)(1/d2); the attempt to break it tests the natural guess 1/d1 for the base part and refutes it, the base part being a digit-string count, with the identity itself pinned against direct enumeration at base = 6, d = 10. Witness: mobius.md digit strings across divisors, lab/rs/rho-decoupling.
2026-09-01 Proved The digit-gcd hypothesis is a wall: if gcd(d, Delta_F) > 1 there is no equidistribution, witness base = 3, F = {0,2}, d = 2, where every value is even, N_F(level; 2) = fill^level and the normalized error d |N_F(level; d) - fill^level/d| / fill^level is exactly 1 at every level; the attempt to break the wall by sweeping the whole range instead of one divisor leaves it standing, the unrestricted worst error over d <= 200 reading 1.0483 at level = 32 pinned at d = 164 against 0.019166 once d is required coprime to Delta_F, and such families reduce to a primitive one through the scaling bijection S_(aF') = a S_(F'). Witness: mobius.md digit strings across divisors, lab/rs/rho-decoupling.
2026-09-01 Proved The second moment across residue classes: sum_{r mod d} (N_F(level; d, r) - fill^level/d)^2 = (1/d) sum_{a not 0 mod d} prod_{j < level} |g_F(a base^j/d)|^2, by Parseval mod d on the orthogonality identity, the mean being the a = 0 term and no cross terms surviving; the attempt to break it looks for a hidden hypothesis and finds none, the identity holding for every d >= 1 and every F, including the walls where the supremum bound is worthless, which is what makes it the one handle left at a pinned divisor. Witness: mobius.md digit strings across divisors, lab/rs/rho-decoupling.
2026-09-01 Verified The divisor census of digit strings: exact dynamic-programming counts of N_F(level; d) over 2 <= d <= D for ten families at base = 3, 4, 5, 10, 100, depths to level = 96 and D to 500, printing the worst normalized error, the multiplicative order of base at the argmax, the per-factor ceiling gamma_F(d) and the slack against the proved bound; the counts are pinned against brute-force string enumeration at four bases, the residue vector totals fill^level, and the argmax is a pinned divisor of base^t - 1 with t <= 8 at every family's deepest level, d = 164 at base = 3, d = 143 at base = 10, d = 101, 303 at base = 100, with shallow depths straying (d = 199, ord = 99, at base = 10, level = 6). The slow column is the sparse one: F = {0,1} at base = 100 reads worst normalized error 28.593, 14.590, 9.0340, 7.2034 at level = 16, 32, 64, 96, per-digit factor 0.9929. Witness: lab/rs/rho-decoupling, mobius.md digit strings across divisors.
2026-09-01 Conjecture The orbit-mean law at a pinned divisor: for F = {0..base-1} minus one digit and d = base^t - 1, the worst orbit-mean damping is fill^(-1/t) (1 + o(1)), the orbit a base^j mod d carrying t - 1 undamped points and one damped by ~ 1/fill; at base = 100, level = 12 the single-divisor probes read orbit mean 0.1059 at d = base^2 - 1 against fill^(-1/2) and 0.2369 at d = base^3 - 1 against fill^(-1/3), with the proper divisor d = 3367 | base^3 - 1 better at 0.0549 and d = 101 | base + 1 pinned but harmless at 0.0261, the kernel being flat across that whole orbit. Two values of t on one base with one dominant character are a check and not a law, and the o(1) is untested; t = 4 needs the orbit product analysed rather than counted, the exact count at d = base^4 - 1 being out of reach of the census. Witness: lab/rs/rho-decoupling, mobius.md digit strings across divisors.
2026-09-05 Verified The signed pinned sum against its absolute sum: over the squarefree moduli e = (base^t - 1)/g, g | base - 1, e >= 2, t <= level <= 40, with T_level(e) = N_F(level; e) - fill^level/e, the ratio sum mu(e) T_level(e) / sum |T_level(e)| reads -0.211, -0.123, +0.069, -0.498 at level = 10, 20, 30, 40 for F = {0,1}, base = 3, and +0.812, -0.495, -0.127, -0.192 for one excluded digit at base = 10, swinging across [-1, 1] with no decay, Abs_level/fill^level at 2.1 * 10^-4 and 3.9 * 10^-12 at level = 40; counts exact by the carry DP pinned against brute force and the residue DP at every reachable e <= 30000, mu from a complete certified factorisation with zero unknown cofactors. Witness: mobius.md digit strings across divisors; lab/rs/rho-decoupling, the carry sweep and its five pinned tests.
2026-09-06 Refuted The adversarial pass on the divisor census: the geometric bound was attacked as an exact integer inequality at every census cell where its hypotheses hold, five bases and depths to level = 96, with zero failures and the closest cell at observed-to-bound ratio 0.187; the hypothesis edges were attacked one at a time, d = 2 with k = 2 holding at bound factor (1 - 1/2)^level, the digit-gcd hypothesis breaking exactly where the proof says it must (base = 3, F = {0,2}, d = 2, normalized error 1 at every level, sweep worst 1.0483 at level = 32), and the base-coprimality hypothesis handled by the exact split rather than dropped; the search for decay at fixed digit count failed and is recorded as the slow column rather than smoothed away. The printed floats truncate at forty decimal digits, so every claim-bearing comparison runs in exact integers or fractions and no rate is quoted past what the exact columns carry. Witness: lab/rs/rho-decoupling.
2026-09-14 Proved The pinned-orbit law, two-sided at one excluded digit. For t >= 1, every d dividing base^t - 1 with d >= 2 and every a nonzero mod d, the full Dirichlet kernel's orbit product telescopes to Prod_(j < t) abs(D_base(a base^j/d)) = 1 exactly, since a base^t = a mod d and no factor degenerates, so a closed shift orbit is invisible to the full digit set and every damping comes from the excluded digits. With abs(g_F) <= abs(D_base) + m and abs(D_base(a base^j/d)) <= B = min(base, d/2), convexity of log(e^y + m) puts the maximum at a vertex of {Sum_j log D_j = 0, log D_j <= log B} and gives Prod_(j < t) abs(g_F(a base^j/d)) <= (B + m)^(t-1) (m + B^(1-t)). At one excluded digit that is sharp both ways: for d = base^t - 1, t >= 2, base >= 10 and any single excluded digit, fill^(-1/t) (1 - 9/base) <= max_(a not 0 mod d) (Prod_(j < t) abs(g_F(a base^j/d))/fill^t)^(1/t) <= fill^(-1/t) (1 + 3/(base-1)) uniformly in t, the lower bound witnessed by a = 1. So t - 1 undamped positions and one damped by ~ 1/fill is the truth and the 1 + o(1) is a two-sided O(1/base) that does not grow with t; the constants 9/base and 3/(base-1) are stated at base >= 10. This supersedes the orbit-mean Conjecture row in OPEN, whose base = 100, level = 12 probes 0.1059 and 0.2369 read the finite-depth error rate (abs(N_F(level; d) - fill^level/d)/fill^level)^(1/level) d^(1/level) and not the orbit maximum. Witness: mobius.md digit strings across divisors, lab/rs/rho-decoupling.
2026-09-14 Proved The a-average at a full pinned modulus is exact at every digit set and every number of excluded digits: for d = base^t - 1, Sum_(a mod d) Prod_(j < t) abs(g_F(a base^j/d))^2 = d (fill^t + 2w) with w = 1 when both 0 and base - 1 lie in F and w = 0 otherwise, since val is injective on length-t strings with range [0, d], so the congruent pairs are the diagonal plus the single wraparound pair of the all-0 and all-(base-1) strings when both lie over F. Under the pinned-orbit law's hypotheses, one excluded digit and base >= 10, the worst orbit exceeds the average over all a, which is fill^t + 2w, by fill^(t-2) e^(O(t/base)); most of that average is its own a = 0 term fill^(2t)/d, 970299/101 of 9803 at base = 100, t = 2, so the average a second moment sees, over a nonzero, is (d (fill^t + 2w) - fill^(2t))/(d - 1), smaller again by ~ t/base and 980298/4999 at the same cell, and the spread is wider than the exponent states rather than narrower. From t = 3 on at most a fill^(2-t) fraction of residues sits near the worst orbit: the bad mass at a pinned divisor is spread, which is what the supremum over r gives up and an average over a buys. Witness: mobius.md digit strings across divisors.
2026-09-14 Proved The bisection bound and a level of distribution x^(alpha/2) for digit strings, up to one factor. For every F with fill >= 1, every d >= 2 coprime to base, every level >= 1 and uniformly in r, abs(N_F(level; d, r) - fill^level/d) <= fill^(level/2) (1 + 2 base^((level+1)/2)/d): cut the string in the middle and bound each half's variance by fill^b (1 + 2 base^b/d), an off-diagonal congruent pair of length-b strings needing val(f) - val(f') = j d with 0 < abs(j) <= (base^b - 1)/d and each pair (f', j) fixing at most one f. Summed against Sum_(d <= D) 1/d <= 1 + log D this gives Sum_(2 <= d <= D, (d,base) = 1) max_r abs(N_F(level; d, r) - fill^level/d) <= fill^level (D fill^(-level/2) + 2 sqrt(base) (1 + log D) (base/fill)^(level/2)) at every base >= 3, level >= 1 and D >= 2, every d and not only the squarefree ones, supremum over the target residue and not only the residue 0. At D = x^theta with theta <= alpha/2 and m = base - fill excluded digits the whole sum is at most 3 sqrt(base) (1 + log x) fill^level x^(m/(2 fill log base)): a level x^(alpha/2 - o(1)) at every theta up to alpha/2 at once, with a defect sub-power in base and a positive power in x, the exponent 1/(2(base-1) log base) at one excluded digit sitting under 0.0011 at base = 100. At d >= sqrt(base x) the same bound reads max_r abs(N_F(level; d, r) - fill^level/d) <= 3 fill^level x^(-alpha/2), asking nothing of F at all. The defect is the pair count's own overshoot 2 (base/fill)^b over its mean fill^(2b)/d at the balanced depth b = level/2, and no cut point removes it. Witness: mobius.md digit strings across divisors.
2026-09-14 Proved The assembled level-of-distribution theorem for digit strings, and the one family it leaves. Fix eps in (0,1) and an integer T_0 >= 2 and put base_0(eps, T_0) = max(4^(1/eps), base_1) with base_1 any base satisfying 3 base_1^(-eps)/log base_1 <= eps/(16 T_0). For every base >= base_0, every F = {0..base-1} minus E with 1 <= m <= base^(1-eps)/2, every level >= max(6 T_0, 4/eps) and x = base^level: at every level D <= x the sum of max_r abs(N_F(level; d, r) - fill^level/d) over 2 <= d <= D coprime to base with d <= base^(1-eps) or ord_d(base) <= T_0 is at most (T_0 + 2)(1 + log x) fill^level x^(-eps/(8 T_0)); at level x^(alpha/2) the full sum is at most 3 sqrt(base) (1 + log x) fill^level x^(m/(2 fill log base)); every d coprime to base with sqrt(base x) <= d <= x has max_r abs(N_F(level; d, r) - fill^level/d) <= 3 fill^level x^(-alpha/2); and at one excluded digit the full pinned moduli are damped together by level fill^(2 - sqrt(2 level)) e^(4 level/base) fill^level, superpolynomial in level, worst at t ~ sqrt(2 level), and never a fixed power of x. No clause asks d squarefree and every clause is a supremum over the target residue. One family survives at a fixed level, the generic-order middle moduli base^(1-eps) < d <= x^theta coprime to base with ord_d(base) > T_0, where only the level-alpha/2 clause applies and its single factor x^(m/(2 fill log base)) is the whole distance to a fixed power; that factor is the wraparound overshoot shared by the pair-count certificate, the orbit moment and the additive large sieve over the Farey points, and no rearrangement of cuts, Cauchy-Schwarz or divisor bookkeeping tried here removes it. Witness: mobius.md digit strings across divisors.
2026-09-19 Proved The uniform geometric bound summed over a divisor range is microscopic: D exp(-8 level/(fill^2 D^2)) < 1 fails past D ~ sqrt(level)/fill, so it certifies a level of that size and nothing like a power of x. Witness: exact arithmetic on the geometric bound of mobius.md DIGIT STRINGS ACROSS DIVISORS.
2026-09-19 Proved No moment past the second helps at a pinned divisor: at d = base^t - 1 the 2r-th orbit moment is d times an additive energy of length-t strings, and the pair-count certificate places it a factor 4 (base/fill)^(r t) above its own mean fill^(2 r t)/d, a loss growing in r. Witness: the orthogonality and bisection bullets of mobius.md DIGIT STRINGS ACROSS DIVISORS, lab/rs/rho-decoupling.
2026-09-19 Proved The signed Type I weight is itself a Mertens-type sum: with T_level(d) = N_F(level; d) - fill^level/d and P_level(e) the primitive frequency sum, sum_{d <= U} mu(d) T_level(d) = sum_{e >= 2} (mu(e)/e) M_e(U/e) P_level(e) with M_e(y) = sum_{f <= y, (f,e) = 1} mu(f)/f, so mu(e) fixes only the sign and the signed route restates the wall one layer down. Witness: Mobius inversion over reduced denominators, carried out in the sentence that prints it.
2026-09-19 Refuted Any bound on M_e(y) uniform in e and tending to zero in y: at the primorial e of all primes up to P and y = P the only f <= y coprime to e is f = 1, so M_e(P) = 1 exactly. Witness: that witness, exact.
2026-09-19 Proved No clause in the supremum norm is a fixed power of x, and clause (iv) is not slack: at one excluded digit and base >= 10, d = base^t - 1 with t = ceil(sqrt(level)) carries max_r abs(N_F(level; d, r) - fill^level/d) >= fill^level x^(-O(1/sqrt(level))) by the two-sided orbit law and the second moment across residues, so the sum over 2 <= d <= x^theta is at least that at every fixed theta > 0 and every level >= max(9, 4/theta^2). Witness: the two-sided pinned orbit law of mobius.md DIGIT STRINGS ACROSS DIVISORS, lab/rs/rho-decoupling.
2026-09-19 Proved The wraparound defect is circular and not loose: with the pair-count certificate alone both halves of a cut need base^b <= d, so b <= 2 log_base d and the certificate reads 3 d^(1 - alpha) > 1, while winning asks the depth b ~ (2/alpha) log_base d at which fill^b ~ d^2 strings meet d classes and equidistribution there is the statement being proved. Witness: the bisection certificate of mobius.md DIGIT STRINGS ACROSS DIVISORS.

Dimension one

2026-08-28 Proved The burst certificate moves the pincer's top edge to 0.6402122 unconditionally: when 3^k divides a ray coordinate (div) or the coordinate sum (opp), the branching type of a carry state is predetermined k steps ahead and the two digits of a 2-branching state force distinct types exactly k steps downstream, so admissible paths inject into the subsets of {1..w} avoiding distance exactly k, a Fibonacci product bounded by phi^(w+k) and attained on the shifts (9,1) and (27,1), hence rho <= phi for every primitive ray, one certificate for the infinite family with no computation; the opp designation rule at states whose predetermined window hits the dead type is fixed by a digit-order tie rule never consulted in the argument, the div ray count is per orientation with constant 61.6 rather than 600, and the edge rounding 0.640212 sits 1.9e-7 on the unsafe side. Witness: lemma-b-pincer.
2026-08-28 Proved Theorem R: the second moment Z(n) = sum_y M_n(y)^2 over primitive rays satisfying Z <= C 3^(gamma n) closes every beta > gamma/2, with 1/2 the method's own wall since the diagonal alone forces Z >= 3^n; Z(n) decomposes exactly as diagonal plus multiplier triples (s, t, z) with Q_n(1, 3^j) in closed form, Z/3^n peaks at 2.676455 at n = 10 and falls monotonically to 2.226210 at n = 18 (limit 2 conjectured, which would shrink the window to (0.4475978, 1/2]); directions are reduced from the origin per point, never as pairwise displacements, the six-digit x_j + y_j <= 2 construction is a different 6^n gasket and not G_n, and the 1D coprime pairs of the {0,1} base-3 Cantor set (Z/4^n -> 0.513358) are a different object. Witness: lemma-b-pincer.
2026-08-28 Proved Two named ways past 0.6402122 are shut: keeping the exact Fibonacci burst product D_k(w) = prod_r F_(m_r + 2) cannot lower the octave exponent, since at k = 1 already D_1(w) = F_(w+2) = Theta(phi^w) while the depth-one octave-j census is at most a constant times 3^(2j-1), reproducing psi_phi(c) = 2c + (1 - c) log_3 phi and beta = 1/(2 - log_3 phi); and averaging spectral radii cannot replace the supremum, since ray mass is governed by rho^w and Jensen gives int rho^w dnu >= (int rho dnu)^w; the height-40 catalogue (486 primitive non-shift rays plus the shifts (1,3), (1,9), (1,27)) has mean rho = 1.0228285, median 1, standard deviation 0.0907378, inverse-square-weighted mean 1.0639086 (a 34.25% deficit below phi, 1.0481989 if (1,1) were wrongly counted as a shift) and certified weighted upper mean 1.0997454, and substituting them into (2 - log_3 lambda)^(-1) yields 0.5145062, which is not a coprimality bound and must never be quoted as one; the inverse-square weight is a probability measure only after a height cutoff since sum 1/(a^2 + b^2) diverges logarithmically, and the shifts are removable because their count is O(1) per octave. Witness: lemma-b-pincer.
2026-08-28 Verified Paley-Zygmund, Bonferroni and Cauchy-Schwarz on the first two Fourier moments are structurally unavailable for the dimension-one lower edge: Paley-Zygmund lower-bounds the heavy rays while the proof needs an upper bound on the total bad mass sum_y M_n(y); Bonferroni needs the signed intersection counts T*_pq, T*_pqr, ... with no uniform estimate over the exponentially growing modulus range; and (sum_t F_a(t))^2 <= p^2 sum_t F_a(t)^2 is an upper bound on the first absolute moment, the direction the ladder already uses, so M_1^2/M_2 cannot improve 0.4479; the cap 0.447930987882 is purely the absolute-Fourier-moment wall from the low-frequency peak E_2K >= 3^((2K-2)a)/K^2, forcing kappa_2K < 2 at every finite K, not a Cauchy-Schwarz artifact and not Mobius truncation (which needs the separate tail control A_z(n) - A(n) <= G(n)/log z, divergent at dimension one); no universal cap holds for "any moment-based method", since a complete moment sequence determines the distribution. Witness: lemma-b-pincer.
2026-08-28 Verified Higher Fourier moments cost polynomial time in the moment order: once the carry transfer matrix is built, E_2K(G_a) = (M_K^a)_{0,0}, polynomial in the level a by matrix powering; the bounded carry radius is about K/2, giving S_K = (2 c_K + 1)^2 = O(K^2) states, and the naive construction is about O(K^6) operations before bit complexity, with 9, 25, 25, 49, 49, 81 states at orders 6, 8, 10, 12, 14, 16; exact characteristic-polynomial algebra still grows with integer size, and a numerical Perron root is not a master inequality. Witness: lab/rs/dimension-one-ladder.
2026-08-28 Verified The multiplier pairs have a spectral gap at 2: lambda(s,t) = 3 only on the shift pairs (1, 3^j) and their reverses, every other coprime pair obeys P_w <= (3/2)^K 2^w at every state with K = v_3(st) + v_3(t' - s'), and the interval (2, 3) is empty over the certified domain max(s,t) <= 52 only, universality being the lane's open con:gap and no theorem; aligned 3-way splits land their penalty exactly k steps later, giving G_m = G_(m-1) + 2 G_(m-2) with Perron root exactly 2 at (1,4) (P_w = (2^(w+2) + (-1)^(w+1))/3, characteristic polynomial lambda (lambda - 2)(lambda + 1)); over all 829 coprime unordered pairs with max(s,t) <= 52, 20 non-shift pairs attain 2 and the largest non-shift radius below 2 is 1.6956207695598 for the gasket-digit automaton A but 1.8488475886485 for the free-digit B the census actually needs, on (4,13), (4,39), (12,13), (13,36); the 9-divisible classes (9,2), (2,9), (18,1), (1,18) have lambda = 1; the closest ratios to the bound are 0.8888893 at (3,4), (3,7), (1,12) and 0.8888887 at (1,4), (1,7); the gap means no radius strictly between 2 and 3, not a gap below 2, and alone gives only E <= C 9^n. Witness: gasket-ray-machine.
2026-08-28 Verified The heavy gasket rays carry named run-length counts: M_n(3,1) = F(n+1) - 1, the Fibonacci product prod_r F(m_r + 2) - 1 at (3^j, 1), Narayana's cows A000930(n) - 1 at the supergolden ray (1, 12), and c(n-3) - 1 with c(m) = c(m-1) + c(m-4) at (7, 3), exact to n = 140 (M_140(3,1) = 131151201344081895336534324865, M_140(1,12) = 106502839316458556100416, M_140(7,3) = 21561294536157802712); M_14(7,3) = 49 = 7^2 is a coincidence, x^4 = x^3 + 1 being irreducible and the quartic sequence square only at n = 4, 7, 9, 12, 14 (1, 4, 9, 25, 49) through n = 140; max M_13 = 376 = F(14) - 1. Witness: gasket-ray-machine, A000930.
2026-08-28 Proved Codes 98, 140, 266 and the fourth permutation design {(0,2),(1,1),(2,0)} (code 84 under the 3a + b indexing) are diagonal: Z_F(n) = 3^n - 2 for every n >= 1 (the identity fails at n = 0, where the sides are 0 and -1), no two distinct points ever collinear with the origin, by a 3-adic cross lemma: weights all in one unit residue class mod 3 against weights injective mod 3 pin the cross determinant's valuation to the first differing digit position; the three named codes are the permutation graphs j -> j+1, j -> j+2 and the swap of 0 and 1, so the lemma has instances and not separate proofs; for these designs ray mass is ray occupancy and the window problem is pure divisor rarity. Witness: gasket-ray-machine.
2026-08-28 Conjecture Occupancy is the one door left in the 1/2 wall: the Cauchy-Schwarz bound saturates at beta = 1/2 against the trivial ray count, but occupied rays number only 3^(0.5416 n) to 3^(0.5798 n) at the critical band against the trivial 3^n under the pinned threshold reading, n = 10..18 (the earlier band 0.543 to 0.557 does not reproduce), every occupied ray has exactly one coordinate divisible by 3, and Theorem R+ closes the entire window under Conjectures Z and O while every bootstrap from Z to O collapses to the trivial fixed point; minimal witnesses are not unique (four tied rays at n = 12, repaired by a least-multiplier tie-break) and prefix-newness is necessary but not sufficient, overcounting occupied rays by a stable 1.51x. Witness: gasket-ray-machine, lab/rs/dimension-one-ladder.
2026-08-28 Conjecture Higher ray-mass moments make the Holder conversion strictly worse, capping the ray power-moment route at the second-moment edge 1/2: at n = 12 the 345318 occupied rays have S_1 = 523250, S_2 = 1374038, S_3 = 46380938, S_4 = 8145428822, max M = 232, and the Holder bound S_1 <= N^(1 - 1/r) S_r^(1/r) overshoots by factors 1.316, 3.380, 8.179 at r = 2, 3, 4, because Fibonacci-heavy shift rays dominate the high moments (phi > 3^(1/(2K)) at the critical half-scale); the ladder n = 8..12 lists occupied rays 3904, 12170, 37298, 113836, 345318 with max M 33, 54, 88, 143, 232.
2026-08-31 Proved The pincer at dimension one: Lemma G, the gasket case of Lemma B, hence all 36 lines by the reduction above, holds at level n for every prime exponent beta = log_3(p)/n below 0.4475978 and above 0.6402122: below by exact gasket moment identities (carry-free additive energy exactly 15^a, 6th, 8th and 10th moment growths the exact algebraic numbers 57 + 6 sqrt(46), 456 + 3 sqrt(11017) and the largest root of x^4 - 7833x^3 + 7916949x^2 - 850684437x + 13054946580 from nine- and twenty-five-state transfer matrices) fed through a Holder ladder that never uses ord_p(3), above by the ray decomposition (fibres 2^(n+1) - 2, shift rays at most 2n phi^n, every carry state of every primitive ray at most 2 admissible digits) with the regime bookkeeping on the trichotomy of n against 3a and 4a, a = floor(log_3(p/2)), which closes the belt of primes near p ~ 3^(n/3); the ladder saturates at 2/(3 + log_3 5) = 0.447931 < 1/2 and per-ray-maximum methods stop at 1/2; the master bounds hold against exact L_n(p) for every prime 5 <= p <= 199; the eighth rung 0.446717 is the row the shelf lane still imports and is now one row stale. Witness: lemma-b-pincer, lab/rs/dimension-one-ladder.
2026-08-31 Proved The tenth rung of the moment ladder moves the bottom edge to 0.4475978 (beta_0^(10) = 0.4475978134...), leaving the standing window (0.4475978, 0.6402122] with both edges unconditional; the eighth rung at 0.446717 is now only a table row. Witness: lab/rs/dimension-one-ladder.
2026-08-31 Proved Conjecture O has no content below alpha = 1/2 - the rays of height at most 3^(alpha n), occupied or not, number at most 3^(2 alpha n) under the threshold reading and 9 * 3^(2 alpha n) under the octave cut, so the box alone gives delta = 1 - 2 alpha with no occupancy input, and the whole conjecture lives in alpha in [1/2, 0.5533]. Witness: lab/py/occupancy-decay.
2026-08-31 Proved The first moment of occupancy is the window itself, so no proof of O may pass through it - with F(n, X) the non-fibre gasket points of primitive height at most X, Sum_{p > 3^(beta n)} N_n(p) <= (F(n, 3^((1-beta) n)) + 2^(n+1)) / beta at target zero, each such x carrying at most 1/beta primes above 3^(beta n); a first-moment bound at alpha > 0.3597878 moves the standing window and at alpha >= 0.5524022 closes it with no Conjecture Z, and the inequality holds with ratio 0.0846 to 0.1517 against the sieved prime sum at n = 10, 12, 14, beta = 0.45, 0.5, 0.6. Witness: lab/py/occupancy-decay.
2026-08-31 Verified Occupancy pays no exponent for the multiplicity, so O carries the full weight of the window and is no cheap half of Theorem R+ - F/A at alpha = 0.5533 reads 5.41, 5.20, 5.52, 5.64, 5.63, 5.86, 5.79, 6.08, 5.92 at n = 10..18 while log_3 F / n falls 0.7645 to 0.7201 against log_3 A / n inside [0.6109, 0.6345], the exponents converging at log(F/A)/(n log 3); only at fixed height do the shift rays split them, A(n, 3^5) = 384 .. 474 against F(n, 3^5) = 2728 .. 51694. Witness: lab/py/occupancy-decay.
2026-08-31 Verified The digit-congruence seed is measured out as a route to O - the proved bound A(n, X) <= 2 sigma_k X^2 + 2 sigma_k 3^k X + 2 X^2 3^(-k) + 3^k + 2 X for 3^k <= X collects every digit-class constraint, the mod-3 dichotomy being k = 1, but sigma_k = |R_k|/3^k falls only polynomially through k = 18, |R_k| = 73440, 206149, 580920, 1643545, 4663382, 13272515 at k = 13..18 with growth rising 2.794 to 2.8461 and k(1 - log_3 growth) inside [0.8418, 0.8628], so the route buys n^(-0.86) and no exponent; on the measured hypothesis M_2(k) = O(4^k) (M_2/4^k = 0.4098, 0.4077, 0.4071, 0.4029 at k = 13..16, still falling) Cauchy-Schwarz caps any congruence-only decay at c = 0.2618596, alpha = 0.575328, excluding neither 0.5533 nor 0.5524022, and no exponential floor is proved either way. Witness: lab/py/occupancy-decay.
2026-08-31 Verified Occupied non-fibre ray totals 1044840, 3151656, 9491964, 28545340 at n = 13, 14, 15, 16 from a second builder, two below the earlier totals at every level, exactly the two fibre rays. Witness: lab/py/occupancy-decay, lab/rs/dimension-one-ladder.
2026-08-31 Proved The golden ceiling M_n(z) <= F(n+1) - 1 holds for every direction of the 13158-box at every level, promoted from an enumeration to n <= 40 - in the direction coordinate a multiplier word is a word over the increments {0, z_2, -z_1} summing to zero, its carry automaton has out-degree at most 2 with the branch states in one residue class mod 3, and the two successors of a branch state differ by q/3 for the unique q in {z_1, z_2, z_1+z_2} divisible by 3, so when no branch state has two branching successors (in particular when v_3(q) = 1) the state maximum obeys G(n) <= G(n-1) + G(n-2) and the ceiling follows; that settles 206 of the 218 occupied directions, 107 by v_3(q) = 1, three of the twelve left are shift rays closed by F(p+2) F(q+2) = F(p+q+3) - F(p+1) F(q+1), and nine carry rational Fibonacci certificates of denominators 18, 40, 381, 18, 2013, 18, 40, 2013, 34. The hypothesis z_1, z_2 >= 1 is load-bearing: on the fibre ray (0,1) two digits share the increment 0, a set-valued reading sees no branch state, and M_6(0,1) = 63 against F(7) - 1 = 12. Refutation attempt: ground truth rebuilt independently from the ray definition for 20 directions including all twelve hard ones, zero mismatch; all nine certificates re-verified in exact rational arithmetic with domination checked to n = 60; the box census, the renewal criterion, the (1,9) profile, the weight bound and the -1 path accounting all recomputed exact. Witness: gasket-ray-machine, lab/py/gasket-witness-weights.
2026-08-31 Proved The whole case list collapses into one algebraic inequality per direction. Weight the first returns of the direction automaton by phi^-1 a step: with g(c,m) the paths from a live state c to the start meeting it only at the end, u(c) = Sum_m g(c,m) phi^-m and U(z) = Sum u(c') over the start's successors other than itself, so Sum_{j>=2} f_j phi^-j = phi^-1 U; any pi > 0 with Sum_succ pi <= phi pi(c) at every live c != 0 and Sum_(c' != 0 succ 0) pi(c') <= phi^-2 pi(0) forces U(z) <= phi^-2 by a maximum principle on the truncated sums, and then M_n(z) <= F(n+1) - 1 at every n by renewal against the envelope phi^(m-2) <= F(m) <= phi^(m-1), with pi = u admissible as soon as U(z) <= phi^-2. It proves 45 directions no earlier case reached, the nine hand-tuned rational certificates and the 36 that rested on enumeration alone; 3 nmid z_1 z_2 gives M_n = 0 outright, settling 6566 box directions on residues against 3284 before; f_1 = 1 always and f_2 = 1 only at {a,b} = {1,3}, both from the increments. Refutation attempt, briefed to break it: the proof read line by line for convergence, normalisation, S = phi^-1 U and both envelopes; an independent Q(sqrt5) rebuild reproduced every count (218 occupied and 214 passing in the box with the four shift-ray failures, 647 and 644 outside it with three, 865 and 858 in total with seven, 57 distinct U values, the exact attainers); a 600 x 1800 box census with 3866 occupied directions, 4.5 times the shipped range, plus 19681 stressors including 52 in the open v_3(q) >= 2 ground, found failures only at shift rays; and every shipped solve is confirmed strictly positive and against both criterion inequalities, not merely against the linear system. Witness: gasket-ray-machine, lab/py/gasket-witness-weights.
2026-08-31 Verified The golden potential misses exactly the shift rays and, on every censused range, nothing else: U(1,3^j) = phi^-1 exactly because the shift mass grows at rate phi, U = phi^-2 only on the supergolden (1,12), (3,10), (4,9), and no direction of any range censused has U in the open interval (phi^-2, phi^-1) - the box, the six adversarial families, a 36037-direction lab sweep with high-v_3 stressors, and the independent 600 x 1800 recompute. The gap is empirical only: a legal-looking first-return profile f_3 = f_5 = 1 gives S = 0.3262 inside it, so nothing arithmetic excludes the interval and the observation is never a theorem. The open conjecture is U(a,b) <= phi^-2 for every non-shift primitive direction, which with the theorem and the Fibonacci product identity is the whole golden ceiling. Witness: gasket-ray-machine, lab/py/gasket-witness-weights.
2026-08-31 Proved M_n(z) <= D_n(z_1 + z_2), Conjecture W's owed first move, in one line - disjoint binary supports make m(z_1+z_2) binary below 3^n and m -> m(z_1+z_2) injective - and it is the wrong half: D_n(w) grows at rate 2, not phi, reading 4196351, 1683971, 613817, 228519 at n = 24, w = 4, 10, 28, 82 against the ceiling F(25) - 1 = 75024, so the weight enters only through the constant. Witness: gasket-ray-machine, lab/py/gasket-witness-weights.
2026-08-31 Proved gasket-ray-machine stated M_n(a,b) = (T^n)_{00} where its own proof gives M_n(a,b) + 1 closed paths; corrected to (T^n)_{00} - 1, and the carry bound |c| <= max(a,b) sharpened to c in [-a/2, b/2], which ties the live state count to the witness weight at floor(a/2) + floor(b/2) + 1. Witness: gasket-ray-machine.
2026-08-31 Refuted The occupancy band 3^(0.543 n) to 3^(0.557 n) at c = 1/2 - it reproduces under no cut convention at n = 12..15, the threshold reading giving [0.5416, 0.5798] over n = 10..18 and the integer octave cut giving the paired readings 0.5249 / 0.6052 at n = 13; the band was stale, not a convention difference, and the adversarial pass that killed it also killed a pruning bug in the new census, A(9, 3^7) = 1176 printed where the truth is 2818, the tracked-direction cut sitting below the requested threshold, now pinned as a regression. Witness: lab/py/occupancy-decay, lab/rs/dimension-one-ladder.
2026-08-31 Refuted The state maximum G(n) = max_c N(c,n) does not obey G(n) <= G(n-1) + G(n-2), so the branch argument does not extend to v_3(q) >= 2 - at (1,9) the profile runs 1, 1, 1, 2, 4, 6, 9 and G(4) = 4 > G(3) + G(2) = 3, and 8 directions of the box break it, every one with v_3(q) >= 2; the sharp reformulation is the renewal criterion Sum_{j>=2} f_j F(n+1-j) <= F(n-1) on first-return counts, with f_1 = 1 always and f_2 = 1 only at (1,3), holding on all 218 occupied directions to n = 46, both f facts now proved from the increments and the whole criterion subsumed by the golden potential through Sum_{j>=2} f_j phi^-j = phi^-1 U. Witness: lab/py/gasket-witness-weights.
2026-08-31 Refuted The ceiling's adversarial family census double-counted: the six families overlap, the no-adjacent-ones family sitting inside the binary one, so the shelf's 11369 coprime members are 10862 distinct directions, 717 already in the box and 10145 genuinely further, of which 9498 carry no mass, 608 fall to the branch argument and 3 are shift rays, leaving 36 on the enumeration alone and not the 70 first claimed; the same overlap inflated the lab's widened sweep from 23435 distinct to 24088 with multiplicity. The 36 hold to n = 60, worst ratio below 0.1516, and are now proved outright by the golden potential, so no direction of the six families rests on enumeration alone. Witness: gasket-ray-machine, lab/py/gasket-witness-weights.

Divisor avatars

2026-08-28 Proved The Avatar Theorem: for x = prod p_i^(a_i) with p_1 < ... < p_dim and every a_i >= 1, the dim-axis design with f_w = e_w(a_1 - 1, ..., a_dim - 1) has P(n) = d(x^n) at every n >= 0, by the substitution a_i n + 1 = b_i n + (n + 1) and prod (b_i t + 1) = sum_w e_w(b) t^w at t = n/(n+1); the map is injective by Newton's identities and surjective onto the f_0 = 1 signatures whose fill splits completely into linear factors over Q, and a design exists iff sum_i (a_i - 1) <= dim; the first ten colossally abundant numbers 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720 have avatars while the first without one is 21621600; on the seven dim = 3 ladders x = 30, 60, 120, 180, 240, 360, 900 to n = 20, all 131 of the 140 powers exceeding 5040 satisfy Robin's inequality, the largest ratio R(14400) = 1.5732599059 against e^gamma = 1.7810724180, margin 0.2078125121, strictly decreasing in n on every ladder, which proves nothing about Robin beyond them; sigma(m)/m < sigma(N)/N for every m < N defines a superabundant number, not a highly abundant one. Witness: divisor-avatars.
2026-08-28 Proved The sigma-hunt is closed negatively for polynomial census laws: no census law of a design at odd side 2n + 1 equals sigma_k(x^n) for k >= 1, x > 1, since that grows at least like x^(kn) while every such law is a polynomial in n of degree at most dim, and none equals sigma(x^n)/x^n, strictly increasing and bounded hence not constant; at x = 1 both collapse to the nine constant identities O(n) = 1 = d(1^n); over the nine observables (fill, voids, surface, touched vertices, edges, faces, Euler characteristic, components, cycle rank) on all 22 least-mask representatives of the 256 base-2 3D designs counted through side 21, exactly eight non-fill strict divisor avatars survive, only for voids, Euler characteristic and components; E and R vanish at n = 0 while d(x^0) = 1, so they are never strict identities, and the four graph laws for codes 30 and 126 hold only for n >= 1; if sigma is a design observable at all it lives among geometrically growing counts. Witness: divisor-avatars.
2026-08-28 Verified The eight non-fill divisor avatars come from topology, not measure: the empty design's voids (2n+1)^3 = d(900^n); Euler characteristic and components both (n+1)^3 = d(30^n) for {000}; both (n+1)^2 = d(6^n) for {000, 100}; components n + 1 = d(2^n) for the three-corner path {000, 100, 010}; both n + 1 = d(2^n) for the square face {000, 100, 010, 110}; plus nine constant identities O(n) = 1 = d(1^n), components for codes 23, 27, 31, 61, 63, 111, 127, 255 and Euler for 255; the statement is for the canonical least-mask representatives, since cube symmetry does not preserve the odd/even origin under coordinate reversal; codes 15 and 27 share a weight signature and a fill law, so eight orbits carry seven weight signatures. Witness: divisor-avatars.

Exposure

2026-08-30 Proved The exposed faces of a design's level power obey V(level + 1) = occ V(level) - 2 sum over the axes of P S^level, with occ the tile's filled cells, P its adjacent filled pairs along the axis and S the cross positions whose two end cells are both filled, because two adjacent blocks bury one face per spanning position and the spanning positions multiply by S a level; so V(level) is a sum of the powers occ^level and S^level in every dimension, the carpet perimeter closes as (4*8^level + 16*3^level)/5 (A381517) and the sponge surface as 2*20^level + 4*8^level (A332705), and all four counts fold from the residue corners without rendering the tile. Witness: mrlymath::formulas::surface prediction_matches_census_on_every_cube_code and the_corners_fold_what_the_tile_shows, sequences.

Farey stack

2026-08-28 Verified The odd-carpet stack renders an RH-equivalent object and no route to a proof: the lit nodes are exactly the Farey fractions, each scale n contributing phi(n) new nodes, their discrepancy is the object of the Franel and Landau 1924 theorems, and the measured S2 * Q flattens near 0.656 (0.6560 at 2000, 0.6564 at 8000) with the local exponent walking to -1. Witness: mrlynum::lattice::new_nodes, lab/rs/farey-discrepancy.
2026-08-28 Proved The stack is an address, not a construction: the odd-carpet stack's brightness at x = (a_1/q, a_2/q) is the residue count B_N(x) = ceil(N/2) - sum_{r in S(x), r <= N} (floor((N - r)/(2q)) + 1) over the bad residues mod 2q, per-point cost independent of N, and the line stack's is floor(N/b); a stack of 5 * 10^17 layers, N = 10^18, evaluates exactly in a tenth of a second, and at N = 55 the Farey table holds 940 nodes summing to 1540 = N(N+1)/2; the closed form's proved boundaries are per-point only (an R x R raster costs R^2), exact representations only (a real-oracle input is undecidable on {n x integer}, irrationals with known continued fractions stay computable via Ostrowski), finite N only (infinite-depth membership is undecidable) and unweighted only. Witness: lab/py/carpet-stack-address, mrlynum::lattice::farey.
2026-08-28 Verified Immediacy buys no RH content: the Mobius-weighted node is the Mertens-type sum Sum_{k <= N/b} mu(kb), equal to M(N) at b = 1 and to M(floor(N/b)) at only 64 of 200 denominators at N = 200, with no polynomial-time algorithm for the Mertens function at binary input and the best known near x^(2/3); the rank closed form sum_d mu(d) sum_e floor(x e) re-imports Mobius, the meter's global readout collapses to sum_{n <= N} M(floor(N/n)) = 1 identically (checked exactly to N = 20000), the divisibility incidence array is the Redheffer matrix up to its first column, and Franel 1924 is already the symbolic all-Q reduction, so the route ends at Mertens. Witness: lab/py/mertens-meter.
2026-08-28 Verified The stack's complexity frontier is the sharing of its scales: per-pixel brightness with binary inputs is in P by fixed-dimension lattice-point counting (Barvinok 1994, two parity branches summed), destroying the shared scales makes "does any point reach maximum brightness" NP-complete (Simultaneous Incongruences, Garey and Johnson SP3), and making the ambient dimension part of the input makes "is any layer lit at this fixed point" NP-complete (Lagarias 1985) while polynomial at every fixed dimension, so a no-shortcut theorem for this stack could never separate P from NP; d(n) is not factoring-hard by the sigma route (sigma(pq) = pq + p + q + 1 recovers the factors while d(pq) = 4 carries nothing), and the O(q) residue sweep is polynomial in the denominator q, hence a unary-input algorithm. Witness: REFS.md.
2026-08-28 Conjecture The Baez-Duarte coefficients c_k = sum_n mu(n) n^{-2} (1 - n^{-2})^k = sum_{j=0}^k (-1)^j C(k,j)/zeta(2j+2) from an independent Mobius sieve to n = 10,000 read -0.316011506 at k = 1 to -1.68003e-5 at k = 1000 against a 450-digit reference -1.65958e-5, difference 2.04521e-7; the sieve agrees with the test vector [1,-1,-1,0,-1,1,-1,0,0,1] and with a second linear sieve at every integer through 10,000, counts 3053 minus-ones, 3917 zeros, 3030 plus-ones; consistent with the criterion and evidence for the Riemann hypothesis of exactly nothing.
2026-08-28 Conjecture S_N = sum_{k=1}^N (-1)^k C(N,k)/zeta(2k) is not a Riemann-hypothesis criterion tending to zero: it has the wrong zeta shift, omits the j = 0 term and tends to 2 - S_100 = 1.843329, S_500 = 1.967518, S_1000 = 1.983699; the sequential Baez-Duarte coefficient is c_k = sum_{j=0}^k (-1)^j C(k,j)/zeta(2j+2), the Nyman-Beurling distance d_N = inf ||1 - D_N||^2 is not the coefficient sequence and needs its own basis and Gram matrix, sum_{j | k} mu(j) is the Mobius inversion identity (1 at k = 1, 0 after), and direct binomial evaluation at 80 digits is nonsense by k = 500, so 450-digit arithmetic or the Mobius series is required.
2026-08-28 Conjecture Under the convention Q = 3^level the Landau discrepancy reads 0.166667, 0.549206, 1.150760, 2.118500, 3.187070 at Q = 3, 9, 27, 81, 243, computed in exact rationals by two routes that agree; D_Q/sqrt(Q) stays in [0.0962, 0.2354] and the last-three log-log slope is 0.464, consistent with O(Q^{1/2+eps}) and discriminating nothing, since five nested deterministic points cannot test a statement quantified over every positive epsilon.
2026-08-28 Refuted Stack brightness encodes the Mobius function, so a Baez-Duarte meter can replace the Franel table - neither brightness carries factorization data: the Farey stack gives B_Q(a/b) = floor(Q/b), 199 distinct values for the 10,000 denominators at Q = 10,000, every denominator from 5,001 to 10,000 sharing brightness 1 while mu runs -1, 0, +1; the gramstack gives 1 + K(a/b) with K = (-1)^a/b^2 for odd b and K = 0 for every even b, so all even denominators coincide; joining a node to a factorization through its denominator puts the arithmetic in the factorization. Witness: lab/py/carpet-stack-address.
2026-08-28 Refuted A design's Farey order is determined by its fill count - the stack of grid scales n = 1..Q lights exactly F_Q = {a/b : 1 <= a <= b <= Q, gcd(a,b) = 1} with brightness floor(Q/b), a boundary coordinate k/n reducing to a/b and recurring at every scale divisible by b, checked by literal stacking at Q = 30 on all 278 lit fractions; Farey order is Q, fill count plays no part, and every design gives the same sequence at fixed Q. Witness: lab/py/carpet-stack-address.
2026-09-11 Proved The Farey sequence restricted to a digit design counts without enumerating a fraction: with S_F the whole numbers whose every base digit lies in the digit set, card {a/b reduced : 0 < a <= b <= Q, b in S_F} = sum_{b in S_F, b <= Q} phi(b) and card {a/b reduced : 0 < a <= b <= Q, a and b in S_F} = sum_{b in S_F, b <= Q} sum over d dividing b of mu(d) #{multiples of d in S_F up to b}, the second by inclusion-exclusion on the divisors of b. Witness: lab/rs/farey-discrepancy.
2026-09-11 Verified Both restricted counts agree with a Stern-Brocot enumeration at every rung of both ladders, base 3 {0,1} to Q = 3^11 = 177147 and base 10 without the digit 9 to Q = 10^5, 45 checks and no failure, the largest being 9538759028 nodes on the full-set control. Witness: lab/rs/farey-discrepancy.
2026-09-11 Verified On the strict convention the discrepancy sums ride the node count instead of cancelling against it: the local exponents e_2 and e_1, single ratios between consecutive rungs, agree with the mass exponent to two decimals at both designs, +1.259 and +1.262 against +1.263 at base 3 {0,1} and +1.904 and +1.906 against +1.908 at base 10 without 9, while the same lane's S1/card holds two figures from Q = 2187 (card 4286 to 1080458) at base 3 and from Q = 10000 (card 11890654 to 963170938) at base 10, at 9.4e-2 and 5.2e-3, and S2/card likewise at 1.3e-2 and 3.6e-5, the base 10 rung below moving the first figure of S2/card from 4.1e-5. Witness: lab/rs/farey-discrepancy.
2026-09-11 Proved The strict digit-restricted Farey sequence at base 3 {0,1} misses the closed interval [1/2, 2/3] at every Q, so it does not equidistribute: a denominator with leading digit at position L satisfies 3^L <= b <= (3^(L+1) - 1)/2, a numerator with leading digit at the same position gives a/b >= 2*3^L/(3^(L+1) - 1) > 2/3, and one with leading digit at L - 1 or below gives a <= (3^L - 1)/2 < b/2, hence a/b < 1/2; the measured widest gap contains that interval at every finite Q and shrinks onto it from outside, 0.16827, 0.16720, 0.16684, 0.16673, 0.16669, 0.16667 at Q = 3^6 .. 3^11 from left endpoints 0.49931, 0.49977, 0.49992, 0.49997, 0.49999, 0.50000. Witness: lab/rs/farey-discrepancy.
2026-09-11 Conjecture The strict digit-restricted Farey sequence at base 10 without the digit 9 does not equidistribute either, on the settled constants alone and with no interval to argue from: its widest gap falls like 1/Q, 0.01136, 0.00113, 0.00011, 0.00001 at Q = 10^2 .. 10^5 against the control's 0.01000, 0.00100, 0.00010, 0.00001, so the base 3 emptiness argument does not transfer. Witness: lab/rs/farey-discrepancy.
2026-09-11 Conjecture Restricting only the denominator to a digit design keeps the square-root shape transplanted to that design: with D_Q = #{b in S_F, b <= Q} ~ Q^alpha and card ~ Q^(1+alpha), a node-count error of order sqrt(D_Q) puts e_2 at -1 and caps e_1 at alpha/2, and the measured e_2 reads -0.959 and -0.899 while S2*Q reads 0.8926 and 0.8536 against the control's 0.6782 and 0.6684 and S1/Q^(alpha/2) reads 0.243, 0.281, 0.267, 0.268, 0.274 at Q = 3^7 .. 3^11 and 0.213, 0.222, 0.207, 0.265 at Q = 10^2 .. 10^5, flat where the control's S1/sqrt(Q) falls, so the reading is S2 = O(Q^(-1+eps)) and S1 = O(Q^(alpha/2+eps)). Witness: lab/rs/farey-discrepancy.
2026-09-19 Proved The stack is the Farey resonance diagram up to the floor: normalising the brightness law gives the node a/b the height floor(Q/b)/Q, which lies in (1/b - 1/Q, 1/b] at every depth and equals 1/b exactly when b divides Q. Witness: lab/rs/farey-discrepancy.

Fill polynomials

2026-08-28 Conjecture The discriminant staircase is gapless and counts 2(dim-1) new discriminants per dimension: the negative fundamental discriminants (d = 0 or 1 mod 4) carried by the irreducible quadratic factors of fill polynomials at dimension dim form a gapless initial segment of length dim(dim-1), dim = 2 giving -3, -4, dim = 3 adding -7, -8, -11, -12 for 6, dim = 4 adding -15, -16, -19, -20, -23, -24 for 12, with 20 at dim = 5, exhaustive over 17424 signatures, gapless to -40; at dim = 6, exhaustive over 1053696 signatures, the peeled-remainder reading is gapless from -3 to -63 with length 31 and the reading over every irreducible quadratic factor gives length 80 to -160, deepest -899, so the dim(dim-1) count law predicting 30 is Refuted at dim = 6 under both readings while the run stays gapless; gapless and unbounded forces every imaginary quadratic order to appear at some finite dim, the geometric content of the imaginary completeness conjecture; exhaustive at dim = 2..6 by exact factorization. Witness: lab/py/fill-polynomials.
2026-08-28 Refuted Seven dim = 4 fill-polynomial remainders labelled irreducible over Q of degree 4 - palindromic signatures (1,0,1,0,1), (1,0,2,0,1), (1,1,2,1,1), (1,2,3,2,1), (1,3,2,3,1), (1,4,2,4,1), (1,4,5,4,1) - split over Q into two centered-polygonal quadratics k n^2 + k n + 1; rational-root peeling proves irreducibility only through degree 3, and no qualify/fail verdict moves since no factor is linear. Witness: lab/py/fill-polynomials.
2026-09-11 Proved The odd-side fill is a product of norm forms, one per irreducible factor of the weight enumerator W(t) = sum_j s_j t^j: with m = deg W and W = cont(W) prod_i g_i^(e_i) over Z, primitive irreducible g_i, the fill P(n) = (n+1)^dim W(n/(n+1)) is (n+1)^(dim-m) cont(W) prod_i g_i*(n)^(e_i) with g*(n) = (n+1)^(deg g) g(n/(n+1)) = lc(g) prod_theta ((1-theta) n - theta), the norm form of Q(theta); the constant is the content, never the leading coefficient, and is 1 on the origin-filled box. Exact on all 16, 256 and 65536 designs at dim = 2, 3, 4, in the factored and the resultant form. Witness: lab/py/field-ladder.
2026-09-11 Proved The norm form of a factor has the discriminant of the factor: the map t = n/(n+1) is the Mobius map of [[1,0],[1,1]] in SL_2(Z), so Q(theta/(1-theta)) = Q(theta) and disc(g*) = disc(g) for every factor with g(1) != 0, which is deg g* = deg g and is automatic for an irreducible factor of degree at least 2; checked over the 6, 32 and 350 origin-filled signatures at dim = 2, 3, 4, 0 mismatches on the 2, 25 and 343 factor slots of degree at least 2. Witness: lab/py/field-ladder.
2026-09-11 Proved The bare product form P = s_dim prod_theta ((1-theta) n - theta) needs the lift (n+1)^(dim - deg W) on exactly half the designs: deg W < dim iff s_dim = 0 iff the all-odd corner is empty, and the complement in that corner is a fixed-point-free involution, so the count is 2^(2^dim - 1), which is 8 of 16, 128 of 256 and 32768 of 65536 at dim = 2, 3, 4. Witness: lab/py/field-ladder.
2026-09-11 Proved With the origin filled every rational root of W is -1/k and every linear factor of the fill is (a n + 1): W has nonnegative coefficients and W(0) = 1 so no factor has a positive real root, W is primitive with constant term 1 so every irreducible factor has g(0) = 1, and (1 + k t)* = (k+1) n + 1; this is why the divisor tribe is the all-rational floor and why its factors are never (a n + b) with b > 1, and with the origin empty the law fails, signature (0,2,1) at dim = 2 having W = t^2 + 2t and fill n(3n + 2). Witness: lab/py/field-ladder.
2026-09-11 Proved The pure quadratic layer realizes exactly the imaginary quadratic fields of discriminant at least -2 dim (dim-1): the pure signature (1, b, c) with 0 <= b <= dim and 0 <= c <= C(dim, 2) has W = 1 + b t + c t^2, fill (n+1)^(dim-2)((1+b+c) n^2 + (b+2) n + 1) and discriminant b^2 - 4c on both sides, every d = 0 or 1 mod 4 in [-4C(dim,2), -1] occurs at b = 0 or b = 1 and nothing deeper occurs, and dividing out the conductor leaves exactly the fundamental discriminants of absolute value at most 4C(dim,2) = 2 dim (dim-1), which is 2, 5, 10, 14, 21 fields at dim = 2..6. Witness: lab/py/field-ladder.
2026-09-11 Verified The whole origin-filled box adds no further imaginary quadratic field at dim <= 6: the imaginary quadratic field discriminants carried by every irreducible factor of every signature are exactly the fundamental discriminants of absolute value at most 2 dim (dim-1), so the run is gapless and its first gap is the next fundamental discriminant, 7, 15, 31, 43, 67 at dim = 2..6, exhaustive over 6, 32, 350, 8712 and 526848 signatures. Witness: lab/py/field-ladder.
2026-09-11 Proved Amendment to the discriminant staircase row: its dim(dim-1) count law is exact under the order reading on the origin-filled box, and what needs amending is the gloss "fundamental", since -12 = -3 * 2^2, -16, -20 and -24 are discriminants of orders and not fundamental. The pure layer gives exactly the dim(dim-1) values 0 or 1 mod 4 in [-4C(dim,2), -1], and exactly 2, 5, 10, 14, 21 fields at dim = 2..6. The two rows sweep different boxes, 1053696 signatures with s_0 free against 526848 with s_0 = 1, so -63, -160, -899 and -60 do not meet: -899 sits at the origin-empty (0,0,15,1,15,0,0), W = t^2 (15t^2 + t + 15). Witness: lab/py/field-ladder.
2026-09-11 Proved Imaginary completeness holds for the origin-filled box: every imaginary quadratic order of discriminant d and every imaginary quadratic field of discriminant d occurs as a quadratic factor of a fill polynomial at every dim with 2D(dim-1) >= abs(d), so at every dim at least (1 + sqrt(1 + 2 abs(d)))/2. Witness: lab/py/field-ladder.
2026-09-11 Verified Positive-disc completeness does not follow the imaginary law: the pure layer reaches only d = b^2 - 4c <= dim^2 - 4 on the real side, the census at dim = 6 runs 5, 8, 12, 13, 17, 21, 24, 28, 29, 33 and stops at 37 with the last two values coming from signatures of degree above 2, and the real quadratic run is the binding constraint on the degree-2 Hunter bound at every dimension. Witness: lab/py/field-ladder.
2026-09-11 Verified The ladder by degree over the origin-filled box: signatures by the top degree of the irreducible factors of W are 4 rational and 2 quadratic at dim = 2, 7, 13, 12 at dim = 3, 12, 62, 130, 146 at dim = 4, 19, 266, 955, 3522, 3950 at dim = 5, and 30, 1173, 7305, 45292, 222437, 250611 at dim = 6, exhaustive over the 6, 32, 350, 8712 and 526848 signatures of the box, the last carrying 2^63 oriented designs; counted by oriented design the dim = 4 row is 504 rational, 6884 quadratic, 13241 cubic and 12139 quartic of 32768. Witness: lab/py/field-ladder.
2026-09-11 Verified Cubic ownership: every cubic field of absolute discriminant at most 307 is a norm form of a fill polynomial at dim = 6, the signature (1,1) run being the whole 100-entry table 23 to 815 with no gap and the totally real run 49, 81, 148, 169, 229, 257 stopping at 316, against (1,1) runs stopping at 44, 244 and 652 at dim = 3, 4, 5. Witness: lab/py/field-ladder.
2026-09-11 Verified The field-discriminant run of the box stops at a first gap per degree and field signature, reading dim = 3, 4, 5, 6 where the class is nonempty: degree 2 at 15, 31, 43, 67 for (0,1) and 8, 13, 24, 37 for (2,0); degree 3 at 44, 244, 652 and past 815 for (1,1) and 81, 316 for (3,0); degree 4 at 225, 981, past 2156 for (0,2), 400, 1423, 3275 for (2,1) and 1125 for (4,0); degree 5 at 7684 and past 12752 for (1,2), 5783 and past 13883 for (3,1), and nothing at all for (5,0). Witness: lab/py/field-ladder.
2026-09-11 Verified The Hunter-type bound of the box: with B(d, dim) the largest bound such that every field of degree d and absolute discriminant at most B is reached, B(2, dim) = 7, 12, 23, 35, B(3, dim) = 31, 44, 76, 307, B(4, dim) = none, 189, 697, 1107 and B(5, dim) = none, none, 5753 with B(5, 6) at least 12752 at dim = 3, 4, 5, 6, the first miss at dim = 6 being 37 at signature (2,0), 316 at (3,0) and 1125 at (4,0), while the box height max_j C(dim, j) is only 3, 6, 10, 20. The merge is by field: 8 is two fields and only -8 is reached at dim = 3, and a discriminant the table lists twice is credited only on two non-isomorphic factors. Witness: lab/py/field-ladder.
2026-09-11 Proved No field signature is excluded by the sign condition: no irreducible factor of W has a positive real root, and for any field K with generator gamma the element theta = -1/(gamma + N) with N above every real conjugate generates K, has all real conjugates negative and has 1/theta an algebraic integer, so its primitive minimal polynomial meets both conditions a factor of W meets. Witness: lab/py/field-ladder.
2026-09-11 Verified The totally real classes are the sparse side of the ladder: signature (3,0) first occurs at dim = 5 with the single field 49, (4,0) at dim = 6 with the single field 725, and (5,0) does not occur at dim <= 6, the smallest totally real quintic field being 14641. Witness: lab/py/field-ladder.
2026-09-11 Verified Every field discriminant of the census is computed by PARI nfdisc on the reversed monic model and guarded against the polynomial discriminant, which must be a square multiple of it, and against 0 or 1 mod 4: all 256179 distinct irreducible factors of degree 2 to 5 over dim = 2..6 pass, so no run rests on an unchecked value and no factor is unresolved. A run counts discriminants, since two fields can share one, the first repeat inside a printed run being 576 twice in degree 4 signature (0,2); the Hunter bounds are lifted to fields by nfisisom. Witness: lab/py/field-ladder.
2026-09-11 Conjecture Every number field appears at some finite dim and its discriminants arrive in order: the box s_0 = 1, 0 <= s_j <= C(dim, j) reaches every field of degree 2, 3, 4 of absolute discriminant at most 35, 307, 1107 at dim = 6 and every quintic field of the tables read, to 12752, each a gapless initial run, and the two necessary conditions on a factor, constant term 1 and no positive real root, are met by a generator of every field. Witness: lab/py/field-ladder.
2026-09-11 Refuted The swap clause, that when one side factors completely over Q the other carries an irreducible factor, forbids the P+ V+ cell alone, and that cell is not empty: over origin-filled oriented designs the table by fill split and void-core split reads 17, 4, 67, 40 at dim = 3 and 413, 91, 4994, 27270 at dim = 4, so 0.13 of the designs at dim = 3 sit in the forbidden cell, the smallest on corners 000 and 001 with fill (n+1)^2 (2n+1) and void core (2n+1)(3n+2). The exception class was already named with the clause; the count is the news, and the P- V- cell is not forbidden. Witness: lab/py/field-ladder.
2026-09-19 Proved The sponge rule, keep a cell with at most one odd coordinate, is the signature (1, dim, 0, ..., 0) and fills (n+1)^dim + dim n (n+1)^(dim-1) = (n+1)^(dim-1)((dim+1) n + 1) at odd side 2n+1, a product of dim linear factors of exponent pattern (dim+1, 1, ..., 1), hence d(x^n) for x = 2^(dim+1) 3 * 5 * ... * p_dim, the tower 4, 24, 240, 3360 at dim 1..4, with n = 1 column (dim+2) 2^(dim-1) = A001792. Witness: lab/py/field-ladder, divisor-avatars. Claims heading: Divisor avatars.
2026-09-19 Verified The Q column of the ladder over the origin-filled box, 4, 7, 12 signatures at dim 2, 3, 4, is in bijection with the exponent patterns of prod_i (a_i n + 1) under the avatar map, so the rational floor is named integer by integer: 30, 60, 120, 180, 240, 360, 900 at dim 3 and 210, 420, 840, 1260, 1680, 2520, 3360, 5040, 6300, 7560, 12600, 44100 at dim 4, and it is closed under products. Witness: lab/py/field-ladder. Claims heading: Divisor avatars.
2026-09-19 Verified At dim 3 the signature (1,3,1,0), the sponge with one weight-2 corner added, has W = 1 + 3 t + t^2 of discriminant 5 and fill (n+1)(5 n^2 + 5 n + 1), the norm form of the real quadratic field of discriminant 5. Witness: lab/py/field-ladder. Claims heading: Fill polynomials.
2026-09-19 Verified Over the 350 origin-filled signatures of the box at dim 4 the 6884 quadratic oriented designs split into 6518 imaginary, 105 real and 261 mixed, carried by 55, 4 and 3 signatures. Witness: lab/py/field-ladder verb ladder.
2026-09-19 Verified Over the 8712 origin-filled signatures of the box at dim 5 the 2147483648 oriented designs read 1209703 rational, 102969641 quadratic (92090824 imaginary, 85372 real, 10793445 mixed), 213022933 cubic, 956166567 quartic and 874114804 quintic by top irreducible factor degree. Witness: lab/py/field-ladder verb ladder.

Flake band gap

2026-08-28 Verified The base-2 flake's Laplacian has an interior band gap whose upper edge is exactly 4: on the bang dim 3, code 23 flake (4^level nodes, 4^level - 1 edges, connected), exact rational elimination of Lap - 4I has a single zero pivot, 3 * 4^(level-1) eigenvalues lie below 2 and none in [2, 4), the lower edge climbs 1.000000, 1.827520, 1.975680, 1.996862, 1.999605, 1.999950 at level = 1..6, and the top eigenvalue is 3 + sqrt(5) = 5.2360679775 at level = 2 climbing to 5.7090316570 at level = 6. Witness: lab/py/flake-band-gap.
2026-08-28 Conjecture The lower edge closes at a fitted c * 8^(-level) with c near 12.9868 over ten levels, (2 - lo) * 8^level reaching 12.984807 and 12.986289 at level = 7, 8, a fit with no mechanism. Witness: lab/py/flake-band-gap.

Flat carpet stack

2026-08-28 Proved The moire correlation law: for odd m, n the mean of s(mu) s(nu) with s(x) = (-1)^floor(x) is exactly gcd(m,n)^2/(mn) (for general integers it needs m/g and n/g both odd, else the integral is 0), and the Pearson correlation of the 1D parity indicators is exactly (gcd^2 - 1)/sqrt((m^2 - 1)(n^2 - 1)), so the correlation is exactly 0 if and only if the scales are coprime, and zero covariance is independence for Bernoulli pictures; exact rational integration over all odd pairs to 99 matches the closed forms to 5.6e-17. Witness: moire-correlation-laws.
2026-08-28 Proved The stack is an exact prime detector: an odd n >= 3 is prime exactly when its carpet is uncorrelated with every earlier carpet; over odd 3..199 all 45 primes sit at exactly 0 and all 54 composites strictly positive, minimum 0.0517383 at n = 169 = 13^2; the finite-window corollary "the zero-redundancy layers of the 1..55 stack are the primes above 55/3" is a window artifact and not the statement. Witness: moire-correlation-laws.
2026-08-28 Proved Pi cancels out of every visible brightness of the stack: ray strengths, crosshair steps, hot-spot values, layer correlations and per-layer means are all rational, because the square wave's (4/pi)^2 meets the odd Basel sum pi^2/8; the diagonal is exactly twice the background in paper coverage in the limit, and the anti-diagonal is its pixel-for-pixel copy by the palindrome symmetry. Witness: moire-correlation-laws.
2026-08-28 Proved The moire rays obey a 2-adic law, not a Farey law: the slope-one family at offset a/b carries (-1)^a/b^2 for odd b and exactly nothing for even b, and the slope q/p ray through the origin carries exactly 1/(pq), the same number as the correlation of grams p and q. Witness: moire-correlation-laws.
2026-08-28 Verified The stack fades at the random rate in L^2: RMS contrast falls as c/sqrt(L) in the layer count L with c^2 = lim L * Var, c = 0.522, a constant factor 1.2054 above independent layers, so "does not fade like random noise" is false in L^2; the exact variance is a finite rational sum at every L. Witness: moire-correlation-laws.
2026-08-28 Proved Coprime independence holds exactly in all four flat families: carpet, net, tree and void layer pairs have covariance identically 0 at every coprime odd pair, checked exhaustively to 201 and in exact rationals at (3,5), (5,7). Witness: moire-correlation-laws.
2026-08-28 Proved The void flat stack obeys a gcd-to-the-fourth law, Pearson_void(m,n) = (g^4 - 1)/sqrt((m^4 - 1)(n^4 - 1)) (void being the pure pair field (1 + s(mu) s(mv))/2), tree obeys (g^2 - 1)/sqrt((m^2 - 1)(n^2 - 1)), net's closed form is carpet's under m, n -> -m, -n with slightly larger correlations, and the gcd echo orders tree > net > carpet > void; all four match exact lcm-grid counting on (3,9), (5,15), (9,15) with zero error. Witness: moire-correlation-laws.
2026-08-28 Verified The flat variance constants reduce to two gcd sums, S2(N) = sum g^2/(mn) and S4(N) = sum g^4/(m^2 n^2) over odd pairs: lim L * Var is the S2/(2N) limit for tree, the S4/(2N) limit for void (0.2768062) and S2/(4N) + S4/(8N) for carpet and net (their difference dying like ln^2 N / N), with the identity Var_L(carpet) = Var_L(tree)/2 + Var_L(void)/4, measured to 5 digits. Witness: moire-correlation-laws.
2026-08-28 Proved The S4 limit is a theorem: lim S4(N)/N = (16/31) T/zeta(5) with T = sum_{k,l odd} 1/(k^2 l^2 max(k,l)) = 1.1122336970, value 0.5536124372, measured 0.5536124482 at N = 3 * 10^6, by a bounded coprime tail plus Mobius over odd moduli, with an independent Jordan-totient recomputation. Witness: moire-correlation-laws.
2026-08-28 Conjecture The rendered diagonal-to-background ratio reads 2.1189 at N = 55 walking to 2.0000252 at N = 10^6, and the rendered grey ratio is 16/9 because ink is 17.
2026-08-28 Conjecture A primitive integer line alpha u + beta v = gamma is a ray iff alpha and beta are both odd, and the crosshairs at u = a/b carry 1/(4b), a factor b stronger than the diagonal family.
2026-08-28 Conjecture Pi survives only in the census and the decay arithmetic: distinct rays are indexed by odd-denominator reduced fractions, counted by sum phi(b) ~ (2/pi^2) B^2 through the odd-prime Euler product 8/pi^2, and the pairwise gcd sum obeys S2(N)/N -> pi^2 ln 2/(7 zeta(3)) = 0.8130217 (6 digits at N = 10^6 two ways), whence the tree constant lim L * Var = pi^2 ln 2/(14 zeta(3)) = 0.4065108521 and the carpet and net constant 0.2724570.
2026-08-28 Conjecture The sup-norm never fades: diagonal, crosshairs and the four brightest points (paper 9/14 at the inner-thirds crossings, grey exactly 170) hold their values forever while their width shrinks like 2/(N + 1) on an exactly triangular profile.
2026-08-28 Conjecture Chaining 27 8-bit blends and truncating each step shifts the rendered stack by six grey levels, the saved image's mean 61.9 against the true 67.8 and its brightest pixel 163 against 170; quote paper-coverage fractions, never absolute greys.
2026-08-28 Conjecture The 2D ray law transfers to the void flat stack at double contrast: void has no crosshairs (no single-wave terms) but carries the full odd/odd 2-adic diagonal web at twice the carpet's strength (offset a/b: (-1)^a/(2b^2); slope q/p through the origin: 1/(2pq)), with u = v and u + v = 1 solid ink, a black X on mid-grey, the negative of the carpet's paper X.

Franel on a digit design

2026-09-14 Proved At frequency 1 the exponential sum of the denominator-restricted Farey set IS the design's Mertens meter: with S_F the whole numbers whose every digit lies in a digit set, sum of e(r) over r in {a/b reduced, b in S_F, b <= Q, 1 <= a <= b} equals M_F(Q) = sum of mu(b) over b in S_F, b <= Q, since sum over a mod b coprime to b of e(a/b) = mu(b) by Mobius inversion against the complete sums; every denominator in S_F up to Q = 10^5 has its literal sum of phi(b) roots of unity equal to mu(b), worst deviation 1.09e-11 at b = 86293 (base 3 digits {0,1}) and 1.36e-12 at b = 7247 (base 10 without 9), 0 wrong roundings. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel ladder_check)
2026-09-14 Proved At frequency m the same sum is sum over d dividing m of d M_F(Q/d; d), where M_F(x; d) = sum of mu(c) over c <= x with dc in S_F is the Mertens function of the DILATED design d^{-1} S_F; the classical divisor-shifted Mertens sums are the case S_F = Z, where every dilate is Z and all of them collapse to M, while for a digit design d^{-1} S_F is not S_F, is not a digit design and carries no digit test, so each d > 1 brings a new function; checked at m = 1, 2, 3, 4, 5, 6, 12 on base 3 {0,1} at Q = 2187, base 10 without 9 at Q = 1000 and the full-set control at Q = 300, exact integer against literal sum at every cell. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel verb_denominator)
2026-09-14 Proved The digit-restricted Franel identity, Fourier form: sum over k nonzero of abs(S_F(k,Q))^2 / k^2 = (pi^2/3) G_F(Q) with G_F(Q) = sum over d, e of (gcd(d,e)^2/(d e)) M_F(Q/d; d) M_F(Q/e; e), a finite sum of exact rationals; the kernel is the Smith gcd matrix that already carries the moire correlation law and the Gaussian identity, so digit restriction moves the entries and never the kernel; checked against the literal Fourier side truncated at abs(k) <= 200000 with the printed tail bound 2 m^2/K, gap inside bound at base 3 Q = 81 and 243, base 10 Q = 40 and the control Q = 40. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel kernel_sum and fourier_side)
2026-09-14 Proved The digit-restricted Franel identity, rank form: if the node set has top node 1 and mean value sum of rho_r = (m_F(Q)+1)/2, then G_F(Q) - 1 = 12 m_F(Q) sum_j delta_j^2 as exact rationals, with m_F(Q) = sum of phi(b) over b in S_F, b <= Q and delta_j = rho_j - j/m_F(Q); closure under r -> 1 - r away from the node 1 is one sufficient condition for that mean value, holding for every denominator-restricted set and failing for every proper strict set; the proof is Parseval plus piecewise integration of (A(v) - m v)^2; True at base 3 Q = 81, 243, base 10 Q = 40 and the control Q = 40, which regenerates Edwards section 12.2. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel farey_delta_square)
2026-09-14 Proved The denominator lane's CONJECTURED shape implies the square-root ceiling for the design's Mertens meter: dropping every term but k = 1 and k = -1 from a sum of nonnegative terms gives 2 M_F(Q)^2 <= (pi^2/3) G_F(Q), which by the rank form is 4 pi^2 m_F(Q) sum_j delta_j^2 + pi^2/3; with A_F(Q) << Q^alpha (the block count) and m_F(Q) <= Q A_F(Q) << Q^(1+alpha), the conjecture sum_j delta_j^2 = O(Q^(-1+eps)) forces abs(M_F(Q)) = O(Q^(alpha/2+eps)), the constant absorbed and no unproved input entering; the measured exponent is -0.959 and -0.899, not -1, so only the conjecture yields the ceiling. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel scan_backward)
2026-09-14 Proved The strict set's frequency-1 sum has an exact divisor form: it equals sum over b in S_F, b <= Q of sum over d dividing b of mu(d) times sum of e(a/(b/d)) over a <= b/d with da in S_F, by Mobius inversion of the coprimality condition followed by a -> da; the inner sum is a digit-restricted exponential sum over an arithmetic progression, the Type II object with no bound on the tree, so the identity is exact and inert; literal summation against the divisor route agrees to 6.28e-15 over the 64 denominators of base 3 {0,1} below 729 and to 1.95e-14 over the 162 of base 10 without 9 below 200. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel strict_ramanujan_divisor)
2026-09-14 Verified Literal enumeration of the strict digit-restricted Farey set reaches the counts the sieve prints without enumerating a fraction: 278, 4286, 67561, 1080458 at Q = 3^5, 3^7, 3^9, 3^11 on base 3 {0,1} and 1830, 147096, 11890654 at Q = 10^2, 10^3, 10^4 on base 10 without 9, 7 rungs and no disagreement. (witness: lab/py/restricted-franel strict_literal against lab/rs/farey-discrepancy design)
2026-09-14 Proved For a design carrying the digit 0 the dilate d^(-1) S_F = {c : dc in S_F} is a regular language recognised least-significant-digit-first by a deterministic automaton whose d states are the carries of long multiplication by d: reading digit e from carry r writes the output digit (de + r) mod base, which must lie in F, and moves to carry floor((de + r)/base), which stays below d by induction, and after level digits dc is the level output digits with the terminal carry r_level written above them, so acceptance is exactly that r_level has all its digits in F; the accepting set is Acc_d = {0} union (S_F intersect [1, d)), of size A_F(d-1) + 1. The hypothesis 0 in F is load-bearing and not cosmetic: without it the run tests all level PADDED output digits, a leading output digit 0 is not a digit of dc, and the automaton recognises the padded set of the Mobius page instead, reading 8 at base 3 with F = {1,2}, d = 1 and level = 3 where the true count is 14. Regularity of the dilate itself survives without the hypothesis; the count identity does not. So the dilated Mertens sums of the restricted Franel identity run over regular sets, not over digit designs. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel dilate_matrix, verb_converse 0 mismatches of 840)
2026-09-14 Proved The dilate's transfer matrix T_d(r, r') = #{e < base : (de + r) mod base in F, floor((de + r)/base) = r'} counts it, #{c < base^level : dc in S_F} = e_0 T_d^level 1_(Acc_d) for a design carrying 0, and EVERY column of T_d sums to exactly #F, in every base, at every digit set and every d, with no hypothesis at all: the pairs (e, r) in [0,base) x [0,d) are in bijection with v = de + r in [0, dq) by the division algorithm, the column at r' counts the v with v - base r' in F, and the window [base r', base r' + base) lies inside [0, dq) for every r' < d, so exactly #F of them qualify. Hence the all-ones vector is a positive left eigenvector and the spectral radius of T_d is #F for every d: the dilate carries the design's own mass exponent as its Perron root. The rows sum to g times #(F intersect (r + gZ)) with g = gcd(d, base), so they equal #F whenever gcd(d, base) = 1, giving #{c < base^level : dc in S_F} <= (#F)^level at those d with constant 1, again only for a design carrying 0: base 3 with F = {1,2} and d = 1 reads 14 at level = 3 against (#F)^level = 8. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel verb_dilate, verb_converse)
2026-09-14 Verified The dilate automaton and its transfer matrix are checked against brute-force enumeration: over d <= 64 at base 3 {0,1} and base 10 without 9 no column of T_d is off #F, while rows are off #F at 21 and 38 of the 64 respectively, every one of them at a d sharing a factor with the base; at base 3 {0,1} with d = 2 the transfer matrix [[1,1],[1,1]] with both carries accepting counts 2^level - 1 at every level <= 12, agreeing with literal enumeration of {c : 2c in S_F} at every rung and reading 4095 at x = 3^12 = 531441; and the count identity's scope is exact on the sweep over base = 3, 4, 5, every F, every d <= 6 and every level <= 5, with 0 mismatches in the 840 cases carrying the digit 0 and 399 in the 750 without it. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel verb_dilate, verb_converse)
2026-09-14 Proved The dilated meter is blind to the base's own powers: if 0 in F then M_F(x; base^j d) = M_F(x; d) for every j >= 0 and every d, since appending j zero digits neither leaves nor enters S_F, so the dilates repeat along every base-power ladder and only the base-prime part of d can move them. At base 3 {0,1} the d = 3 column reproduces the d = 1 column exactly, M_F(3^12; 3) = M_F(3^12) = 56 with both peaks 61. It is the lever that fixes the rate in the converse's hypothesis and that refutes the mass saving uniformly in d. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel verb_dilate)
2026-09-14 Proved The digit transform of a dilate is the transfer matrix in place of the digit symbol: for a design carrying 0, sum of e(ct) over c < base^level with dc in S_F equals e_0 M(t) M(qt) ... M(base^(level-1) t) 1_(Acc_d) with M(t)(r, r') = sum of e(et) over the digits e carrying r to r', and M(0) = T_d, by decomposing over automaton paths. That is the ladder of the Mobius page with the scalar symbol g_F(base^j t) replaced by a matrix, and the replacement is exactly what the route costs: the product no longer factors, so the sup-over-shift l^1 exponent that carries a Type I estimate for a digit design has no scalar analogue here. The matrix form gives the exact count at t = 0 and exact evaluation at any t, and gives no cancellation in mu; the Type II wall stands where it stands at d = 1. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel verb_dilate)
2026-09-14 Proved The converse of the restricted Franel identity, from (U') and with the dependence on d explicit. (U') gives G_F(Q) = O_eps(Q^(alpha + eps)) and hence sum_j delta_j^2 = O_eps(Q^(-1+eps)) on the denominator-restricted set, which is the denominator lane's conjecture. Write d = a d_base and e = b e_base with a and b supported on the primes dividing base; the two parts have disjoint prime support, so gcd(d,e) = gcd(a,b) gcd(d_base,e_base) and the kernel sum FACTORS. Each term is at most gcd(d,e)^2 (de)^(-1-alpha/2-eps) (d_base e_base)^((alpha-1)/2) Q^(alpha+2eps); the coprime factor carries exponent -3/2-eps and, writing d_base = g u and e_base = g v with gcd(u,v) = 1, is at most zeta(1 + 2eps) zeta(3/2 + eps)^2; the base factor is the product over p dividing base of sum over i, j >= 0 of p^(2 min(i,j) - (i+j)s) with s = 1 + alpha/2 + eps, which sums in closed form to the product of (1 + p^(-s))/((1 - p^(-s))(1 - p^(-alpha-2eps))) and is FINITE because alpha > 0. Then m_F(Q) >> Q^(1+alpha)/log log Q, the >> Q^alpha members of S_F in the top block below Q each exceeding Q/base with phi(b) >> b/log log b, so the rank form divides the bound down to Q^(-1+3eps). The exponent (alpha-1)/2 on d_base is critical, not chosen: at (alpha-1)/2 + delta on d_base the same argument gives only G_F(Q) = O(Q^(alpha + 2delta + eps)) and no threshold, the coprime g sum becoming sum of g^(-1+2delta) of size Q^(2delta), and at delta = 0 it is the harmonic sum, convergent only through the eps; the base factor never sees the exponent. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel verb_dilate)
2026-09-14 Verified The raw exponent readings on the dilates separate nothing and are not exponents: over base 3 {0,1} to x = 3^12 = 531441 the reading log max abs M_F(x;d) over log x is 0.311823 at d = 1 and at most 0.292046 over d = 2, 4, 5, 7, 8, 11, 13, 16, 22, 31, the d = 3 row being the d = 1 row by the free base powers rather than an independent reading; over base 10 without 9 to x = 10^7 the reading is 0.484570 at d = 1 against 0.489199 at d = 7 and 0.472377 at d = 2, and the crossing seen at x = 10^6, 0.495982 at d = 2 against 0.444731 at d = 1, reverses by x = 10^7, peaks 2026 against 2466. The local exponents between consecutive rungs swing over 0.24 to 0.845, so none of these readings is an exponent and none of them tests the converse's hypothesis, which is a statement about the ratio to d_base^((alpha-1)/2) x^(alpha/2) and is metered separately. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel verb_dilate)
2026-09-14 Conjecture The denominator-restricted set's Franel analogue is sum_j delta_j^2 = O(Q^(-1+eps)), equivalently G_F(Q) = O(Q^(alpha+eps)); the forward half of an equivalence with the square-root conjecture for M_F is open and needs the dilated sums M_F(x; d) for d > 1, for which the desk has no bound, so only the implication above is proved and no exponent is claimed here. (witness: farey.md, The restricted Franel identity)
2026-09-14 Conjecture The strict set's frequency-1 sum divided by its node count converges to the first Fourier coefficient of a limit measure of the strict set, nonzero; the readings are 0.335693, 0.343837, 0.345905, 0.346338 at Q = 3^5, 3^7, 3^9, 3^11 and 0.015138, 0.012250, 0.011561 at Q = 10^2, 10^3, 10^4, four and three nested rungs and no exponent claimed; that limit measure has no definition on the tree, and until it is named there is no Franel-type equivalence to state on the strict set. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel verb_strict)
2026-09-14 Conjecture (U'), the surviving hypothesis with the dilate's true mass: abs M_F(x; d) = O_eps(d_base^((alpha-1)/2) x^(alpha/2 + eps)) uniform in d >= 1 and x >= 1, with d_base the part of d coprime to the base. It is square-root cancellation in each dilate's own mass read correctly, since d_base^((alpha-1)/2) is the square root of the accepting-set constant at d_base and the base-smooth inflation is bounded; it is consistent with the free base powers by construction because (base^j d)_base = d_base; and its d = 1 case is exactly the square-root ceiling the forward implication already delivers. Metered as a ratio it does not fire: max abs M_F(y;d) over y <= x divided by d_base^((alpha-1)/2) x^(alpha/2) reads 0.9531, 0.7991, 0.9531, 0.6054, 0.9883, 0.3580, 0.5504, 0.8269, 1.0535, 0.4431, 0.5528, 0.3828 at d = 1, 2, 3, 4, 5, 7, 8, 11, 13, 16, 22, 31 on base 3 {0,1} at x = 3^12, and 1.1276, 0.9264, 0.5523, 0.6173, 0.8583, 1.2702, 0.4594 at d = 1, 2, 3, 4, 5, 7, 11 on base 10 without 9 at x = 10^7, while the refuted exponent puts 1.1673 at d = 3 against 1.0535 as the coprime maximum. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel verb_dilate)
2026-09-14 Conjecture The dilate's mass constant is read off the automaton's accepting set under a SECOND coprimality: writing Delta_F for the gcd of the differences of the digits in F, for gcd(d, base Delta_F) = 1 the matrix T_d over #F is doubly stochastic, the carry chain is irreducible, its stationary law is uniform, and A_d(base^level)/(#F)^level converges to #Acc_d/d = (A_F(d-1) + 1)/d, which is O(d^(alpha-1)) and is exactly the saving a level of distribution for S_F at the modulus d would give. Verified to three decimals at level = 24 at every printed d coprime to the base, both metered designs having Delta_F = 1: base 3 {0,1} reads 1.0000, 1.0000, 0.7501, 0.8000, 0.5714, 0.5001, 0.5455, 0.5394, 0.5001, 0.3636, 0.3548 at d = 1, 2, 4, 5, 7, 8, 11, 13, 16, 22, 31 against 1, 1, 0.75, 0.8, 0.571429, 0.5, 0.545455, 0.538462, 0.5, 0.363636, 0.354839, and base 10 without 9 reads 1.0000, 1.0000, 1.0000, 0.9091, 0.9231, 0.8264, 0.8272, 0.8148 at d = 1, 3, 7, 11, 13, 121, 243, 729 against 1, 1, 1, 0.909091, 0.923077, 0.826446, 0.827160, 0.814815. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel verb_dilate)
2026-09-14 Conjecture The converse's hypothesis has a Mobius-free surrogate the lab can meter: square-root cancellation in each dilate's own mass is abs M_F(Q/d; d) <= N_F(Q; d)^(1/2+eps) with N_F(Q; d) = #{m in S_F : m <= Q, d divides m}, since the sum for M_F(Q/d; d) runs over exactly those m, so under that hypothesis the converse reduces to the divisor statement that B(Q) = sum over d, e of gcd(d,e)^2/(de) times sqrt(N_F(Q;d) N_F(Q;e)) is O(Q^(alpha+eps)), which mentions no Mobius function at all. This form stays consistent where the d-uniform bound above does not, reading abs M_F(Q/base^j) <= A_F(Q/base^j)^(1/2+eps) at d = base^j, which is the d = 1 ceiling again. Measured at base 3 {0,1}: B(Q)/Q^alpha reads 12.5146, 17.8640, 24.7369, 31.5935, 39.0671 at Q = 3^4 to 3^8 with local exponents 0.955, 0.927, 0.854, 0.824 falling toward alpha = 0.630930, and B(Q) over Q^alpha (ln Q)^2 falls 0.6480, 0.5920, 0.5693, 0.5342, 0.5058, so the range is consistent with Q^alpha times a power of a logarithm and no exponent is claimed. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel smith_bilinear)
2026-09-14 Refuted No Mertens-type sum over S_F equals the strict set's frequency-1 sum, because that sum is not real: at base 3 {0,1} and Q = 3 it is 1 + e(1/3) = 0.5 + (sqrt 3/2) i and at base 10 without 9 and Q = 10 it is -1.809016994 + 0.587785252 i, both exact algebraic sums evaluated past 1e-9, while M_F(Q), the count-weighted sum of mu(b) phi_F(b) and the normalised sum of mu(b) phi_F(b)/phi(b) are all real; the two witnesses carry the refutation alone; beside them sits an observation and not a mechanism, that the strict set also fails the pairing a -> b - a, failure of which is not shown to force a non-real sum. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel verb_strict)
2026-09-14 Refuted (U), the d-uniform dilated bound abs M_F(x; d) = O_eps(d^((alpha-1)/2) x^(alpha/2 + eps)), holds for NO design carrying both 0 and 1, so it cannot be the hypothesis of the converse. Base powers being free gives M_F(x; base^j) = M_F(x), so (U) at d = base^j demands abs M_F(x) <= C_eps base^(j(alpha-1)/2) x^(alpha/2+eps) at every j >= 0, and alpha < 1 drives the right side to 0 at fixed x, forcing M_F identically zero against M_F(1) = mu(1) = 1. Base 3 {0,1} at x = 3^12: the left side is 56 at every j = 0 to 12 while d^((alpha-1)/2) x^(alpha/2) falls 64.0000, 52.2558, 42.6667, 34.8372, 28.4444, 23.2248, 18.9630, 15.4832, 12.6420, 10.3221, 8.4280, 6.8814, 5.6187 over d = 3^0 to 3^12 and the ratio climbs 0.875, 1.072, 1.313, 1.607, 1.969, 2.411, 2.953, 3.617, 4.430, 5.425, 6.645, 8.138, 9.967, unbounded in j. The cause is that (alpha-1)/2 is the square root of the dilate's mass constant only where that constant is d^(alpha-1), and on the base-power ladder the constant is 1. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel verb_converse)
2026-09-14 Refuted The converse is one implication and not an equivalence: the threshold does not give (U') by the natural route. From G_F(Q) = O(Q^(alpha+eps)) the Fourier form gives termwise abs S_F(fill,Q) <= fill (pi^2 G_F(Q)/6)^(1/2), and Mobius inversion of S_F(m,Q) = sum over d dividing m of d M_F(Q/d;d) gives d M_F(Q/d;d) = sum over c dividing d of mu(d/c) S_F(c,Q), hence only abs M_F(Q/d;d) <= (sigma(d)/d)(pi^2 G_F(Q)/6)^(1/2), which is << log log d times Q^(alpha/2+eps) and GROWS in d where (U') needs d_base^(-1/2-eps) decay. So no biconditional is available, and none is claimed. (witness: farey.md, The restricted Franel identity)
2026-09-14 Refuted Coprimality to the base alone does NOT give the accepting-set constant. At base 3 with F = {0,2}, where Delta_F = 2, the dilate d = 2 is coprime to the base and carries T_2 = [[2,0],[0,2]], so carry 1 is unreachable from carry 0, the closed class is {0} and the uniform stationary law is read on the wrong class: exhaustive counts are 2, 4, 8, 16, 32, 64, 128, 256 at level = 1 to 8, exactly (#F)^level, so the constant is 1 against #Acc_2/2 = 1/2. The split is exact where it is swept, over every base <= 7, every F carrying 0 and every 2 <= d <= 24 coprime to base, read at level = 400: 1747 agreements and 0 failures at gcd(d, Delta_F) = 1, 0 agreements and 148 failures at gcd(d, Delta_F) > 1. That same constant 1 sits at a d coprime to the base, so it also kills O(d^(alpha-1)) there, 1 against 2^(alpha-1) = 0.6444, and the base-smooth mechanism is therefore one cause and not the only one. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel verb_converse)
2026-09-14 Refuted The mass saving A_d(x) = O(d^(alpha-1) x^alpha) does not hold uniformly in d, and the base-power ladder is what refutes it: with 0 in F the dilate at d = base^j is S_F itself, so K_d = 1 exactly at every j while the ceiling base^(j(alpha-1)) tends to 0, and A_d(x)/(d^(alpha-1) x^alpha) is at least base^(j(1-alpha)), UNBOUNDED. Off the ladder the base-smooth dilates are denser than the design in the same way: at base 10 without 9 K_d = A_d(base^level)/(#F)^level reads 1.1111, 1.1358, 1.1111, 1.1413, 1.0700, 1.0343 at d = 2, 4, 5, 8, 16, 32 against the claimed ceilings 0.968781, 0.938537, 0.929003, 0.909237, 0.880851, 0.853352, and the accepting-set law fails there too, those same d carrying #Acc_d/d = 1, 1, 1, 1, 0.9375, 0.90625. The dilate is denser because the last digit of an element of S_F is uniform on F and F is not balanced modulo a prime dividing the base, so no equidistribution of S_F modulo d is available at base-smooth d. What it costs is the converse's first hypothesis and not its conclusion, the repaired hypothesis asking the rate on the coprime part only: over the 29 base-smooth d <= 1000 at base 10 without 9 the constant lies in [0.9273 at d = 512, 1.1637 at d = 625] and over every d <= 200 its inflation over the value at the coprime part of d lies in [0.9375 at d = 112, 1.1413 at d = 88], bounded on the metered range and unmeasured past it. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franel verb_dilate, verb_converse)

Franel one field up

2026-09-09 Proved Kluyver's identity in Z[i]: the Gaussian Ramanujan sum obeys c_d(lambda) = sum_{e | gcd(d, lambda)} mu_G(d/e) N(e) over ideal divisors, so the exponential sum of the Gaussian Farey set is S_N(lambda) = sum_{[e] | lambda, N(e) <= N} N(e) M_G(N/N(e)) with M_G the Gaussian Mertens function over associate classes; 2720 exact sums at norm bound 50 with 0 mismatches, 68 literal node sums agreeing to 1.281e-13, and the node set identified with the complex Farey set of the literature at T = 2 to 6 (4, 24, 64, 176, 320 points). Witness: lab/py/gaussian-franel check_theorem_1, check_sayous.
2026-09-09 Proved Franel's identity one field up: the Fourier L^2 discrepancy of the Gaussian Farey set on C/Z[i] is m^2 D_2(N)^2 = 4 zeta_K(2) sum_{[a],[b]} N(gcd(a,b))^2/(N(a) N(b)) M_G(N/N(a)) M_G(N/N(b)), a finite gcd-weighted quadratic form in Gaussian Mertens sums whose kernel is the Smith gcd matrix of the layer Gram; checked against the Fourier side truncated at N(lambda) <= 200000 with gaps 0.001883 and 0.010546 inside the printed tail bounds 0.226192 and 7.093392 at norm bounds 20 and 50; the same identity regenerates the classical Farey discrepancy with no Farey enumeration, C(Q) - 1 = 12 Phi(Q) sum delta_v^2 exactly at Q = 40 and S2 Q reading 0.5395, 0.5848, 0.6241, 0.6387, 0.6560, 0.6538, 0.6564 at Q = 125 to 8000, the Farey page's table digit for digit; the Gaussian global readout collapses like the rational one, sum_{N(a) <= x} M_G(x/N(a)) = 1 at every x to 2000. Witness: lab/py/gaussian-franel check_theorem_2, classical_exact, franel_form, readout.
2026-09-09 Proved F(N) = O(N^{1+eps}) for every eps > 0, equivalently D_2(N) = O(N^{-3/2+eps}), is equivalent to the Riemann hypothesis for zeta_{Q(i)}(s) = zeta(s) L(s, chi_-4): backward, the four units give M_G(N)^2 <= zeta_K(2) F(N) and partial summation makes 1/zeta_K analytic right of the critical line; forward, by divisor splitting from Littlewood's bound on M_G transcribed to zeta_K by Perron with the bounds of Hu, Kaneko, Martin and Schildkraut (Lemma 5.4 and Lemma 2.4), the biconditional being assembled here and written in neither source; the rational template is Huxley 1971 Lemma 10 and Huxley 2012 Theorem 1 over Q, the announced number-field part abandoned by its author's account, and the Gaussian identity not found in the sources read; an equivalence is exactly as hard as the hypothesis it names and the meter renders, it does not measure. Witness: lab/py/gaussian-franel main (the backward inequality alone, ratio at most 0.017 over norm bounds 100 to 64000), REFS.md.

Gasket rays and the window

2026-08-28 Verified Occupied-ray totals differ by exactly 2 between the two counting conventions at every level, 1044842 against 1044840 at n = 13, because one counts the two fibre rays (1,0) and (0,1) and the other does not; nothing else in either table moves, so any occupancy total must say which convention it uses. Witness: lab/rs/dimension-one-ladder; gasket-ray-machine.
2026-08-28 Verified The multiplier-count census of the gasket at n = 13: over all 1,044,840 occupied non-fibre rays, 699,508 carry M_n = 1 (17% of Z), 339,530 carry M_n in [2,5] (55% of Z), the ten heaviest rays are exactly the shifts (1, 3^j) and their reverses for j = 1..5 and carry 14% of Z, sum M_n = 1,577,940 = 3^13 - 2^14 + 1 exactly, and max M_13 = 376 = F(14) - 1 reproduces the mass law M_n(3,1) = F(n+1) - 1 from a generator that never mentions Fibonacci, all by exact exhaustive enumeration of the 3^13 gasket points with gcd reduction into a hash table. Witness: lab/rs/dimension-one-ladder; gasket-ray-machine.
2026-08-28 Conjecture Conjecture N: every primitive ray automaton of the gasket has spectral radius at most the golden ratio with equality exactly on the shift rays (3^j, 1), and the supremum off them is the supergolden ratio 1.4655713, the root of x^3 = x^2 + 1; the bound half holds for every primitive ray by the burst certificate (top edge 0.6402122 unconditionally, from 0.730424), and what stays open is strictness rho < phi off shifts and the supergolden supremum that would move the edge to 0.605303, measured on all 490 primitive rays of height <= 40 plus a fixed 766-ray sample to height 200 where every observed radius is a root of x^k = x^(k-1) + 1 or x^k = x + 1. Witness: lemma-b-pincer.
2026-08-28 Conjecture Shear class 26 carries a flat mid-octave Chebyshev residue near 1.5e-4 at n = 14, roughly 40 times its peers 98, 176 and 416, with no degenerate fibre to blame, being the class that is a graph of nothing, parametrized by the balanced-ternary integer x_2 - x_1; the excess is collinear shift-ray mass Mertens-smeared flat: code 26 builds its points as c_n minus a disjoint-support binary pair, so the golden shift family survives with M_n(3,1) = F_(n-1) exactly where its peers carry zero, and the deep excess at n = 14 is 65.7% shift rays plus 2.9% supergolden against a Mertens-predicted flat height 676 ln 3 / 3^14 = 1.55e-4 versus the recorded 1.5e-4, the carriers flat across octaves j = 7..12 including inside the proved top range, so the window-mass-in-disguise reading is dead; the "3995 points" of the first count are 3993 non-fibre plus 2 axis points.
2026-08-28 Conjecture Conjecture W and its ray twin Conjecture O: the weighted active multiplier census per octave is C 3^j, measured C ~ 120 at n = 14, unimodal in j, with activity concentrated at 3-adic depth (attainer families (3^a, 3^b +- 1), per-pair activity decaying like 0.65^K against the weight 1.5^K, so per-octave convergence is delicate); W implies Conjecture Z, hence the window (0.4475978, 1/2], and W with O closes the window entirely, both implications exact; the precursor A_(j,K) <= C 3^(j-K) fails on the deep-K families, the universal pair-prefix transfer matrix has Perron root 4, not 3, and the unweighted octave census C = 1.042, 1.136, 1.244, 1.356 at n = 13..16 is a different quantity from the weighted (3/2)^K sum W names, so W is neither supported nor damaged by it.
2026-08-28 Conjecture Statement (A) is the exact averaged theorem the ray machine needs: for primitive non-shift (a,b) with 3^j <= max(a,b) < 3^(j+1), sum M_n(a,b) <= C 3^(2j) lambda^(n-j) poly(n) with lambda < phi inserts into the octave sum and gives beta > 1/(2 - log_3 lambda); it is weaker than a uniform non-shift spectral gap and far stronger than any average of rho, and its tail form requires the octave-j count of rays with rho >= t to be at most 3^(2j - I(t)j + o(j)) followed by an optimization over t; two routes are named, a large sieve on a bounded local deficit observable Fourier-expanded over the ray's residue modulus and a finite-state fractional-moment operator, with three obstructions to the sieve (varying state spaces with no common separated frequency family, the Cauchy-Schwarz loss of the square root of the ray count in passing from an L^2 average to L^1 octave mass, and an unweighted octave count the sieve's measure must match), and Turan power sums are ruled out, since they lower-bound maxima where an upper bound for a positive sum over many nonnegative matrices of varying dimension is needed. Witness: lemma-b-pincer.
2026-08-28 Conjecture The weighted active multiplier census W_j(n) <= C 3^j is not numerically stable across the two known levels: at n = 9 the exhaustive active-pair census gives A_j/3^j = 4.000, 6.667, 6.370, 4.025, 1.794, 1.141, 0.368 and W_j/3^j = 7.500, 17.750, 23.719, 25.041, 12.841, 12.097, 5.837 for j = 1..7, a peak of 25.0, while the n = 14 summary reports a peak near 113, so one level supports a constant and the two together do not, and the n = 14 tally is not in this tree; this weakens but does not refute Conjecture W, since W and O together closing the window is exact and W and O themselves remain untested.
2026-08-28 Conjecture The 3-power family of the second moment sums in closed form: S(n) = 2 Sum_(j=1..n-1) Q_n(1, 3^j) = 3^n - 4*2^n + 2n + 3 from Q_n(1, 3^j) = 3^(n-j) - 2^(n-j+1) + 1, so E(n) = T(n) + S(n) + R(n) with T + S = 2*3^n - 6*2^n + 2n + 4 exactly and the whole of Conjecture Z's constant 2 is accounted for before any residual is measured, leaving one statement about R; the identity is machine-checked at n = 3..17 and R = Z - T - S re-subtracted against the E list at n = 13, 14, 16 agrees.
2026-08-28 Conjecture The Pair Census Bound R(n) = o(3^n) over ordered collinear non-fibre gasket pairs whose multiplier ratio is not a power of 3 is the only unproved step to E(n) = O(3^n), hence to Conjecture Z and the window (0.4475978, 1/2]; the dominant carrier is the shift-ray family pairing with itself, now closed at ((13 + 5 sqrt5)/11) phi^(2n) and carrying only 0.65 to 0.84 of R, every other ray having spectral radius below phi, and any proof must use the multiplier-specific automata, since the universal pair-prefix transfer matrix has Perron root 4 and not 3. Witness: lemma-b-pincer.
2026-08-28 Conjecture Per-octave occupied ray counts by exact exhaustive enumeration at n = 14, 15, 16 total 3,151,658, 9,491,966 and 28,545,342, with occ(j,n)/3^j peaking at j = 8 in all three at 2.217, 2.492 and 2.740 and low octaves j <= 4 identical at all three levels; the table cannot be joined to the n = 13 per-octave table under the half-open convention 3^(j-1) <= max(a,b) < 3^j, the discrepancy not being a uniform label shift (4 rays with max = 3 placed at octave 1 where the interval forces octave 2; 336 against a recomputed 294 in the next bin), and the fibre-ray convention (1,044,842 with the two axes, 1,044,840 without) is not part of this mismatch.
2026-08-31 Proved The gasket residual changes coordinate. Every off-diagonal collinear pair of the level-n gasket G_n is (sz, tz) for a unique coprime (s,t) and a unique witness z, so R(n) = Sum_z P_n(z) where P_n(z) counts the coprime non-shift pairs one witness realises; the per-pair route needed a constant summable against an active-pair count growing 2.77 a level and is dead by construction, while the per-witness route has its constant. No witness weighs less than 4, hence max(s,t) <= (3^n-1)/8, sharp: the largest multiplier is exactly floor(3^n/8) at n = 4..13. Weight layers scale exactly, R_(3w)(n) = R_w(n-1), because 3 | z_1+z_2 with 3 dividing neither coordinate forces v_3(m z_1) = v_3(m z_2) and the supports collide; checked on all 1869 layers at n = 5..13. Every pair above (3^n-1)/10 carries exactly 4 ordered pairs, its only witnesses being (1,3) and (3,1), verified on all 30028 such pairs at n = 6..13. Witness: gasket-ray-machine, lab/py/gasket-witness-weights.
2026-08-31 Proved The weight-four layer is closed in Fibonacci, the second layer of the residual to close after the shift family. With F_n = {m : (m,3m) in G_n} the no-adjacent-ones set, #F_n = F(n+1) - 1 and R_4(n) = 2 #{(a,b) in F_n^2 : a != b, gcd(a,b) = 1, b/a != 3^j} < 1.0473 phi^(2n), so the whole 3-power orbit obeys Sum_j R_4(n-j) < 1.6945 phi^(2n); exact at n = 4..12 where R_4(n) = 12, 36, 108, 336, 988, 2596, 6672, 17480, 45720, carrying 194096 of R(13) = 863848. Constants safe: 2 phi^2/5 = 1.0472136 and 2 phi^3/5 = 1.6944272. Witness: gasket-ray-machine.
2026-08-31 Verified The golden ceiling: M_n(z) <= M_n(1,3) = F(n+1) - 1 for every direction, so the shift ray (1,3) is the heaviest ray of the gasket at every level, and this is the per-witness constant the per-pair route never had. Refutation attempt, briefed to break it: 13158 coprime directions with z_1 <= 120 and z_1 <= z_2 <= 240 at every n <= 40, (1,3) the sole attainer at n = 40; plus six families chosen to favour a breach at every n <= 45 - all binary base-3 pairs below 3^7 (4221 coprime), all no-adjacent-ones pairs below 3^7 (253), all (1,t) with t < 3000 (2998), all consecutive below 1500 (1499), (s,3s-1) (1199) and (s,3s+1) (1199) with s < 1200, 11369 directions in the shelf script and 24088 with the binary family widened to 3^8 in the lab; plus an independent enumeration on a larger box. Zero breaches anywhere. Next rate down is the supergolden 1.4655, the root of x^3 = x^2 + 1, at (1,12), (3,10), (4,9). Witness: gasket-ray-machine, lab/py/gasket-witness-weights.
2026-08-31 Proved Two cheap constructions for the free-digit automaton B(s,t), which used to blow up at large multipliers. It is a constrained tensor square T = S (x) S - U (x) U - V (x) V + W (x) W of a one-coordinate carry automaton with at most (s+1)(t+1) states, so the four-tuple state graph is never built: 729 carry states against 26931 reachable at (365,1094), 81 against 835 at (41,122), agreeing on all 473 coprime pairs below 40 at every level to 9. And at large multipliers the witness box z_1 + z_2 <= floor((3^n-1)/(2 max(s,t))) replaces the automaton entirely in O(W^2 n) digit tests, agreeing on 812 coprime pairs at n = 9 - cheapest exactly where a forward build is most expensive. Witness: gasket-ray-machine.
2026-08-31 Verified Conjecture Z evidence to level 17: R(n) = 863848, 2211960, 5549452, 14100688, 35354824 at n = 13..17. R/3^n peaks at 0.8401158 at n = 8 and falls at every level to 0.2737709; R/phi^(2n) peaks at 3.2378233 at n = 12 and falls at five consecutive levels to 2.7724831; the level ratio R(n+1)/R(n) reads 2.5073119 at n = 17, below phi^2 = 2.6180339. Witness: lab/py/gasket-witness-weights.
2026-08-31 Proved The golden partition bound U(z) <= phi^-2 is proved outright on an infinite arithmetic family, not checked direction by direction. Write q = 3^k q_1 for the coordinate divisible by 3 and p for the other. For k = 1 and t = v_3(q_1 - p): U(z) <= phi^-1 (1 - phi^-max(t,2)), so U <= phi^-2 on the whole class k = 1, t <= 2 - 261 of the 360 occupied k = 1 directions of the census - sharply at (1,12) where t = 1 and (3,10) where t = 2, and U < phi^-1 for every such ray but (1,3), the first proof that a whole family of gasket rays grows strictly slower than phi. Refutation attempt: 5422 occupied k = 1 directions picked in the hard corners (deep t, q_1 - p = +-m 3^e for e <= 7, 2q <= p so the far predecessor is live) gave zero violations with equality only at (1,12) and (3,10), the independent first-return series was dominated by the exact solve at 5420 of 5420 checked, and an independent adversary sweep of 910 stratified k = 1 directions in exact Q(sqrt5) found zero violations with equality again only at (1,12) and (3,10); a global U hunt over 17624 occupied directions to weight 6000 found only shift rays above phi^-2. Witness: gasket-ray-machine, lab/py/gasket-witness-weights.
2026-08-31 Proved Occupancy of a gasket ray is a congruence before it is anything else: M_n(z) > 0 for some n forces q_1 = p mod 3, by two lines on last digits with no automaton built - a multiplier m = 3^s m' makes m' p and m' q_1 binary in base 3 and prime to 3, so both end in digit 1. It empties 4588 of the 11691 census directions with 3 | z_1 z_2, and it is only necessary: just 865 of the 7103 matching directions carry mass. Refutation attempt: zero violations over a 400 x 2500 sweep and over both censuses, 1995 occupied directions in the lab universe and 865 on the shelf. Witness: gasket-ray-machine, lab/py/gasket-witness-weights.
2026-08-31 Proved The short first returns of a ray automaton are classified. No first return has length between 2 and v_3(q); f_2 != 0 only at {1,3} and f_3 != 0 only at {1,9}, {1,12}, {3,10}, {4,9}, each equal to 1. Hence U = phi^-2 Sum_(j>=3) f_j phi^(3-j), so U <= phi^-2 says exactly Sum_(j>=3) f_j phi^(3-j) <= 1 and forces f_4 <= 1; the supergolden trio is exactly f_3 = 1 with every later f_j zero. Refutation attempt: a 420 x 2600 sweep over 5281 occupied directions returned exactly those five directions and zero burst failures, and the adversary's exhaustive check to weight 12000 found no fifth f_3 direction. Witness: gasket-ray-machine, lab/py/gasket-witness-weights.
2026-08-31 Proved A two-valued potential read off the out-degrees replaces the exact linear solve: pi = 1 where a live state branches, phi^-1 where it does not, pi(0) = 1, is a super-solution of the golden criterion whenever no branch state has two branching successors, and sweeping it under the same operator gives a decreasing chain of exact Q(sqrt5) upper bounds on U. It settles 849 of the 865 occupied shelf directions - least sweep depth 1 on 760, 3 on 48, 4 on 31, 5 on 7, 6 on 3 - and 1966 of the lab's 1995, reaching 37 and 66 directions outside the branch case. What is left is the 7 shift rays and (1,756), (1,2196), (1,2214), (1,2268), (1,2430), (13,1080), (27,730), (28,729), (40,1053). Refutation attempt: an earlier depth split of 760, 48, 31, 10 was wrong because the sweep skipped depth 5 and the expected tuple had been fitted to that grid, a circular self-check that stayed green; the sweep now runs consecutive depths and the split is the least depth that works. Witness: gasket-ray-machine, lab/py/gasket-witness-weights.
2026-08-31 Proved The golden partition bound restated twice with no automaton in it: Sum_n (M_n(z) + 1) phi^-n <= phi^4 = 3 phi + 2, and equivalently Sum_m phi^-l(m) <= phi over the multipliers m of z, where l(m) is the number of base-3 digits of (z_1 + z_2) m. And the obstruction beyond v_3(q) = 1 is now exact rather than heuristic: the burst forces phi^-2 >= pi(c_0) >= phi^-(k-1) Sum_m pi(q_1 m) over 2^(k-1) burst-floor states of valuation 0 while pi(p) >= phi^-1 at the valuation-0 state p, so any valid potential must separate states of equal valuation by phi^2 (2/phi)^(k-1), which grows without bound - no potential constant on the level sets of v_3, and none constant on the out-degree classes, can work once k >= 2. Refutation attempt: both restatements checked exactly against the linear solve on 111 directions, and the burst identity together with u(p) = phi^-1 checked exactly on all 865 shelf and 1995 lab occupied directions. Witness: gasket-ray-machine, lab/py/gasket-witness-weights.
2026-08-31 Conjecture Conjecture Z, E(n)/3^n -> 2 exactly for the second moment E(n) = Sum_y M_n(y)^2, written E to keep it clear of the lane's ray count Z_F(n): E = T + S + R with T = 3^n - 2^(n+1) + 1 and S = 3^n - 4*2^n + 2n + 3 in closed form, so the constant 2 is exact before measurement and only R is open; R at n = 8..16 grows at about 2.54 per level, below phi^2 = 2.618, so R/phi^(2n) peaks at 3.238 at n = 12 and decays thereafter, three generators agree to n = 18, and no counterexample is known; the limit 2 has no proof and the residual's exact rate is undecided. Witness: lemma-b-pincer.
2026-08-31 Proved The shift-ray family of the gasket second moment is closed in exact form: M_n(3^j,1) = M_n(1,3^j) = prod_(r<j) F(m_r+2) - 1 with m_r = #{i in [0,n-j) : i == r mod j}, because z(3^j,1) in G_n says exactly that z is a binary string of length n-j with no two ones at distance j, which factors into j independent no-two-adjacent chains; from F(k) <= 2 phi^(k-3) for k >= 2, a two-step induction with equality at k = 3, follow M_n(3^j,1) < (3-sqrt5)^j phi^n and Sh(n) < ((4+12 sqrt5)/11) phi^(2n) < 2.803 phi^(2n) at every level, with Sh(n)/phi^(2n) -> (13+5 sqrt5)/11 = 2.198212717 by dominated convergence; the break attempt ran the family in exact Z[sqrt5] arithmetic to n = 160 and against literal enumeration of all 3^n points to n = 12, where the shift-ray share matched the closed form at every level, and an independent re-enumeration reproduced the closed form with no mismatch to n = 13 at every j and the constants to 80 digits. Witness: gasket-ray-machine.
2026-08-31 Proved The gasket ray mass laws are theorems at every level, not checks to a finite range: the live carry automata of (3,1), (1,12) and (7,3) have 2, 3 and 4 states with characteristic polynomials x^2-x-1, x^3-x^2-1 and x^4-x^3-1, so Cayley-Hamilton gives each recurrence and the first 2, 3 and 4 return counts pin it; the break attempt exhibited every reachable state by hand and by script, found the dead state -1 at (7,3) that makes reachable 5 against live 4, and confirmed the annihilator residuals vanish well beyond the automaton order. Witness: gasket-ray-machine.
2026-08-31 Proved The shelf pair automaton is not the multiplier-decomposition summand: A(s,t) counts #{z in G_n : sz, tz in G_n} while Q_n(s,t) in E(n) = T(n) + Sum Q_n(s,t) counts #{z : sz, tz in G_n}, and min(s,t) = 1 forces the two to agree, since s = 1 makes the first carry stay zero and drives every admissible digit into G; an exhaustive census at n = 9 of all 33552 ordered off-diagonal collinear pairs shows 482 of the 2656 active ordered multiplier pairs disagree, 2540 pairs (7.57%) having a witness off the gasket, the extremes (41,122), (122,41), (122,123), (123,122) with 50 witnesses each and none inside; independent re-enumeration reproduced the census from scratch. Witness: gasket-ray-machine.
2026-08-31 Verified The spectral gap survives that correction but its ceiling below 2 does not: the free-digit automaton B(s,t), reading z over all of {0,1,2}^2, has radius 3 on exactly (1,3), (1,9), (1,27), nothing in (2,3), and exactly 2 on the same twenty pairs over all 829 coprime pairs with max(s,t) <= 52, by the same exact Faddeev-LeVerrier charpoly and nonnegative-shift certificates; its largest radius strictly below 2 is 1.8488475886485 on (4,13), (4,39), (12,13), (13,36), against the A ceiling theta = 1.6956207695598, the real root of x^3 - x^2 - 2, which 44 strictly-sub-2 pairs reach or beat in the sharp split 19 strictly above theta and 25 exactly at it, the latter carrying x^3 - x^2 - 2 as a charpoly factor and the former never, with live sets reaching 167 states at both (25,52) and (31,40) against 33, so the claim that B matches A item for item is Refuted; the eigenvalue-free witness is B(4,13) having 4583352807133551 closed paths at n = 60 against 1.6956207695598^60 < 5.76e13. Witness: gasket-ray-machine.
2026-08-31 Verified The n = 9 active-pair census, previously claimed with no generator on disk, is 2656 ordered coprime multiplier pairs and 1328 unordered, 14 ordered of them the shift pairs (1,3^j) and (3^j,1) for j = 1..7; E(9) = 52212, T(9) = 18660, S(9) = 17656, R(9) = 15896, on 12170 occupied non-fibre rays carrying 18660 points, largest multiplier 2460 = floor(3^9/8), and A(s,t) return counts agree with brute force on all 1328 unordered pairs; E(n) and R(n) are regenerated for n = 1..12, filling the skipped levels R(9) = 15896 and R(11) = 124928, and an independent re-enumeration reproduced both lists. Witness: gasket-ray-machine.
2026-08-31 Conjecture The blocking lemma for the Pair Census Bound is Sum over non-shift primitive rays M_n(a,b)^2 = o(3^n): the shift rays are closed at ((13+5 sqrt5)/11) phi^(2n) and carry between 0.65 and 0.84 of R(n) at n = 6..12, so between a sixth and 0.35 of R is untouched; on the multiplier side that residue is a sum over non-shift active pairs numbering 10, 30, 106, 332, 1010, 2642, 7564, 20934, 57858 at n = 4..12, growth about 2.77 a level with largest multiplier exactly floor(3^n/8), so lambda <= 2 per pair buys nothing without a constant C(s,t) summable against that count. That per-pair summability is now Proved dead by construction and the door is restated in the witness coordinate as Conjecture W sharp, R(n) = O(phi^(2n)), with the weight-four orbit and the shift family both closed and only summability over the witness weight owed. Witness: gasket-ray-machine.

Half-ball chords

2026-08-28 Verified The half-disk chord constant (Zerr) decomposes into an integer and an area: P(the chord through two uniform points of the upper unit half-disk crosses the diameter) = I_diam/(3 Area(H)^2) by Blaschke-Petkantschin, the flat-face chord-cube integral is the integer I_diam = 4 and Area(H)^2 = Pi^2/4, giving 16/(3 Pi^2); the whole computation collapses to Integral_(-1)^(1) (-a u + sqrt(1 - a^2 + a^2 u^2))^3 du = 2 for every a, the even part of the cube being the exact derivative d/du [u (1 - a^2 + a^2 u^2)^(3/2)] with R(+-1) = 1, checked by the exact derivative, by differentiation under the integral in a, at 50 digits on 50 values of a, and with a symbolic residual of exactly zero at every step. Witness: lab/py/half-ball-mismatch.
2026-08-28 Conjecture The mismatch theorem: a design's coprime density equals a half-ball flat-face probability at dim = d = 2 and nowhere else, since design densities are rational multiples of 1/zeta(dim) while Version level gives rational/Pi^2 at even d and a pure rational at odd d and Version H gives Q + Q/Pi^2 at even d and Q + Q Pi at odd d, so even dim >= 4 is blocked by Lindemann and dim = 3 against Version level by Apery, both unconditionally; dim = 3 against Version H is conditional on zeta(3) not being algebraic over Q(Pi), odd dim >= 5 on zeta(dim) irrational (Rivoal and Zudilin give it only for infinitely many odd dim), and the dim = 2 uniqueness half is numerical, 11 base-2 and 502 base-3 designs against every Version level value to d = 24, base-2 numerators 4, 16/3, 6, 8, exactly one match, 16/3 at d = 2 carried by 3 designs. Witness: lab/py/half-ball-mismatch; coprime-density-above-dimension-one; A395134.
2026-08-28 Conjecture The Version H half-ball family, d uniform points and the hyperplane through them, has exact values 4 - 19845 Pi/16384 at d = 3, 4 - 549978112/(14189175 Pi^2) at d = 4 and 16 - 178919214166875 Pi/35184372088832 at d = 5, so odd d carries Pi^1 where Version L carries a pure rational, by an unoriented-normal Blaschke-Petkantschin reduction integrated in closed form, quadrature at 60 against 80 digits agreeing to 2.3e-62, 7.2e-64 and 1.5e-63, and an independent 10^8-sample random-point estimate whose deviations 2.39e-6, -6.74e-6, -3.55e-6 sit inside one sigma of 3.96e-5, 2.6e-5, 1.54e-5; d = 6 and d = 7 are exact too, 16 - 10363195833496113250304/(65656392092180764875 Pi^2) = 0.0074784083 and 64 - 403492347953923610203877211975 Pi/19807040628566084398385987584 = 0.0021206659, so the parity law "even d gives Q + Q/Pi^2, odd d gives Q + Q Pi" rests on six terms and no proof. Witness: lab/py/half-ball-mismatch.
2026-08-28 Conjecture No Euclidean body reproduces the bracket: a sweep of composite bases finds B(F) taking 1/2, 5/8, 3/4, 7/8, 1 across twenty-five base-4 designs of identical dimension 1.5, so B(F) is where a design's geometry lives, but no body whose chord-power integral reproduces B(F) was found, and the mismatch theorem makes the search futile above dim = 2; a failed search, not a proof of nonexistence, and the base-4 sweep has no generator in lab/. Witness: lab/py/half-ball-mismatch for the futility only.

Hexagram provenance

2026-08-28 Verified The hexagon-triangle substitution matrix [[6,1],[6,3]] (an encoding of published prose, "replace each hexagon with 6 hexagons and 6 triangles, and replace each triangle with 1 hexagon and 3 triangles", not a published matrix) iterated from (1,0) gives 1, 6, 42, 306, 2250, 16578, the A299916 row term for term, by matrix iteration and by the (9,-12) recurrence alike; one side counts tiles and the other holes, so the shared row is not by itself an identity of objects. Witness: A299916; slice-recurrence-order.
2026-08-28 Conjecture No peer-reviewed source states the hexagram count: the upstream is a photograph, a video, three blog posts and one OEIS comment, and the one peer-reviewed item upstream, the Bridges proceedings paper on three-dimensional diagonal cross-sections of four-dimensional sponges, states the cut and the star-of-David holes but gives no count and no sequence (its full text searched for hexagram, A299916, 306, 2250 and 1.8184).
2026-08-28 Conjecture The hexagram slice is not scooped at any base but 3: the adjacent published work generalises the cut along dimension n and a type-k hole parameter M^n_k or draws it on a different solid, fixed to the base-three expansion throughout, and none of it treats base 5, 7 or 9; the sequel on pentagon, dodecahedron and 120-cell slices frames the Menger slice as a two-tile closed fractal family and names the directed-graph iterated function system, the same structural reading at base = 3, so the correct statement is "generalises along dimension and hole type rather than base".
2026-08-28 Conjecture The seed's anchor is self-confirming and cannot falsify anything: "regenerate the A299916 row from two independent generators" holds whether or not the slice's object is A299916's object, since two generators of the same tile census agree by construction (matrix iteration and the (9,-12) recurrence agree exactly so); the real cold check is reproducing the matrix [[6,1],[6,3]] from the published prose.

Integer census and avatars

2026-08-28 Conjecture No geometric observable beyond dimension tracks Robin's inequality along the colossally abundant numbers: over the first 50 the Robin ratio is governed by dim alone (Spearman rho = 0.9906 on indices 13-50, adjusted p = 2.93e-32) while the fill polynomial adds nothing (normalized-fill coefficient 0.0190, p = 0.649, AIC worsening from -116.34 to -114.56), because every exponent equal to 1 contributes zero to (a_i - 1), so a new largest prime raises dim and doubles k while fixing every nonzero coefficient, the maximum nonzero degree being 6 while dim reaches 34; 38 dependent points over 6 <= dim <= 34 are a corridor, the fit 1 - R = 0.03611 exp(-0.03089 dim) at R2 = 0.960 is descriptive only, the ratio is not monotone (minimum 0.964531 at index 13, n = 21621600, 5 of 37 later transitions non-increasing), and nothing here bears on the Riemann hypothesis.
2026-08-28 Conjecture 10 is the smallest positive integer that never occurs as a base-2 design fill count at side number 2, the fill counts through 1000 being 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 125, 128, 216, 243, 256, 343, 512, 625, 729; fill 10 does occur at side 4 for a base-3 2D design; the catalog behind the original count had no generator.
2026-08-28 Refuted The dim = 4 integer census has 350 qualifying signatures among 65536 designs - there are 12 distinct qualifying signatures, A000070(4), realized by 504 oriented designs of which 503 have k >= 2, meeting 33 full B_4 classes; 350 is none of 12, 504, 503, 402 or 33 and has no recoverable definition. Witness: divisor-avatars, A000070.
2026-08-28 Refuted The census lock predicate identifies 13 designs - it identifies 14: 9 on the P5 clause (62, 94, 110, 118, 122, 124, 188, 218, 230) and 5 on the edgeless clause (128, 134, 146, 148, 150), the recount to 13 dropping code 128, F = {111}, origin-free, edgeless, containing 111, of size 1. Witness: lab/py/fill-polynomials.

Kronecker word order

2026-08-28 Verified Boundary is order-blind at word length 2 and order-sensitive from length 3: on the 4 x 4 grid only the four central cells can be interior and their requirements pair up under the factor swap, (S_1, S_4) against (S_4, S_1) and (S_2, S_3) against (S_3, S_2), so interior and hence boundary is symmetric in the two factors, exact on all 256 code pairs with the interior formula matching direct computation on all 256; at length 3 it fails on 36 of 210 multisets over the ten-code library of every code of fill 2 or 3. Witness: lab/rs/magic-words.
2026-08-28 Verified Connected components of a mixed Kronecker word are order-sensitive, with minimal witness the multiset {3, 6}: comp(A_3 (x) A_6) = 4 against comp(A_6 (x) A_3) = 2, both factors of fill 2, because the inner tile's contacts decide whether adjacent outer copies merge (A_6 is two isolated cells with no boundary contact, so four cells stay apart, while A_3 is a connected vertical pair sitting in two non-adjacent outer copies), enumerated by two independently written renderers, the study's substitution pass and the crate's Kronecker factory, agreeing cell for cell. Witness: lab/rs/magic-words.
2026-08-28 Proved The fill = 2 designs at base 2, dim = 2 split into adjacent (codes 3, 5, 10, 12) and diagonal (codes 6, 9), and commutativity of the component count follows the split: adjacent times adjacent commutes, diagonal times diagonal gives 4 in both orders, and adjacent times diagonal never commutes, always 4 against 2, by contact geometry, the diagonal pair having neither a face-adjacent cell nor a contact in either direction, and on all 15 pairs among the six codes with zero violations. Witness: connectivity.md, lab/rs/magic-words.
2026-08-28 Verified Any two designs with fill >= 3 at base 2, dim = 2 commute and give exactly one component in either order, since every such tile is connected and carries both a top-bottom and a left-right contact, so every pair of adjacent outer copies merges through the inner tile and connectedness of the outer tile merges them all; exhaustive over the four fill = 3 tiles plus the full tile, all 10 pairs among codes 7, 11, 13, 14, 15. Witness: lab/rs/magic-words.
2026-08-28 Proved Block reduction: every periodic schedule (c_1, ..., c_p)^level equals the level-fold self-similar product of its one-period composite tile A_(c_1) (x) ... (x) A_(c_p), of base prod_i side_i and fill prod_i fill_i, by associativity of the Kronecker product alone, so periodic mixed words carry no new theory and the first genuinely non-stationary behaviour requires an aperiodic word; cell for cell on six test cases at periods 2 and 3 and lengths to 6, the study's flat rendering against Tensor::fractal of the composite. Witness: dimensions.md, lab/rs/magic-words.
2026-08-28 Verified Contact counts of a mixed Kronecker word are exactly multiplicative, h(A_w) = prod_i h(A_(c_i)) and v(A_w) = prod_i v(A_(c_i)) for the row and column contact counts, because L(A (x) B) = L(A) (x) L(B) on outer columns and rows and the inner product of Kronecker products is the product of inner products, so whether adjacent copies touch is order-blind and decided factor by factor even where the component count is not; induction on length, exact on all 15^3 words with zero mismatches, the correct strengthening of the length-2 boundary theorem. Witness: lab/rs/magic-words.
2026-08-28 Verified The naive geometric transfer state for mixed products is unbounded: kappa(A_w), the number of components meeting a contact position and so still able to merge with a neighbouring copy, reaches 2^(level-1) on the family w_level = (15^(level-1), 3), whose product is 2^(level-1) disjoint full-width rows each meeting both the left and right column, from the full-tile Kronecker power, exact at level = 2..10, with exhaustive search over all 15 codes showing w_level is a maximiser at level <= 4 with maxima 1, 2, 4, 8; the same family's component count is exactly 2^(level-1) too. Witness: lab/rs/magic-words.
2026-08-28 Verified Order sensitivity of mixed products is exactly the noncommutation of the cocycle matrices: 14 of the 15 pairs of component-matrix classes fail to commute, the only commuting pair being the two zero-contact classes, fill = 1 and the diagonal pairs, whose matrices have rank 1 with M_(6,9) = 2 M_(1,2,4,8); the vertical and horizontal domino classes carry different matrices, so the representation sees more than the square's full symmetry group; checked on all 15 class pairs. Witness: lab/rs/magic-words.
2026-08-31 Proved The scale dimension of a magic word over a finite alphabet is a frequency functional: log side and log fill of a prefix are sums of per-letter values from a finite set, so whenever each letter's frequency exists the ratio log fill / log side converges to the frequency-weighted average, and in particular every uniquely ergodic word (Thue-Morse, period-doubling, Fibonacci) has a scale dimension equal to that average; open are finite alphabets without letter frequencies and all unbounded alphabets, where frequencies can exist while the dimension oscillates. Witness: magic.md.
2026-08-31 Proved The fill assumption lab/py/slice-ladder-controls states before printing its five staircase dimensions is discharged: the parity-carpet code at odd side side fills E^2 + 2EO = side^2 - ((side-1)/2)^2 with E = (side+1)/2 and O = (side-1)/2, the octagonal fill 3k^2 - 2k at side = 2k - 1 already on this ledger for the same rule, so the staircase dimensions stand without the assumption. Witness: magic.md, sequences.md.
2026-08-31 Proved The hyperoctahedral group acts diagonally through the Kronecker product, g . (A (x) B) = (g . A) (x) (g . B), because reflecting a mixed-radix coordinate reflects every digit at once, so a magic word canonicalises under one shared symmetry applied to all letters and never letter by letter. Witness: magic.md.
2026-08-31 Proved Every value in [0, log 8 / log 3] is the scale dimension of some word over the two letters carpet(3) and c8(3): rational carpet frequencies by periodic words, irrational by Sturmian words, endpoints by constant words, all through the frequency functional. Witness: magic.md.
2026-09-06 Proved The component count of a mixed Kronecker word is a rational series of Hankel rank 4: with lambda = (1,0,0,0), gamma = (1,1,1,1)^T and one 4 x 4 integer matrix per code in six classes, comp(A_w) = lambda M_(c_1) ... M_(c_level) gamma at every word, by the four transfer laws in the observable frame (gamma, h, v, phi) and induction from (comp, H, V, fill)(A_e) = (1,1,1,1); the same induction reads H, V and the fill off the same matrices. Witness: connectivity.md THE JOINT SPECTRAL RADIUS OF THE COCYCLE, lab/py/jsr-schedules.
2026-09-06 Verified Euler characteristic, boundary and holes of a mixed Kronecker word are rational series in the word, of Hankel rank 4, 8 and 11: there are lambda, gamma and one matrix per code with phi(A_w) = lambda M_(c_1) ... M_(c_level) gamma, exhaustive on all 54240 words of length at most 4 over the 15 non-empty codes plus 120 seeded words of length 5 to 7, all four observables, zero mismatches, built by Hankel-basis elimination in exact rational arithmetic; the rank is unexplained, staying 4 while the geometric state grows like 2^(level-1) and the component count reaches 2 * 4^(level-1) on (15^(level-1), 6), the checkerboard and the largest component count any subset of the 2^level grid can carry. Witness: lab/rs/magic-words.
2026-09-19 Proved Only a trailing solid letter is a magnification: A (x) full(n) replaces each filled cell of A by a solid block while full(n) (x) A lays n^dim copies of A in a grid, so c15(2), carpet(3) and carpet(3), c15(2) differ cell for cell at equal side and equal fill. Witness: magic.md, mrlymath::bang::magic.
2026-09-19 Proved A count along a periodic word satisfies a linear recurrence, being a fixed vector times the powers of one integer matrix in the rank-4 representation, so a match against a table of sequences is a recurrence and never an arithmetic fact. Witness: lab/rs/magic-words, connectivity.md.

Kronecker words

2026-08-28 Proved Fill, side, density and the main-diagonal count of a mixed Kronecker word are order-blind at every word length, the first three as products of per-factor quantities and the diagonal by diag(A (x) B) = diag(A) (x) diag(B), exhaustive on all 15^3 words of length 3 at base 2, dim = 2; the whole anti-diagonal profile is order-sensitive on 99 of 105 multisets at length 2 over the 15 non-empty codes and on 204 of 210 at length 3 over the ten-code library of every code of fill 2 or 3, its peak on 23 of 105 and its support on 27 of 105, minimal witness the one-cell codes 1 and 2 with profiles (0,1,0,0,0,0,0) against (0,0,1,0,0,0,0); take the whole profile, never one coefficient. An earlier length-3 reading of 110 of 112 is withdrawn, not confirmed: 112 is the multiset count of an eight-code library that was never recorded, so the figure names no sweep anyone can rerun. Witness: lab/rs/magic-words.
2026-09-19 Proved A decoration on a letter folds into a code: the fill swap renders the complementary code at every base and side, and at base 2 and odd side the half turn of the rendered tile is the identity and a quarter turn is the render of the transposed corner rule, because side - 1 is even. Witness: magic.md.
2026-09-19 Proved A unit letter is a property of the code and the side together: at odd side 2k - 1 a base-2 letter fills sum over its corners of k^(zeros) (k - 1)^(ones), so fill 1 forces one corner with every coordinate odd and k = 2, giving c8 as the only letter of fill 1 at side 3 and a smallest fill of 4 at side 5. Witness: mrlymath::formulas::counting, magic.md.
2026-09-19 Proved No periodic word carries irrational letter frequencies, a word of period p having every letter frequency in (1/p)Z, so at an irrational frequency vector the control for a schedule is the value of the frequency functional and never a periodic word. Witness: magic.md.
2026-09-19 Proved A mosaic whose palette is the empty cell and one design is a magic pair M (x) c, since the block at a mask site is the palette cell the mask value indexes, so a census over both families counts it once on the word side. Witness: mrlycore::cell::mosaic, mrlymath::bang::magic.
2026-09-19 Proved Magic meets special at a plain tiling and nowhere else: every mask site of a special carries a block, so a special is a magic pair exactly when all its blocks agree, which is full(f) (x) c at a constant mask or at a design c fixed by the quarter turn. Witness: mrlymath::two::geometry::special, mrlymath::bang::magic.
2026-09-19 Proved The fill of a mosaic is linear in the mask histogram, fill = sum_i h_i(M) fill(P_i) over disjoint blocks, collapsing to the product law of a word only at a palette of the empty cell and one design, so block reduction and the Kronecker law are given up the moment a palette holds two designs. Witness: mrlycore::cell::mosaic.
2026-09-19 Proved A special is fill-invariant, fill(special(M, c)) = f^dim fill(c) at a mask of side f in any dimension: an orientation permutes the entries of a cell, so every palette copy carries the fill of c, and the mask lays f^dim disjoint blocks. Witness: magic.md, spot-checked by mrlymath::two::geometry test special_rotations_preserve_sum and mrlymath::three::geometry test orientations_preserve_sum_and_shape.

Levels and designs

2026-09-09 Proved The product law: for 1-periodic f, g the mean of f(mx) g(nx) over the unit interval is sum_j hat f(j n') hat g(-j m') with g = gcd(m,n), m' = m/g, n' = n/g, and the covariance is the same sum over j != 0; the parity specialisation returns 1/15, 1/3, 1/3 at (3,5), (3,9), (5,15) and gcd^2/(mn) at all 820 pairs to 40. Witness: lab/py/stack-levels check_parity.
2026-09-09 Proved The base-3 carpet stack correlates 3-adically, not by gcd: the shadow at level level has hat f_level(k) = (-1)^k 2^level sin(pi k/3^level) prod_{i <= level} cos(2 pi k/3^i)/(pi k), zero exactly when 3^level | k, so Cov(f_level(mx), f_level(nx)) = G_level(m' mod 3^level, n' mod 3^level)/(m' n') with G_level(a, 3^level - b) = -G_level(a, b) and G_level(0, b) = 0; at level 1 the closed form is (2/9) chi(m') chi(n')/(m' n') with chi the character mod 3, exact at all 1600 ordered pairs to 40; the covariance vanishes whenever |v_3(m) - v_3(n)| >= level, and the 2D field carries the same zero set through Cov_2D = Cov_1D (M + (2/3)^(2 level)). Witness: lab/py/stack-levels carpet_law_level1, carpet_kernel.
2026-09-09 Proved Levels are products, not layers: f_level(x) = prod_{i < level} f_1(3^i x), so a layer at level level and scale n is the product of level-1 layers at n, 3n, ..., 3^(level-1) n, with f_level(nx) <= f_{level-1}(3nx) pointwise and gap measure (2/3)^(level-1)/3; level level-1 at scale 3n is not redundant against level level at scale n, Cov(f_2(x), f_1(3x)) = 4/27 against Cov(f_1(x), f_2(3x)) = 0. Witness: lab/py/stack-levels check_levels.
2026-09-09 Proved Coprime independence is the half period's: the 1-periodic odd strip p(x) = 1 iff floor(2x) odd carries covariance g^2/(4mn), nonzero at 159 of the 490 coprime pairs to 40 and 1/60 at (3, 5), while the tree's chi_n at odd n carries (g^2 - 1)/(4mn), zero at every coprime pair; p(nx) = chi_{2n}(x), so this is the landed law read at even scales, a sharpening and not a break, and the prime detector is a statement about odd scales under the half-period sampling. Witness: lab/py/stack-levels check_parity.
2026-09-09 Proved Design pairs at level 1: two base-3 one-digit-removed shadows correlate by c/(27 m' n') with c in -6, -3, 3, 6, zero exactly when 3 divides m' n', so no two base-3 designs are coprime-independent; a base-2 strip against a base-3 shadow correlates by c/(18 m' n') with c in -3, 0, 3, and the odd strip against the Sierpinski shadow is exactly zero at all 1600 scale pairs, because the centred middle-thirds indicator is even and the centred strip odd under x -> -x. Witness: lab/py/stack-levels design_law_33, design_law_23.
2026-09-09 Proved The parity layers are linearly independent: the Gram matrix gcd(m,n)^2/(mn) over odd m, n <= 2K + 1 has determinant prod_{k odd <= 2K+1} J_2(k)/k^2 = prod_{k odd} prod_{p | k} (1 - p^-2), from k^2 = sum_{d | k} J_2(d) and a unitriangular incidence factorisation on the factor-closed odd set, exact at K = 1..12, 11399736556781568/21994507608198125 at K = 12, positive. Witness: lab/py/stack-levels check_gram.
2026-09-09 Conjecture The carpet zero law is exact at every level, Cov(f_level(mx), f_level(nx)) = 0 iff 3^level | m' n', and the cross-level law Cov(f_level(mx), f_{level'}(nx)) = 0 iff 3^level | n' or 3^{level'} | m' likewise: Proved at level = 1 from the closed form, Verified only over m, n <= 40 and level <= 3, 14400 ordered triples with zero breaches and zero unpredicted zeros. Witness: lab/py/stack-levels carpet_kernel, check_levels.
2026-09-09 Refuted The gcd law and coprime independence hold outside the parity family: at base 3 level 1 coprime layers carry (2/9) chi(m') chi(n')/(m' n'), witness Cov(f_1(x), f_1(2x)) = -1/9, a coprime pair, anticorrelated; the zero set is v_3(m) != v_3(n), not coprimality. Witness: lab/py/stack-levels carpet_law_level1.
2026-09-09 Refuted The carpet kernel is separable above level 1: G_2(1,1) G_2(4,4) - G_2(1,4)^2 = 76/729, so no theta(a) theta(b) form exists at level >= 2 and the level-1 character law does not lift. Witness: lab/py/stack-levels check_carpet.

Matched random control

2026-08-28 Conjecture A self-similar slab is not a better acoustic barrier than an equally sparse random one: at levels 2 and 3 the matched random mask blocks more in every comparison, four filled-cell counts per level, 96 of 96 at 12 seeds per cell with margins of 4 to 31 sd (tree L3 fractal 0.3720 against a random mean 0.1446, sd 0.0074), every fractal value reproducing exactly (carpet L2 0.3120 / L3 0.2548, net 0.2999 / 0.2619, tree 0.3023 / 0.3720, void 0.3122 / 0.3642), so the suppression is carried by fineness and sparsity rather than by self-similarity; the repeat reused the same simulator and so tests seed-fragility rather than the physics.

Menger slice sources

2026-08-28 Verified Cook's 2011 slice code is a point-sampled raster, not an array-free predicate: the listing allocates a side x side integer array, fills it by sampling the digit predicate and renders it, so the exact-census method is this tree's and not Cook's; its prose puts the normal through (1, 1, 1) while the listing sets (1, 1, 0.5). Witness: REFS.md.
2026-08-28 Verified Hart 2012 is an exact citation for the diagonal Menger slice: the Simons Foundation page carries his byline, the title "Mathematical Impressions: The Surprising Menger Sponge Slice" and a photograph credit. Witness: REFS.md.
2026-08-28 Verified "The Marstrand value is the dimension minus one" is the wrong attribution for a plane cutting a solid in R^3: Marstrand 1954 concerns plane sets and lines in R^2, and the hyperplane generalisation is Mattila 1975; both are almost-everywhere statements, so one maximally arithmetic plane landing above or below that value contradicts neither. Witness: REFS.md.
2026-08-28 Verified A299916's Menger reading is not its definition: the entry's name is a(n) = A299914(2n+1), offset 0, signature (9, -12), terms 1, 6, 42, 306, 2250, 16578, its reference is number theory with no sponge in it, and the hexagram-hole geometry lives in one comment with one uploaded picture, which Wikipedia sources onward beside a newspaper article; it must be cited as a comment, never as the sequence's definition. Witness: A299916.
2026-08-28 Verified Abel 2012 claims no proof of the base-3 slice dimension, only a computation, naming mass distributions and similarity graphs as routes to one, so every generalised dimension inherits that status. Witness: REFS.md.

Mertens stack

2026-08-28 Verified Weighting the Farey stack by mu(n) renders a Mertens-type sum at every node, Sum_{k <= N/b} mu(kb) at a/b and M(N) at b = 1, and the log-space power spectrum of M(x)/sqrt(x) shows the first eight nontrivial zeta zeros, detected at 13.94, 20.90, 24.97, 30.19, 32.52, 37.74, 40.64, 42.97 against 14.1347, 21.0220, 25.0109, 30.4249, 32.9351, 37.5862, 40.9187, 43.3271, errors 0.04 to 0.42 inside one bin of width 0.5806; the zeros are known to far beyond any precision a moire can reach, so this is a rendering and not a measurement. Witness: lab/py/mertens-meter.

Moment ladder and Lemma B

2026-08-28 Conjecture The moment ladder's rows above the tenth approach the wall and stop: beta_0^(2K) reads 0.447838092, 0.447904613, 0.447923402, 0.447928788, 0.447930346 at 2K = 12, 14, 16, 18, 20 from Perron roots 59307.487289, 532101.317617, 4784678.13057, 43051182.4466, 387432198.159, the twentieth row lying only 6.42e-7 below the universal wall 2/(3 + log_3 5) = 0.447930987882; these are floating eigenvalues of exact integer matrices, not interval-certified, and the wall above them is separately settled and unaffected. Witness: lab/rs/dimension-one-ladder.
2026-08-31 Proved The energy cap E_2K(G_a) <= lambda_2K^a holds at every order with constant exactly 1, which is what turns a ladder rung from a growth rate into a master inequality: the carry box {-r,...,r}^2 with r = floor((K-1)/2) is closed because a digit difference lies in [-K, K] and floor((r+K)/3) <= r for every K >= 1, every walk from the zero state back to itself stays inside S, the strongly connected component of that state, M_S is irreducible by the definition of a component and carries a self-loop at the zero state, hence is primitive with Perron root lambda_2K and positive right eigenvector u, and e_0 <= u/u_0 componentwise with M_S >= 0 gives (M_S^a)_(0,0) <= lambda_2K^a; the attempt to break it looked for the constant C > 1 a reducible matrix would force and found none, since the reduction to the component is free and lambda_2K = lim E_2K(G_a)^(1/a) is the component's own root, checked equal to the full matrix's Perron root at 2K = 4, 6, 8, 10 with the component sizes 1, 7, 7, 19 inside 1, 9, 9, 25 states and the ratios E_2K(G_a)/lambda_2K^a falling monotonically to 1, 0.942327, 0.790590, 0.643725 at a = 6. Witness: lab/rs/dimension-one-ladder.
2026-08-31 Proved The master bound at order 2K is one formula for every rung: with a = floor(log_3(p/2)), d_K = ceil(log_3(K/2)), a digit window of n digits with n >= 2a and b = min(a - d_K, n - 2a), Hoelder over three blocks of the digit window of lengths a, a, b at exponents 4K/(2K-1), 4K/(2K-1), 2K gives L_n(p) <= p^2 3^(-((2K-2+kappa)a + kappa_2K b)/(2K)), the outer blocks interpolated between the exact L^2 and L^4 identities and the inner block supplied by the energy cap, both admissible since K 3^(a-d_K) <= 2*3^a <= p; the exponent gain exceeds a exactly when beta < kappa_2K/Lambda_2K with Lambda_2K = 2 - kappa + 2 kappa_2K, which is the rung formula, and the seam 3a against n is the same at every order with the two branches agreeing at n = 3a - d_K, so the feared order-10 crossing does not exist; the adversarial pass tried to break it by hunting a violation over every prime 5 <= p <= 199 and 2 <= n <= 24, running the order-10 block alone in its 833 applicable cases as well as the min over all orders, and found none, worst ratio 0.7839 at (p, n) = (11, 2) for the order-10 block alone and 0.8755 at (13, 4) for the min, and by hunting an uncovered or negative-gain case in the main range 3a > n over eta in 0.001..0.1 and n = 6..400, finding none, worst geometric sum over cap 0.8630. Witness: lab/rs/dimension-one-ladder.
2026-08-31 Proved The order-10 rung is unconditional and the exponent needs no root-finding to be trusted: with Lambda_10 = 4.436585106, main-range decay Lambda_10/10 = 0.443658511 and geometric constant 2/(1 - 3^(-Lambda_10/10)) = 5.1842 rounded to 6, the ladder reads Sum_(z < p <= 3^((beta_0^(10) - eta) n)) T*_p(n)/3^n <= 2/z + 35 z^(1-kappa) + 40 * 3^(-(kappa-1)n/8) + 6 * 3^(-0.443658511 eta n) for eta in (0, beta_0^(10)), z >= 5, n >= 1, the first three terms being the unchanged order-4 bookkeeping; a Sturm count on the exact quartic x^4 - 7833x^3 + 7916949x^2 - 850684437x + 13054946580 places no root above 66641136626/10^7 and exactly one root in the bracket of width 10^-7 below it, so lambda_10 < 6664.1136626, kappa_10 > 1.985805792698 and beta_0^(10) > 0.447597813453, every digit truncated down, never rounded, so the short form printed everywhere is 0.4475978; rows 12 through 20 stay Conjecture for a different reason, their lambda_2K being floating eigenvalues and not certified algebraic numbers, so the energy cap alone does not promote them. Witness: lab/rs/dimension-one-ladder.
2026-08-31 Proved The dimension-one moment ladder has a tenth rung and the lower wall is 0.4475978, not 0.446717: the exact 25-by-25 integer carry matrix M_10 on the box {-2,-1,0,1,2}^2 satisfies E_10(G_a) = (M_10^a)_((0,0),(0,0)) with first energies 1, 4653, 28967859, 190911254427, 1270015973323281, 8461182216374750493 (matched by direct convolution of the digit set at a = 1, 2, 3), its characteristic polynomial factors symbolically as x^6 (x-120)(x^2-450x+12231)(x^3-2190x^2+282096x-5186835)^2 (x^3-990x^2+116154x-2569725)^2 (x^4-7833x^3+7916949x^2-850684437x+13054946580), the Perron root is the largest root of the quartic lambda_10 = 6664.113662506, so kappa_10 = 1.985805792712 and beta_0^(10) = kappa_10/(2 kappa_10 + 2 - (3 - log_3 5)) = 0.447597813454, above the eighth rung by 0.000880502992, with Holder block exponents 20/9, 20/9, 10; "dimension one" means the similarity condition log(fill)/log(base) = 1 at base 3 and not the base-2 gasket of density 16/(3 Pi^2), and the order-10 three-block master bound with explicit constants and checked regime seams is now written, so the rung is a theorem and a re-proof of target-uniform Lemma B at that edge. Witness: lab/rs/dimension-one-ladder; lemma-b-pincer.

Odd-base slice grammar

2026-08-28 Verified The odd-base generalisation rests on a choice of solid: Cook's predicate "at most one coordinate in the middle third" and the bang dim 3, code 23 rule "at most one odd coordinate" agree at base = 3 (20 of 27) and nowhere else, at base = 5 filling 4^3 + 3 * 4^2 = 112 of 125 against 3^3 + 3 * 2 * 3^2 = 81 = 4k^3 - 3k^2 at k = 3, and there is no canonical base-5 Menger sponge. Witness: lab/py/odd-base-slice-grammar.
2026-08-28 Verified In the comparison of the slice dimension against the dimension minus one, the dimension is the solid's own log(fill)/log(base) and never the ambient 3: at the ambient 3 the value 3 - 1 = 2 exceeds all four slice dimensions 1.8184, 1.6869, 1.8026, 1.7204 and the mod-4 split collapses. Witness: lab/py/odd-base-slice-grammar.
2026-08-28 Verified The four printed dimensions are consistent with the four printed rules and this is not evidence for either: the dominant roots (9 + sqrt(33))/2 = 7.37228, (11 + sqrt(369))/2 = 15.1047, (42 + sqrt(612))/2 = 33.3693, (28 + sqrt(3556))/2 = 43.8161 give log(root)/log(base) of 1.8183, 1.6870, 1.8026, 1.7204 at base = 3, 5, 7, 9, while 4k^3 - 3k^2 at k = 2..5 gives 20, 81, 208, 425 and dimension minus one 1.7268, 1.7304, 1.7430, 1.7544; a rule and its own dimension cannot cross-check each other. Witness: lab/py/odd-base-slice-grammar.
2026-08-28 Proved The middle diagonal layer sits at coordinate sum 3(base-1)/2, odd exactly when base = 3 mod 4 (3, 6, 9, 12 at base = 3, 5, 7, 9). Witness: lab/py/odd-base-slice-grammar.
2026-08-28 Conjecture No definition of a "blow-up of 4" for the slice exists in this tree, so the phrase carries no claim.
2026-08-28 Conjecture The two-tile grammar closes at ten odd bases with a 2 x 2 substitution matrix rational in base within each class of base mod 4.
2026-08-28 Conjecture That parity forces structurally different cells into the middle layer in each residue class, which is the mechanism of the mod-4 split.

Odd-side fills

2026-08-30 Proved At odd side n = 2k - 1 the residue split of an axis has k low positions and k - 1 high, so a base-2 flat design fills sum over its corners of k^(zeros) (k - 1)^(ones), and the six designs of the plane read as the polygonal numbers in k: low corner k^2 (A000290), tree k(2k - 1) hexagonal (A000384), carpet k(3k - 2) octagonal (A000567), void 2k^2 - 2k + 1 centered square (A001844), corner and centre 3k^2 - 3k + 1 centered hexagonal (A003215), solid (2k - 1)^2 odd squares (A016754); two_census at sides 3 to 11 returns 8, 21, 40, 65, 96 for the carpet and 6, 15, 28, 45, 66 for the tree. Witness: mrlymath::formulas::counting fill polynomial, mrlydemo two_census, A000567, A000384.

Primes on a design

2026-09-06 Proved Primes on a design at fill = base^dim - 1 are primes with one restricted digit at base base^dim: the Morton code x -> Sum_j (Sum_c base^(c-1) x_(c,j)) base^(Dj) maps S_level bijectively onto the integers at base base^dim whose digits lie in the image of the digit set F, the gasket base 4 missing 3 and the carpet base 9 missing 4; the gcd-prime reading is a positive-density count when B(F) > 0 and empty otherwise, base 32 on {0,4,...,28}^2 having fill = 64 > 32, (E), and every gcd divisible by 4; the x_1-prime reading is a sum-of-digits large deviation. Witness: coprime.md PRIMES ON A DESIGN.
2026-09-06 Proved Lemma A' the window rate: for (E), gcd(d,base) = 1 and nonzero t in (Z/d)^dim, Prod_(l<level) f_l(t) <= c(base,fill)^floor(level/m_d) with m_d = max(1, floor(log(d/2)/log(base)) + 1), and ord_d(base) >= m_d so it is never weaker than Lemma A; with a base part e and t nonzero mod the coprime part m, and |eta|_inf < base^(-2n/3)/(4 base dim (base-1)), the rate is c'(base,fill)^floor(2n/(3 m_d)), which is Maynard Lemma 8.2 in every dimension with an explicit constant and no consecutive-digit hypothesis; the hypothesis on t is sharp, the gasket at d = 6 and t = (3,0) sitting at 1/3 at every level. Witness: lab/py/digit-transform-norms lemma, worst per-digit rate 0.830915 at d = 257 over d <= 301 against Lemma A's 0.986514.
2026-09-06 Proved The 2D Type I saves a power when alpha_1* < dim/2, the dyadic block d ~ Q_1 costing fill^level (Q_1^(2 alpha_1* - dim) + Q_1^dim base^(level(alpha_1* - dim))) and the small moduli going to Lemma A': the carpet certified at alpha_1* < 0.8124 gives Sum_(d <= Q, gcd(d,3) = 1) |#{x in S_level : d | x} - fill^level/d^2| <<_A fill^level level^(-A) at Q = 3^(0.5938 level) level^(-C), and the gasket certified at alpha_1^- >= 1.0126, alpha_1^+ <= 1.1022 closes the route, min_x Sigma_2 > 4.059204 against 4 and min_x Sigma_3 > 8.213932 against 8 in interval arithmetic with directed rounding, Sigma_2(0) = (8 + 2 sqrt(5))/3 exactly. Witness: lab/py/digit-transform-norms certify.
2026-09-06 Verified The carpet misses the one-dimensional criterion at every order: base 9 missing 4 has g(1) = 0.3437 below 27/77 but g(3/2) = 0.1531, g(235/154) = 0.1457, g(1.6) = 0.1262, g(1.7) = 0.1031, g(1.8) = 0.0835 against 0.1473, 0.1397, 0.1179, 0.0884, 0.0589, a gap of 0.0058 on the printed pair at s = 3/2 and 0.005749 in full, and g(3/2) moves 0.154389, 0.153068, 0.152921 over four, five and six digit-vectors, so windows do not close it. Witness: lab/py/digit-transform-norms moments.
2026-09-06 Verified The gasket is out of reach at both numbers: base 4 missing 3 has g(1) = 0.4820 against 27/77 and g(235/154) = 0.3170 against 59/433, so no Type II range opens at any order computed. Witness: lab/py/digit-transform-norms moments.
2026-09-06 Verified The componentwise route is closed at source: Chow, Varju and Yu Remark 6.1 puts the Fourier l^1 dimension below 1/2 for (b,a) in {(3,0),(3,1),(3,2),(4,1),(4,2)} by interval arithmetic at level = 2, so the base-3 design's coordinate marginals fall on the wrong side, while Proposition 2.4 puts base 4 missing 3, the base-2 gasket's Morton code, above 1/2. Witness: arXiv:2402.18395v2 pp.25-26.
2026-09-07 Verified The missing-digit criterion is unreachable for the carpet at every order: dividing by 2 - s the criterion is the single inequality g(s)/(2 - s) < (1/5)*(1 + c/2) on the transform's moment exponents, and for base 9 missing 4 the shift sandwich at a power certifies g(3/2) > 0.149397 and g(235/154) > 0.142274 against the required 0.147320 and 0.139667, with a monotone chain of orders anchored at the exact Sigma_N^(2)(x) = (9/8)^N covering [3/2, 2) in 21 closed cells sharing endpoints and [1, 2) in 87; the pointwise deficit is at least 0.001268 over [3/2, 2) and the decisive cell re-derived independently at N = 5 clears by 0.000840. Witness: lab/py/digit-transform-norms criterion, with an independent recomputation by a digit-tree fold reproducing every printed digit.
2026-09-07 Proved The two-missing-digit transform is (base-2)^2 abs(hat F)^2 = K^2 + 2 + 2 cos(2 pi D t) - 4 K cos(pi S t) cos(pi D t) with K = sin(base pi t)/sin(pi t), D = a - c, S = a + c - (base-1), so a pair enters only through abs(D) and abs(S); that implication does not run backwards, {0,2} and {0,8} at base = 10 reading (2,7) and (8,1) with equal transforms, and the collapse is generated instead by the reflection d -> base-1-d, which flips both signs, together with the integer translation of F available exactly when 0 or base-1 is excluded and identifying {0,c} with {0,base-c}, so the edge family is the one-missing-digit sets of a (base-1)-digit interval read at base base and the number of distinct transforms is (C(base-2,2) + floor((base-2)/2))/2 + floor(base/2). Witness: coprime.md PRIMES ON A DESIGN.
2026-09-07 Verified That pair count reads 7, 16, 21, 31 of the 15, 36, 45, 66 excluded pairs at base = 6, 9, 10, 12, is reproduced by grouping every one of the C(base,2) pairs by its sampled transform at every base 4 <= base <= 41, and sums to 2373 distinct sets over 4 <= base <= 31. Witness: lab/py/digit-transform-norms pairs.
2026-09-07 Verified The least base carrying a certified two-missing-digit set with alpha_1 < 1/4 is base = 32 at the interval class {0,1}, alpha_1 in [0.2499087, 0.2499779] at four window digits, the same class at base = 31 reading [0.2518967, 0.2519717]; over 4 <= base <= 31 the machine certifies alpha_1 > 1/4 at 2363 of the 2373 distinct sets, closest base = 26 missing {2,23} at > 0.2502919, and the ten it cannot bracket from below all have S = 0 or D = base/2 with base/2 odd, a shared shape and not a cause since the clearing headline base = 32 missing {0,1} has a transform vanishing at all 29 points t = j/30, with certified upper bounds 0.2538899 to 0.2826357, above 1/4. Witness: lab/py/digit-transform-norms pairs and pairfail.
2026-09-07 Verified Against the bar 1/3 a two-missing-digit set first clears at base = 13, the interval class at alpha_1 < 0.3318819 on three window digits with base = 12 above at all 31 of its sets to five, and the whole pair family clears from base = 21 on through base = 26, worst base = 23 missing {4,5} at < 0.3333284, every base 4 <= base <= 20 carrying a certified witness above 1/3, base = 20 by {3,11} at [0.3356579, 0.3356674]. Witness: lab/py/digit-transform-norms pairs pairclear pairsome.
2026-09-07 Proved The digit-uniform bound holds at any excluded-digit count: abs(hat F(t)) <= (abs(sin(base pi t)/sin(pi t)) + m)/(base - m), the level product expands with weight m^(N - card E) and telescopes to the same Dirichlet kernels, so a_N = m a_(N-1) + m Sum_(l<N) lambda_l a_(N-1-l) + lambda_N and the growth root solves (z - m)(z - 1)^2 = m(c_1 (log base) z + gamma'(z - 1) + c_1 (z-1)^2/(base z - 1)) on the exact Lebesgue input lambda_l <= c_1 l log base + gamma' + c_1 base^(-l), gamma' = (2/pi)(gamma + log(8/pi)), giving alpha_1 < 1/4 for every base >= 649 at m = 2 with the chain failing at 648, and 125 at m = 1 and 1873 at m = 3, with 32, 105, 230 against 1/3, certified at 120 bits; the coarser c_0 = 0.97 form of the same chain needs base^l >= 86 and gives 126 at m = 1. Witness: lab/py/digit-uniform-bound pairs.
2026-09-07 Proved The threshold 1/4 is the Mertens bar: in the GRH chain steps 1, 2, 4 and 5 never name the digit set and only step 3 substitutes a digit-free kernel bound, so feeding the certified l^1 exponent there gives abs(M_F(x)) <<_(base,eps) A_F(x) x^(alpha_1 - 1/4 + eps), that is A_F(x)^(1 - delta + eps) with delta = (1/4 - alpha_1)/alpha_base > 0, and 1 - b(a) in place of 1/4 under a zero-free half plane; steps 2 and 3 alone force alpha_1 <= 1 - alpha_base + c_base and gap_base(1) > 0 is exactly 1 - alpha_base + c_base < 1/4, so the old certificate implies the new condition and the wall can only fall. Witness: coprime.md PRIMES ON A DESIGN.
2026-09-07 Verified That wall falls from 3690 to 34, on the interval certificates behind the one-missing-digit clearance alpha_1 < 1/4 at base = 34, at every 35 <= base <= 125 and by the uniform chain above. Witness: coprime.md PRIMES ON A DESIGN.

Robin corridor

2026-08-28 Conjecture The Robin corridor is orthogonal to Robin's difficulty, not merely hard: a design of dimension dim is an integer of exactly dim prime factors and level raises the exponents, so the tree varies exponents at fixed prime support, while all of Robin's difficulty lives at growing support omega(n) -> infinity; fixed support is the classical easy half, settled by prod p/(p-1) bounded against a divergent log log, an obstruction read off the definitions. Witness: divisor-avatars.

Slice sign law in every dimension

2026-08-28 Verified The slice sign law holds on every computed range: the central diagonal slice of the dim-axis base-3 Menger analog has sgn(slice dimension - (solid dimension - 1)) = (-1)^(dim+1) for dim = 2..50 at 180 to 210 digits by two generators sharing no code, for dim = 2..100 at 320 digits with 99 of 99 signs, and in exact rational arithmetic through the determinant form sgn det((fill/3) I - M_even) = (-1)^dim for dim = 2..40, with fill = 2^(dim-1)(dim+2) and M_even the reflection-even carry block of size ceil(dim/2); base 5 alternates for dim = 2..15, all four tested non-Menger families alternate, off-centre heights keep the dominant eigenvalue; dim = 3 gives x^2 - 9x + 12 with rho_3 = (9 + sqrt(33))/2 = 7.372281 against fill/3 = 20/3, dim = 4 gives x^2 - 11x - 66 with rho_4 = 15.310708 against 16 and det = 14; the even half and the odd dim != 1 mod 3 half are proved on the shelf, and the class dim = 1 mod 3 beyond the computed range stays open. Witness: slice-sign-even-half, slice-recurrence-order.
2026-08-28 Verified The dim = 3 rung is the base-3 slice dimension: the carry automaton M[c, c'] = P[c + dim - 3c'] prints M_even = [[6, 6], [1, 3]], trace 9, determinant 18 - 6 = 12, characteristic polynomial x^2 - 9x + 12, exactly A299916's signature (9, -12), Perron root (9 + sqrt(33))/2 and log_3 of it 1.818410, against the dimension minus one log_3(20) - 1 = 1.726833 with fill = 20 the sponge's surviving-subcube count; the anchor cuts one way only, saying nothing about higher rungs. Witness: slice-recurrence-order, A299916.
2026-08-28 Proved The digit polynomial P(t) = (1 + t^2)^(dim-1) (1 + dim t + t^2) has B_dim(2k) = C(dim, k) and B_dim(2k+1) = dim C(dim-1, k), P(1) = 2^(dim-1)(dim+2), P(-1) = 2^(dim-1)(2 - dim), P(omega) = (-1)^(dim-1) (dim-1) omega^dim, and root-of-unity filtering gives the full carry matrix's exact row sums sigma(c) = fill/3 + (2/3)(-1)^(dim-1)(dim-1) cos(2 pi c/3); the row-sum identity holds in the carry orientation c -> (c + dim - s)/3 and fails in the transposed even-basis orientation M_even[i,j] = B_dim(dim + j - 3i) + B_dim(dim - j - 3i) for every dim = 3..50, the dim = 3 row sums being (12, 4) against the formula's (8, 6); the coefficient formulas hold at dim = 1..10 three positions past both polynomial endpoints. Witness: slice-recurrence-order.
2026-08-28 Proved The trace of the even carry block is tr(M_even) = 3 dim 2^(dim-3) at odd dim and 3 * 2^(dim-2) - 1 at even dim, reading 2, 9, 11, 60, 47, 336 at dim = 2..7; the even case's -1 is real, starting at dim = 2 where the matrix is [2]. Witness: slice-recurrence-order.
2026-08-28 Verified There is no uniform spectral gap in the slice transfer matrix, so no fixed-epsilon proof of spectral separation can exist: lambda_1/|lambda_2| = (dim+2)/(dim-2) + O(dim^-3), tending to 1, reaching 1.068966 at dim = 60 and 1.04081632653 at dim = 100, 1.0833... at dim = 50 against 13/12 to 2.58e-22, with lambda_1 ~ fill/3 = 2^(dim-1)(dim+2)/3 and |lambda_2| ~ |P(-1)|/3 = 2^(dim-1)(dim-2)/3; the double-precision spectrum agrees with a 180-digit reference over dim = 2..50 to worst relative Perron discrepancy 2.3e-15, median 4.7e-16, every eigenvalue numerically real over dim = 2..60, so any proof of separation must be uniform in a margin of order 4/dim. Witness: slice-recurrence-order.

Spectral spacings

2026-08-28 Conjecture Every mrly fractal spectrum tested clusters rather than repels, excluding GOE and GUE, and the claim is "more clustered than Poisson", not "Poisson": two unfolding maps, both Laplacians and two independent pipelines to 4096 nodes, with random-graph and square-lattice controls separating first; a from-scratch rebuild on the Menger sponge level = 2 cell graph (400 nodes), Sierpinski level = 5 and level = 6, normalised Laplacian, a degree-12 polynomial unfolder and a 20 x 20 square-lattice control gives P(s<0.5) = 0.689 for the sponge against GOE's 1 - exp(-pi/16) = 0.17828, 0.830 for Sierpinski level = 6 and 0.594 for the square lattice; the band 0.44 to 0.57 under one unfolder does not reproduce under a third, and the sponge spectrum is 61.25% repeated eigenvalues at 1e-9, so P(s<0.5) >= 0.61 is forced by degeneracy and GOE exclusion is a corollary of degeneracy rather than an independent measurement. Witness: lab/rs/spectral-spacings.

Spin

2026-08-30 Proved The average of a picture over the q rotations by 2 pi / q keeps exactly the circular harmonics of order divisible by q, the average over all rotations keeps order zero only, and a design of rotation order g shows lcm(q, g) petals under a screen that turns it p/q of a turn per frame. Witness: mrlynum::spin the_harmonics_read_the_rotation_order, spin.
2026-08-30 Proved The rings of a spun square-lattice picture sit at sqrt(n) for n a sum of two squares with weight r2(n) = 4 (d1 - d3), silent exactly where a prime 3 (mod 4) divides n to an odd power, and their Dirichlet series is 4 zeta(s) L(s, chi_4); the hexagonal rings carry 6 zeta(s) L(s, chi_-3); the mass of a spun lattice is the Gauss circle count and Hardy's Bessel series for its error is the ring expansion. Witness: A001481, A004018, A003136, A004016, Hardy 1915, spin.
2026-08-30 Proved The exact ring profile of a raster integrates to its fill, int 2 pi r F(r) dr = fill, 512.0 at level 3 of the carpet; the carpet's profile is zero to side/6. Witness: mrlynum::spin the_mass_of_the_profile_is_the_fill, mrlydemo fixture.
2026-08-30 Refuted The coprime law survives the spin - flat layers at coprime odd scales are exactly uncorrelated, but their ring profiles over the inscribed disc correlate at +0.38 for (3, 5), -0.33 for (5, 7) and +0.38 for (9, 13), no better than gcd pairs; the cancellation is separable in x and y and the spin discards the angle. Witness: mrlylab test the_coprime_law_dies_under_the_spin.
2026-08-31 Proved The spin mass about the fixed point p_d = d/(base-1) of a filled digit d obeys M(r/base) = M(r)/fill exactly, since S(x) = (x+d)/base carries the design onto its d piece and divides the self-similar measure by the fill, so M(r) = r^dim G(log(r)/log(base)) with G of period exactly log base - the ripple's period is an identity and not a fit, valid for r/base below the distance from p_d to the other filled cells, that is r <= side at the corner digit and r <= side/2 at the centre. Witness: mrlynum::spin::mass_within, the_spin_mass_scales_by_the_fill_about_a_filled_corner.
2026-08-31 Verified The spin dimension read about the corner fixed point at level 6 over the window 27 <= r <= 729, three whole periods of log 3, gives slopes 1.465054, 1.649432, 1.783588, 1.761814, 1.897854, 1.879522, 2.000100 for codes 79, 95, 127, 239, 255, 495, 511 against the exact log(fill)/log 3, every gap at or below 1.9e-2 and the discretisation of the identity, max |M(3r)/(fill M(r)) - 1|, running 5.3e-3 to 1.8e-2; the exact integer shell histogram and the crate profile integral agree to 1.7e-6 on the total and 0.5% at partial radii. Witness: lab/rs/spin-census.
2026-08-31 Verified The corner ripple separates both equal-dimension pairs of the census where every density reading is identical: at level 7, 127 against 239 gives ripple gap 0.11984 on drift bar 0.04126 and 255 against the carpet 495 gives 0.12042 on bar 0.01461, with the solid square as the rippleless control at swing 0.00277 under its own bar 0.00578 and a code against its mirror at gap 0.00e0. Witness: lab/rs/spin-census.
2026-08-31 Verified The spin spectrum P_m, m = 0..12, read at levels 1 and 2 over all 511 nonempty base-3 plane codes, splits them into exactly 101 spectra, the number of nonempty orbits of the square group, with no pair outside one orbit agreeing to 1e-9: within the family it is a complete invariant of the dihedral class and no spin-isospectral witness exists. Witness: lab/rs/spin-census.
2026-08-31 Verified The ring-averaged powder of a design is not Porod: every sliding three-period window slope, over every fractal code at level 7 and at both pad 4096 and pad 8192, stays above -2.28 and so at least 0.72 from the -3 of a sharp interface, while the solid square control slides from -2.75781 to -2.35759, within 0.25 of -3 and never near its own -dim = -2. Witness: lab/rs/spin-census.
2026-08-31 Proved The Menger sponge at level level blocks every lattice line down its space diagonal that meets its bounding cube: the shadow obeys S_(level+1) = union_d (3 S_level + proj d), and along (1,1,1) the 27 cube digits and the 20 sponge digits project onto the same 19 classes, so the induction gives equality at every level, the count 3^(2 level + 1) - 3^(level+1) + 1 = 19, 217, 2107, 19441, 176419. Witness: lab/rs/spin-census, A220978, A003215.
2026-08-31 Proved The sponge's axis shadow is exactly the Sierpinski carpet, 8^level against the cube's 9^level, dimension log 8 / log 3 = 1.892789: the 20 sponge digits project along an axis onto the 8 carpet digits, disjoint modulo 3. Witness: lab/rs/spin-census.
2026-08-31 Verified No direction other than the axis is deficient in the searched window - over the 13 directions with 0 <= a <= b <= c <= 3, read to level 4 against the cube, the axis is the only share that falls with the level, every other rising, (1,1,2) to 0.98568 and (0,1,2) to 0.97090 at level = 4. Witness: lab/rs/spin-census.
2026-08-31 Proved A radius sqrt(k)/n of the spun scale-n square lattice, read inside the disc of radius sqrt 2, is new at n exactly when no prime p | n has p^2 | k - the sum-of-two-squares condition at the smaller scale is automatic by a parity argument, so only integrality binds - and hence new(n) = sum_(d | rad n) mu(d) B(2n^2/d^2) with B the counting function of A001481, giving 2, 3, 9, 11, 22, 18, 40, 38, 55, 52, 91, 64, 123, 97, 128, 126, 199, 136, 243, 180, the rule, the identity and a direct union agreeing at every n to 64. Witness: lab/rs/spin-census, A001481.
2026-08-31 Proved The Gaussian Farey's local factor is the Jordan totient J_2(n)/n^2 = prod_(p | n) (1 - 1/p^2), the square-lattice analogue of Farey's phi(n)/n, approached from below at rate 1/ln n because B(X) ~ K X / sqrt(ln X): the radical-6 family climbs 0.56250, 0.59813, 0.61126, 0.62594, 0.63276, 0.63801 at n = 6, 12, 24, 48, 96, 192 toward 2/3. Witness: lab/rs/spin-census, A064533.
2026-08-31 Proved The spin dimension about the raster centre is undefined for a design with an empty centre digit - the empty digit removes the open square of side side/3 about the centre and hence its inscribed disc, so M(r) = 0 for every r <= side/6 and the centre-spun mass carries neither power law nor ripple over a whole factor of base. The bound is attained, in exact integer arithmetic on doubled coordinates rather than cell centres, which would return hole + 1/2 whatever the hole: the squared distance to the nearest filled cell is (side/3)^2 = 6561 at level 5 for both bang dim 2, base 3, code 239 and the carpet 495, that is side/6 = 40.5 exactly, while 79 empties out to 56.572962. Witness: lab/rs/spin-census.
2026-08-31 Conjecture The near-degenerate corner ripples beyond the segment case: de-duplicated to transpose classes, the equal-fill class pairs sitting inside their own drift bar number 13 at level 6 and 6 at level 7, at gap-to-bar ratios 0.71 to 0.95, the tightest 287 against 315 at fill 6, gap 0.03574 on bar 0.04543, two designs that differ by moving one cell from (0,2) to (1,2); six survive both levels, 287-315, 63-123, 123-187, 31-59, 437-485, 37-261, and the count moves with the level and the estimator, so the list is a phenomenon and not a census. The ripple's Fourier coefficient at frequency 2 pi / log 3 should be a linear functional of the digit set whose kernel is what collides. Witness: lab/rs/spin-census.
2026-08-31 Conjecture The powder falls as k^-dim - at level 7 with pad 4096 the slopes -1.37986, -1.51762, -1.73012, -1.83071, -1.97886, -2.00433 sit within 0.24 of -dim over both pads, but the agreement is inside the instrument's own spread: sliding a three-period window a quarter period at a time moves the slope by 0.16 to 0.45, and doubling the pad to 8192 moves 127 from -1.73012 to -1.81607, 255 from -1.97886 to -2.03225 and the carpet from -2.00433 to -2.12289, with no monotone approach to -dim. The log-periodic ripple is not resolved either, the folded residual swinging 1.5 to 4.4 in ln power because the ring average of a lattice point set is spiked on the norms of A001481. Witness: lab/rs/spin-census.
2026-08-31 Conjecture The axes are the sponge's only deficient shadow directions; the window checked is |v| <= 3. Witness: lab/rs/spin-census.
2026-08-31 Conjecture new(n) sqrt(ln n) / n^2 converges to sqrt 2 K prod_(p | n) (1 - 1/p^2) along each radical class, K the Landau-Ramanujan constant; the Mobius sum is proved but B has no closed form, so the Gaussian Farey carries a transcendental constant where the Farey carries none. Witness: lab/rs/spin-census, A064533.
2026-08-31 Refuted The corner ripple as a complete invariant of the transpose class - a design that is a solid segment has M(r) = c r exactly about a fixed point on it, so its ripple vanishes identically, and code 7, the solid row, and code 273, the solid diagonal, are two such designs of dimension exactly 1 in different transpose classes carrying the same zero ripple; the census reads them at swings 0.01114 and 0.01217 and mutual gap 0.01371, all discretisation, and no bar is needed for the conclusion. It does separate both named equal-mass pairs, at 2.9 and 8.2 times the drift bar. Witness: lab/rs/spin-census.
2026-08-31 Refuted The Gaussian Farey counted by primitive representations in Z[i] modulo units - the norms below 2n^2 with a primitive representation run 2, 3, 6, 9, 13, 17, 23, 29, 35, 44 against new(n) = 2, 3, 9, 11, 22, 18, 40, 38, 55, 52, agreeing only at n = 1, 2; primitivity is the wrong condition, since (3,4) is primitive and 25 is a square, so sqrt(25)/5 = 1 is old at 5. The correct criterion is freedom from the squares of the primes of n. Witness: lab/rs/spin-census.
2026-08-31 Refuted The disc and the box read the same Gaussian Farey - restricting to 0 <= a, b <= n instead of the disc of radius sqrt 2 breaks the criterion at n = 3, witness the radius 4/3: 16 is free of 9 and 4/3 < sqrt 2, but 16 = 4^2 + 0^2 needs a coordinate above 3, and the box counts 2, 3, 7, 9, 17, 14, 31, 27, 41, 38 part from the disc counts from n = 3 on. Witness: lab/rs/spin-census.
2026-09-07 Proved The spin spectrum reads a pair census and nothing else: for a constant-valued 0/1 render on a raster of side base^level, every P_m is a quadratic form in the cell indicators whose Gram matrix is constant on the orbits of the raster's symmetry group acting on pairs, because turning a pair by theta multiplies both harmonic coefficients by e^(-i m theta) while a mirror at alpha sends c_m to e^(-2 i m alpha) conj(c_m) and the phase cancels in the real part; so P_m is a linear functional of the pair census Phi_level and equal censuses force equal P_m at every order, ring count and truncation, the base-3 plane carrying 11 pair classes at level 1 and 461 at level 2 and the level-1 coefficients solved from 11 independent censuses reproducing mrlynum::spin::harmonics at 1024 rings and m = 0..12 on all 511 codes at worst relative residual 1.14e-14. In dimension 3 the covariant object is the degree-l power summed over its orders, not a single (l, m). Witness: lab/rs/spin-census shape, spin.md the spin spectrum is a quadratic form.
2026-09-07 Proved The 101 spectra need level 2: Phi_1 takes exactly 97 values on the 101 nonempty base-3 plane orbits, four pairs lying in distinct square-group orbits with all 11 class counts equal, so each pair's level-1 spectrum coincides identically at every order and resolution; reading P_m at level 1 alone and bucketing greedily at 1e-9 returns 97 buckets, with 45-105 at gap 1.30e-16 and level-2 gap 0.151, 61-121 at 1.03e-17 and 0.0689, 78-102 at 6.51e-17 and 0.253, and 94-118 at 1.64e-16 and 0.105. Witness: lab/rs/spin-census shape, spin.md level 1 alone is not complete.
2026-09-07 Verified The 13 orders see 9 of the 11 level-1 census directions, and the odd cap of 3 is exact: the half turn rho is itself in the square group and acts trivially on classes, but half-turning one member, tau: {j, k} -> {rho j, k}, is well defined on classes because rho is central in D4, and g_(m, rho j) = (-1)^m g_(m, j) gives Q_m . tau = (-1)^m Q_m; tau fixes 5 of the 11 classes, so the antisymmetric part has dimension (11 - 5)/2 = 3 and no number of odd orders can exceed rank 3, which the six odd orders reach exactly while the seven even orders reach the proved cap of 6. The level-1 spectrum is strictly coarser than the census it factors through and splits it into the same 97 classes anyway. Witness: lab/rs/spin-census shape, spin.md the 13 orders see 9 of the 11 census directions.
2026-09-07 Verified The completeness is not about base 3, as a statement about the census: over all 2^25 base-5 plane codes the level-1 pair census takes 3993511 values on the 4211743 nonempty square-group orbits with 204856 ties over 423088 orbits and largest tie 8, and over all 2^27 base-3 dim = 3 codes it takes 1461693 values on the 2852287 nonempty orbits of the order-48 cube group with 757066 ties over 2147660 orbits and largest tie 32, and every tie breaks at level 2, the budget-capped weight window failing to bind and covering every group, all 204856 out to weight 21 and all 757066 out to weight 24 against level-2 censuses of 24805 and 6325 classes, with the canonical counts matching the Burnside averages 4211744 and 2852288 computed from the cycle index in the same pass. Witness: lab/rs/spin-census shape, spin.md the completeness is not about base 3.
2026-09-14 Proved The level-1 rank cap is 9 overall, 6 even and 3 odd; the odd cap of 3 is the known tau argument, the even cap of 6 and the total of 9 are new and supersede the earlier even bound of 6 of a possible 8. The common kernel of the coefficient vectors Q_m on the 11 base-3 pair classes holds the corner-centre basis vector, since the centre cell's farthest point and a corner cell's nearest point are both at r = sqrt2/6, so the two radial supports meet only in that null set and the integral vanishes at every order; and it holds (4, 16, 8, 8, 16, 4, 4, 8, 8, 4, 1) less 9 in the centre-centre slot, since the raster is similar to its own centre cell at ratio 1/3 and P_m(S/3) = P_m(S)/9. Both are tau-symmetric, so odd caps at 3 and even at 6. Witness: paper spin-harmonics Theorem 5.4, scripts/verify.py blocks 6 and 7.
2026-09-14 Verified The three caps are attained, on closed-form cell arcs and tanh-sinh quadrature rather than the 1024-ring discrete transform of the tree: 5 and 19 exactly located radial segments at 113 nodes give rank 9, even 6, odd 3 at m = 0..12, stable from 1e-9 to 1e-13, smallest pivot 1.878e-4 against largest coefficient 9.806e-2; the Gram matrix is constant on the 11 classes to 1.44e-17 and on the 461 level-2 classes to 1.52e-18 under both generators of the square group, a quarter turn and a reflection; the same pass returns 97 level-1 and 101 level-2 spectra over all 511 codes, every level-2 bucket one orbit. Witness: paper spin-harmonics scripts/verify.py blocks 5, 7, 9 and 10.
2026-09-14 Proved The four base-3 homometric pairs are two pairs and their centre augmentations, and homometry alone forces it: classes 0, 6 and 10 of the level-1 census are the self-pairs of a corner, an edge cell and the centre, so homometric designs already share their corner count, their edge count and their centre occupancy, and adjoining the centre to two that avoid it adds the corner count to the corner-centre class, the edge count to the edge-centre class and 1 to the centre-centre class: 61 = 45 + centre, 121 = 105 + centre, 94 = 78 + centre, 118 = 102 + centre, the augmented censuses reading 2, 2, 1 there against 0, 0, 0 before. Witness: paper spin-harmonics Lemma 4.4, scripts/verify.py block 3.
2026-09-14 Verified The 13 orders are a genuine truncation, and the solid square is the design the truncation flatters most: at level 1 the orders m = 0..12 hold 0.977541 of the solid square's angular energy against the exact Parseval total 1, the largest share over all 511 codes, tied only by the lone centre cell code 16 where P_m(S/3) = P_m(S)/9 forces it, while the smallest is 0.899436 at the four one-corner codes 1, 4, 64, 256, so every design spills at least 2.2 percent into the unread orders and some spill 10. Witness: paper spin-harmonics Fact 6.3, scripts/verify.py block 11.

Stacked hexagon moire

2026-08-28 Verified The Walsh law's 1/n and 1/n^2 coefficients follow from plane counting: the plane x + y + z = 6n - 2, z even, 0 <= x,y,z < 4n at odd scale n splits by macro parity into counts that depend only on the weight k, N_k(2h+1) = [t^(3h+1)](1 + 6t + t^2) E_h(t)^(3-k) O_h(t)^k with E_h = 1 + t^2 + ... + t^(2h) and O_h = t + t^3 + ... + t^(2h-1) extracts all four exact quadratic quasipolynomials by finite binomial calculation, and ink is linear in the eight counts, so only the constant, 1/n and 1/n^2 orders survive the normalization; exact by direct enumeration of the plane at every odd n <= 55, all eight macro-parity triples, 28 layers, zero discrepancies; the crate rebuilds only the ink, at codes 23 and 11 over n = 1..11, where m_3 = 0 leaves N_3(n) unexercised, so the eight-triple split stays the lane's. Witness: walsh-spectrometer, mrlydemo::walsh_spectrum.
2026-08-28 Conjecture Layer alignment of the hexagon stack is imperfect by construction: the transparent grid fraction of the common box grows from 0.000 at n = 1 to 0.245 at n = 55, so 24.5% of the box differs between the smallest and largest silhouettes, and ray scans on the triangular lattice are aliasing-limited below band half-width ~0.01.
2026-08-28 Conjecture The adjacent-pair limit is -11/135 = -0.0814815 (lab/rs/hexagon-moire measures -0.08116 at (199, 201) and -0.08150 at (249, 251), with residue oscillation) and the gcd-echo limit is 29/135 = 0.2148148 (lab/rs/hexagon-moire measures +0.21473 at (67, 201) and +0.21476 at (99, 297)); both are numerology-grade fits and each needs its lattice integral. Witness: lab/rs/hexagon-moire. Superseded: -11/135 and 29/135 are exact by the phase-map integral, see the layer-pair row under Diagonal slice stack in SETTLED.
2026-08-28 Conjecture The carpet star's exact decay rate, the coefficient on (ln L)/L, is open: one frame measures -1/8 and another -0.18.
2026-09-06 Conjecture The exact doubling constant of the stack is 253/2160, matching the two independent branch extrapolations to 2.6e-6 and 2.2e-6 where 19/162 fails at 1.57e-4; a derivation is missing. Witness: lab/rs/hexagon-moire. Superseded: the constant is exactly 253/2160 by the phase-map integral, see the layer-pair row under Diagonal slice stack in SETTLED.

The 2-adic Smith cascade

2026-08-28 Verified The Smith layers are a Jacobsthal cascade and the excess anticorrelates with the tent: full 2-adic profiles at every odd dim = 5..511 (254 rows) - octave maxima of L_2 = #{a_i >= 2} are exactly the Jacobsthal numbers J(k-2) (about n/6) and of a_max exactly floor(log_2 dim) + 4 on octaves 3 to 8, octave 2 (dim = 5, 7) reading L_2 = 1 against J(0) = 0 and a_max = 7 against 6, L_4 <= 1 and every non-spike divisor has a_i <= 3 (L_5 = 1 at 233 of 254 rows, the spike alone), and the min-of-cones consequences of per-layer tents (interior local minima 1, 1-Lipschitz in steps of 2, no plateaus off 1) hold at layers 1, 2, 3; the excess X = v_2 - nullity grows linearly (octave maxima 6, 6, 7, 8, 10, 17, 28, driven by L_2) but peaks at the tent troughs, so the sum stays small: v_2 <= ceil(n/3) + 9 at every odd dim = 5..511 (per-octave slack 5, 5, 5, 7, 7, 9, 9, growing like log_2 dim, extremal dim = 255, 257, 511), v_2 <= n at every odd dim >= 9 (equality only at 9, 15, the only violations dim = 5, 7), and in the class dim == 1 mod 6 (84 rows, 13..511) v_2 <= n - 3 < dim - 1 everywhere (max ratio v_2/(dim-1) = 7/18 at dim = 19 only, 1/3 at dim = 13), hence rho_dim > fill/3 strictly at every odd dim <= 511; the reading "v_2 <= ceil(n/3) + 13 fails at dim = 511" is arithmetically false (95 < 99), what died is the tent-plus-excess split, the object to bound being the sum; growth laws beyond dim = 511 are unproved. Witness: lab/py/smith-cascade.
2026-08-28 Verified The layer-2 window law: the second 2-adic Smith layer of M_even (base 3, odd dim = 2R+1) is a divisor-plus-ceiling window on the kernel family - with H_i = x^s(1+x^3)^i(1+x)^(2^b), b = ceil(log_2(3R-1)), and kernel elements as coefficient polynomials c(z), the mod-2 kernel vectors that lift mod 4 form V_2 = {c : g_dim | c, deg c <= C_dim}, L_2 = C_dim - deg g_dim + 1, with generator g_dim = z^m c_t(z^(2^e)) (c_t the F_2 Fibonacci polynomials), N = 2t+1 = J(k) Jacobsthal, k + e = b - 1; via the dictionary u^t c_t((1+u)^2/u) = 1 + u + ... + u^(2t) the x-side generator block is the odd-length repunit Rep(N, 2^e)(x^3) of zero (1+x)-valuation, which is why a pure valuation threshold (31/49) and a one-sided ideal (41/49) both fail; Law E gives g = |2R - 2^(b-1) - 1|, e = min{e >= 1 : J(e) >= (g+1)/2}, m = max(0, 2w - c(e)), L_2 = min(w+1, c(e)+1-w) with w = C_dim - t 2^e, c(e) = 2J(e-2) - 1, deriving the L_2 min-of-cones tent of height J(e-2) with octave peaks J(b-4); the explicit element H_2 = x^s(1+x^3)^(i_0+2m)(1+x^(2^b))Rep(N,2^e)(x^3) is derived from the Frobenius identity psi^(2^e) = (1+u^(2^e))^2/u^(2^e) and lifts mod 4 at every row; the mod-4 symbol is P == [(1+t^4)^R + 2Rt^2(1+t^4)^(R-1)](1+Dt+t^2); C_dim = K at 174/199 rows with deficits in 2J({2..5}) constant per (b,e) slot, two trial ceiling laws failing at dim = 249 and b = 9; 199/199 at odd dim = 5..401 and 60/60 at dim = 403..521, with N = 43 = J(7) appearing at dim = 257, 259 and N = 85 = J(8) at the b = 10 peak dim = 513; a peak staircase breaks at dim = 237 (nontrivial Rep(3,32) at b = 9, invisible below by the J(1) = J(2) = 1 collapse), the block-size identity 3J(k) = 2^k - (-1)^k is a tautology, and the family exponent floor(log_2(4R-1)) is wrong at 57/99 rows. Witness: slice-sign-even-half, smith-window.
2026-08-28 Verified det(m_full) = det(m_even) * det(m_odd) exactly, at every base and both parities (the core commutes with carry reflection by palindromy, the symmetric and antisymmetric blocks are the even and odd conventions, conjugation preserves determinants; by Bareiss to dim = 101), so v_2(det m_even) <= v_2(det m_full) and the strictness target v_2 < dim - 1 can be attacked on the core, whose mod-2 kernel is the one-generator shift module; the core is the coefficient-extraction map E: X -> ([x^(3j+1)](PX)) on deg X <= 2R, in polyphase coordinates the striped Sylvester matrix of (P_1, P_0, yP_2) (exact over Z at dim = 5..13), so the window module is a bounded syzygy module, rank-2 free by Hilbert-Burch, and delta_1 + delta_2 = 12R + 5 is the syzygy degree identity, which is why it holds at 400 random symbols. Witness: slice-sign-even-half.
2026-08-28 Verified The cascade holds across [512, 2048) with two fresh octaves attained on the nose: max L_2 = J(k-2) at exactly dim = 3*2^(k-1) + {1,3} (43 at 769/771, 85 at 1537/1539), max L_3 = J(k-4) (11 at 705/707 and 833/835, 21 at 1409/1411), a_max = floor(log_2 dim) + 4 (13 at 1023, 14 at 2047), L_1 = tent(dim) at every row through 2047, v_2 <= n everywhere (worst 0.37); at an L_j maximiser the profile is a flat block a_i = j plus one a_max spike, maximiser sites scale dim -> 2 dim - 1, and the cascade stacks at tent troughs (L_1 = L_2 at 767/769, L_1 = L_2 = L_3 at 701..707), so X peaks at stacking sites (51 at 767/769), not at the tent troughs (X = 7, 3 at 683/685); at 2^k - 1 sites X = v_2 - L_1 = a_max - 1, while v_2 - ceil(n/3) extends 5,5,5,7,7,9,9,11 as ...,11,11, not ...,11,13 (they part at 2047 where L_1 = 340 < 342, tail [1^339, 14]); hence rho_dim > fill/3 strictly at every odd dim <= 583 plus 685, 703, 769, 1021; all 42 adjudicated rows agree between two eliminators sharing no code, dim = 1409 at precision 512. Witness: slice-sign-even-half.
2026-08-28 Verified Base-5 exceptional-class strictness is exact to dim = 511: v_2(det M_even) < 2(dim-1) <= v_2(fill) at every dim == 1 mod 5 - odd class dim = 11..511 complete (51 values, v_2 running 2..105 against thresholds 20..1020, smallest margin 143), even class dim = 166..506 joining 6..156 (35 values, smallest margin 307) - so rho_dim != fill/5 throughout, the range extended from 80; 10 spot rows spanning both classes agree with an independent eliminator on v_2 and full profiles, and the matrix builder agrees entrywise with the graph-search construction at all 10 dim. Witness: slice-sign-even-half.
2026-08-28 Verified The ceiling law: with Law E's slot data (b = ceil(log_2(3R-1)), g = |2R - 2^(b-1) - 1|, e = min{e >= 1 : J(e) >= (g+1)/2}, k = b-1-e) the ceiling deficit is K - C_dim = 2J(e-1) iff k is even, else 0 - k >= 1 for every R (slot-endpoint identity g_max = 2J(b-2) - 1, exact for b = 4..60, no k <= 0 row to R = 60000), so a k >= 2 guard is vacuous, the rows dim = 23, 87 once read as k = 0 are e = 3, 5 with k = 1 (deficit 0 by parity), and dim = 1367 is b = 11, e = 9, k = 1, deficit 0; e = 2 never occurs; 259/259 at odd dim = 5..521 and 643/643 at b = 3..13, dim <= 4779 (the full b = 11 octave of 342 rows plus the b = 12, 13 boundary slots), by a Smith-free extraction with no precision parameter; never-seen deficits predicted and attained: 42 = 2J(6) on the whole slot dim = 429..471, 10 on 493..503, 2 at 511, 517, 0 at the peak, 86 = 2J(7) on all 84 rows of the b = 11 slot (e,k) = (8,2), 170 = 2J(9) at b = 12, 342 at b = 13; realised deficits {0, 2, 6, 10, 22, 42, 86, 170, 342}, 152 of 212 nonzero-deficit rows nondegenerate (L_2 > 1), the b = 12 peak giving N = 341 = J(10); so L_2(dim) is a closed function of R alone through Law E plus the ceiling law. Witness: slice-sign-even-half, smith-window.
2026-08-28 Verified The arithmetic amplitude law: max L_j in octave [2^k, 2^(k+1)) equals J(k + 2 - 2j), attained at dim = 2J(k+1) + 3 + 2(J(k+2-2j) - 1) with the flat-block-plus-spike profile [j x (J(k+2-2j)-1), spike] - 13/13 at k = 6..10, j = 1..4, including dim = 689 (octave 9, L_4 = 3, tail [4,4,6]) and dim = 1377 (L_4 = 5 = J(4), block 4x4+7); each octave carries two block towers, amplitudes J(k-2j+2) at the upper trough and J(k-2j+1) at the lower; the k = 10, j = 4 edge dim = 1379, 1381, 1383 reads L_4 = 5, 4, 3, so max L_4 = J(4) = 5 sits on a length-2 plateau 1377/1379 and is never exceeded; octave 8 (dim = 343..365) gives max L_4 = J(2) = 1 and max L_3 = J(4) = 5 at dim = 353; L_1 = tent(dim) and v_2 <= n hold at all 31 new rows (worst ratio 0.14). Witness: slice-sign-even-half.
2026-08-28 Verified The window-module machinery is classical: the carry core is a generalized (mosaic) Sylvester map of a 1 x 3 polynomial row ("striped Sylvester" is not a term of art), its kernel the truncated first syzygy module, rank-2 freeness is Hilbert-Burch, and delta_1 + delta_2 = 12R + 5 is the mu-basis degree identity mu_1 + mu_2 = n - deg(gcd) (Cox, Sederberg and Chen 1998; the Index Sum Theorem), so neither the identity nor the freeness is claimable and the 400-random-symbol generalisation reproves a 1998 theorem; the one-generator window step is two lines from Forney's predictable-degree property, leaving in-house only the evaluation 12R + 5 for this symbol, which needs the grading stated and polyphase coprimality asserted; claimable after nine recorded empty searches: Lemma M (the closest neighbours bound degrees, never a (1+x)-adic valuation), the Jacobsthal tent rank law (the Jacobsthal literature never uses the sequence as a rank formula's breakpoint set), the 2-adic Smith-layer/window structure (nearest miss: Smith forms over F[y], algebraically closed, no modular treatment), the Bockstein pairing as a layer-2 reader, and F_2 Fibonacci/Dickson kernel generators; three leads open - the full Beckermann-Labahn text, F_2 polyphase filter-bank Bezout twins, and mu-bases in positive characteristic, the last the only plausible threat to Lemma M. Witness: slice-sign-even-half.
2026-08-28 Proved Lemma W, the ceiling mechanism: on family coordinates multiplication by z is multiplication by psi = (1+x^3)^2/x^3 = x^(-3) + 2 + x^3 over Z, so the mod-4 obstruction class obeys ob(zc) = Lambda ob(c) mod im(E mod 2) with Lambda = S + S^(-1) folded at the centre (the raw vector identity fails at dim = 29; only the class is intertwined); hence if Y_0 corrects the generator (E(Y_0) = obraw(g)) with x-valuation cmin, then psi^i Y_0 corrects z^i g while cmin + 3i <= R, so C - deg g >= min(K - deg g, floor(a_0/3)) with a_0 = R - cmin the maximal correction reach and cmin the corrector's half-support extent from the centre, not a valuation; the uncapped C - deg g >= floor(a_0/3) is false at dim = 25 and at 29 of the 115 rows dim = 23..251, exactly the cap-strict rows, and equality L_2 - 1 = C - deg g = floor((R - cmin)/3) holds at the other 86, replacing the fitted ceiling by one linear-algebra invariant of the row; escaper independence is equivalent to ceiling exactness, and rank(phi) <= K - C follows from membership alone; the mod-2 family element's degree does not set the ceiling (dim = 115: all s_j >= 0 yet C = 3) and a_0 has no affine closed form in C - deg g (dim = 47 against 115, a_0 mod 3 varying, the floor load-bearing), so the ceiling law waits on a closed form for a_0 satisfying floor(a_0/3) = K - deg g - 2J(e-1)[k even] plus the single-element membership proof. Witness: slice-sign-even-half.
2026-08-28 Conjecture Layer 2 is read by the Bockstein pairing B(z,w) = (1/2) z^T M w-hat mod 2 with L_2 = nullity - rank(B) and the closed coefficient form (1/2)[x^(6R+1)](P What Zetahat) (49/49 at odd dim = 5..101, the coefficient identity exact at about 1.3k pairs, a corollary of the extraction form), and the layer flag V_k = red_2(ker(M mod 2^k)) is a contiguous step-3 degree run for all k <= 9 at dim <= 201 (99/99), not always top-anchored (witness dim = 29: mod-4 corrections pinned at the window top break shift-closure).
2026-08-28 Conjecture Smith(core) = Smith(even) ∪ Smith(odd) as multisets at base 3 (dim = 5..91); it fails at base 5, dim = 31: even {1,3,4} + odd {1,2,2} against full {1,1,1,2,3,5}.
2026-08-28 Conjecture The base-5 nullity has no bounded tent: the mod-2 nullity valley floors rise linearly, 1, 2, 2, 4, 8, 14 at dim ~ 19 * 2^k, about dim/38, with peaks about 0.1 dim, so the Jacobsthal tent with floor 1 is a base-3 phenomenon; at large odd class-dim the profile is rigidly [1] + [2]^(L_1-2) plus two tail terms, almost all 2-torsion in one layer, a parity split with no mechanism.
2026-08-28 Conjecture Off a maximiser the profile is two-tier, [(j-1)^p, j^q, spike] with p(i) = 2i - 1 marching in from the site and plateau length 2 in the site's own L_j (5 sites, 23 rows; dim = 1379..1383 mirrors dim = 689..693), dim = 1449 (L_2 = 41, tail [2^39, 4, 7]) is an ordinary j = 2 flank row with a one-unit tier-height excess at q = 1, and spikeless rows exist - dim = 1373 is a pure flat block, tail [4,4,4], L_5 = 0, double-sourced at a different precision and Smith-free.
2026-08-28 Conjecture The cascade recursion: the layer-3 law is the layer-2 window law one level down - in the quotient coordinate u = c/g_2 (the layer-2 window is deg u in [0, L_2 - 1]), V_3 is again a divisor-plus-ceiling window with tent parameter two Jacobsthal indices down, c_3 = 2J(e-4) - 1, delta = C_2 - C_3, P = C_3 - deg g_2, j = deg g_3 - deg g_2 = max(0, 2P - c_3), L_3 = min(P + 1, c_3 + 1 - P), tower ladder (delta, c_3) = (0, 2J(e-4)-1) while L_2 <= 2J(e-4) then (2J(e-4), 2J(e-5)-1); 40/40 on rows with L_3 >= 2 (odd dim = 175..401 complete plus 701..707, 735..741, 363, 365, with window, dimension and nesting exact and no contiguity break), six rows predicted before computation (dim = 735..741 with a never-seen delta = 22, seam rows 363 with j = 9 and 365); j is always 0 or odd, the layer-j tent height is J(e - 2(j-1)), which on the k = 1 slot is J(k_oct + 2 - 2j) - the amplitude law derived for j <= 3 - sites tie to the octave troughs (K = (dim - t_k)/2 at 38/38), and c_4 = 2J(e-6) - 1 puts the first L_4 >= 2 at exactly 689; g_3/g_2 is not always a monomial nor Fibonacci-shaped (dim = 481: c_2(z^2); dim = 497: (1+z)^2); further, t > 0 => L_3 = 1 on all 22 rows of the t > 0 region of octave b = 10; unswept: 403..471, 517..699, 709..733, 743+, and the layer-4 window at 689..693. Witness: slice-sign-even-half.
2026-08-28 Refuted The unified amplitude law max L_j = J(k - 1 - T(j-1)) with T triangular - fitted at j = 2, 3 where triangular and arithmetic indices coincide, it fails at j = 1 (true index k) and at j = 4, witnesses dim = 689 and dim = 1377. Witness: slice-sign-even-half.
2026-09-07 Proved Lemma S, the symbol reading of the carry core holds at every 2-adic layer: for odd dim = 2R + 1, with P = (1 + t^2)^(dim-1)(1 + dim t + t^2) and H_x = x_0 t^R + sum_(j >= 1) x_j (t^(R+j) + t^(R-j)), the row at c' of M_even x is the coefficient of t^(3 nu + 1) in H_x P at nu = R - c', and H_x P is palindromic about 3R + 1, so the R + 1 rows are exactly the exponent class 1 mod 3 on [0, 6R + 2]; hence for every r >= 1, M_even x == 0 mod 2^r iff H_x P lies in the Z_2[t^3]-module generated by 1, 2^r t and t^2, equivalently, with u = t^3, H = H_0(u) + t H_1(u) + t^2 H_2(u) and P = P_0 + t P_1 + t^2 P_2, iff H_0 P_1 + H_1 P_0 + u H_2 P_2 == 0 mod 2^r; the r = 1 case is the mosaic Sylvester row already recorded as classical and the mod-4 symbol is already recorded, so what is added is the one row (P_1, P_0, u P_2) carrying every layer, checked as sets and not only as dimensions at dim = 5..13, r = 1, 2, 3. The layers are not truncated-syzygy dimensions of that row over Z_2[u]: the syzygy module of the row is the kernel of M_full, not of M_even, and the two nullities differ by the already-proved halving nullity_even = ceil(nullity_full/2), because u = t^3 does not preserve palindromy; the operator that does is psi = u + u^(-1), as Lemma W states. Witness: smith-window.
2026-09-07 Proved Lemma Lambda, the family shift is intertwined on the nose: write psi = t^3 + 2 + t^(-3), the integer multiplier (1 + t^3)^2/t^3 of Lemma W, and ob(H)(nu) = ((H P)[3 nu + 1] mod 4)/2 on 0/1 palindromic coefficient vectors of the mod-2 kernel; then (psi H P)[3 nu + 1] = (H P)[3 nu - 2] + 2 (H P)[3 nu + 1] + (H P)[3 nu + 4] and (H P)[3 nu + 1] is even, being a mod-2 kernel row, so its doubled term dies mod 4 and ob(psi H) = Lambda ob(H) holds as raw vectors with Lambda = S + S^(-1) folded by nu <-> 2R - nu; the family satisfies H^(j+1) = psi H^(j) - 2 Z_j with Z_j = H^(j) + (t^3 H^(j) AND t^(-3) H^(j)), and Z_j is palindromic and inside the coefficient box because deg H^(K) <= 2R (from i <= (6R + 2 - 2^b)/3) and the overlap sits in [val + 3, deg - 3], so ob(X_(j+1)) = Lambda ob(X_j) + A(Z_j) with A(Z_j) in the image of the mod-2 symbol and the class identity of Lemma W holds at every row with its raw defect named; the pair moves together, and psi = t^3 + t^(-3) with Z_j = H^(j) + AND makes the lift identity false at dim = 29, 31, 47, 115, 251. Witness: smith-window.
2026-09-07 Verified The layer-2 window is a window, and its generator and ceiling regenerate from the symbol: V_2 = g_dim F_2[z]_(<= C_dim - deg g_dim), g_dim = z^m c_t(z^(2^e)) with c_t the F_2 Fibonacci polynomials c_0 = 1, c_1 = 1 + y, c_t = y c_(t-1) + c_(t-2), and C_dim = K - 2J(e-1) at even k, K at odd k, at 199/199 rows of odd dim = 5..401 and 100/100 of odd dim = 403..601; the slot, the window bounds and the closed forms are the shelf lane's arithmetic line for line and only the object side is independent - the kernel family, the mod-4 symbol, the obstruction and the extraction of V_2 - so what this adds is a committed generator for g_dim and C_dim, which the lane's own scripts do not compute, pinning L_1 and L_2 alone. The ceiling is a corrector length: C_dim - deg g_dim = min(K - deg g_dim, floor(reach/3)) with reach = R - jmax, jmax the least index whose mod-2 symbol columns span the generator's obstruction, and reach itself Lemma W's a_0; the min was chosen after the 5..401 overshoot, so honest support is the 100 fresh rows 403..601, where the floor binds strictly at 60, the cap at 38 and they tie at 2, against 108, 99, 92 over all 299 rows, and the floor-strict rows are exactly the C_dim < K rows, both ways. Remark: taking the corrector out of the coefficient box leaves an image of corank exactly 1 in F_2^(R+1) at 129/129 rows of odd dim = 5..261, every family obstruction meeting it, so the unboxed layer-2 window is the whole mod-2 kernel and an argument living in the untruncated module cannot see g_dim or C_dim. Witness: smith-window.
2026-09-07 Verified The reach law, the last unknown of Law E's ceiling: with Law E's slot data (b = ceil(log_2(3R-1)), g = abs(2R - 2^(b-1) - 1), e = min{e >= 1 : J(e) >= (g+1)/2}, k = b - 1 - e) give the slot its length N = J(e) - J(e-1), which is 2J(e-2) at e >= 3 and 1 at e = 1, its offset u = (g+1)/2 - J(e-1) - 1 and its position p = u above the octave centre R = 2^(b-2) and p = N - 1 - u below it; then the tent identity min(p, N - 1 - p) = C_dim - deg g_dim says Law E's window length is the distance to the nearer end of the slot in the slot's own coordinate, and the reach law says reach = R - jmax = 3 min(p, N - 1 - p) + 2 [e even] + [k odd](1 + (p mod 2)), with p == R mod 2 whenever e >= 3 so the parity term is the parity of R; exactly one row per odd octave escapes, the e = 1 row above centre dim = 4^m + 3, where reach = 5 for m >= 2 and reach = 3 at dim = 7. Off those escaping rows floor(reach/3) = C_dim - deg g_dim + [k odd and e even], and on them it reads 1 against C_dim - deg g_dim = 0 with the cap K - deg g_dim = 0 as well, so min(K - deg g_dim, floor(reach/3)) = C_dim - deg g_dim at every row: the corrector law's statement carries no span test and its branch is a slot statistic - the floor binds strictly iff k is even and e >= 2, the cap iff k is odd with e even or dim = 4^m + 3, and they tie otherwise - reproducing the recorded censuses in floor, cap, tie order as 48, 61, 90 at odd dim = 5..401 and 60, 38, 2 at 403..601 with no mismatch, and 2399/2399 to dim = 4801. This repairs Lemma W rather than resting on it: the landed uncapped inequality C_dim - deg g_dim >= floor(a_0/3) is false at dim = 25 (K = C_dim = deg g_dim = 0, jmax = 9, reach = 3, so 0 >= 1) and at 29 of the 115 rows dim = 23..251, exactly the cap-strict rows, while the family-capped psi-orbit bound C_dim - deg g_dim >= min(K - deg g_dim, floor(reach/3)) holds throughout, and Lemma W's a_0 is reach and not jmax, since C_dim - deg g_dim = floor(reach/3) at 86 of those 115 rows and = floor(jmax/3) at none. So only the >= half of the ceiling law is promoted, to a consequence of the Verified reach law and the Verified generator law and not to a proof; the deduction is not span-test-free, since reach is defined by the span test and the psi-orbit needs its corrector valuation maximal; and one half stays open, that z^(C_dim - deg g_dim + 1) g_dim does not lift. jmax is therefore not a 2-adic valuation statistic of R but a slot-tent statistic, and the two rank readings that would replace the span test are Refuted with witnesses, corank(E boxed) = K - C_dim + 1 failing at dim = 15 and the first dependent column index 2J(e) failing at dim = 7. Fit rows are the 399 rows dim = 5..801, read once and unadjusted; out of sample are the 800 rows dim = 803..2401 swept cold plus dim = 4099 and dim = 16387, all clean, and the swept ladder covers every class of R mod 8. Witness: smith-window.
2026-09-14 Proved The slot tent identity min(p, N - 1 - p) = C_dim - deg g_dim holds at every odd dim >= 5, granting Law E's closed forms for C_dim and g_dim, by exact arithmetic in b, e, k, R with no appeal to the module; the proof is a three-case split on e, and e = 2 never occurs (witness: spectra.md, The tent identity, the theorem)
2026-09-14 Proved The window box length is K = J(b-2) - (g+1)/2 in both octave halves, where b is least with 2^b >= 3R - 1 and g = abs(2R - 2^(b-1) - 1) (witness: spectra.md, The tent identity, Lemma 3)
2026-09-14 Proved Law E's offset collapses to C_dim - t 2^e = J(e) - (g+1)/2 whether k is even or odd: the ceiling deficit 2 J(e-1) and the parity of k cancel exactly (witness: spectra.md, The tent identity, Lemma 5)
2026-09-14 Proved The slot length satisfies N = J(e) - J(e-1) = 2 J(e-2) for e >= 2, so Law E's chi = 2 J(e-2) - 1 equals N - 1 for e >= 3; at e = 1 the bridge fails and the case closes because both sides vanish (witness: spectra.md, The tent identity, Lemma 6)
2026-09-14 Proved The slot offset and Law E's offset reflect: u + (C_dim - t 2^e) = N - 1, so {u, C_dim - t 2^e} = {p, N - 1 - p} in both octave halves (witness: spectra.md, The tent identity, Lemma 7)
2026-09-14 Proved s = (g+1)/2 <= J(b-2) at every odd dim >= 5, hence e <= b - 2, hence k >= 1 and t = (J(k) - 1)/2 >= 0; this is Law E's standing hypothesis k >= 1, now proved (witness: spectra.md, The tent identity, Lemma 4)
2026-09-14 Proved p == R mod 2 whenever e >= 3, and the parity fails exactly at the e = 1 rows above centre, R = 2^(b-2) + 1, that is exactly on dim = 2^j + 3 for j >= 2; those rows have k = j - 1, so the reach law's escaping family dim = 4^m + 3 is the k odd half of the set and no more, dim = 11 being a parity-failing row outside it (witness: spectra.md, The tent identity, Lemma 8)
2026-09-14 Proved From the ceiling law alone, with no use of the tent identity, the upper half of the layer-2 window law is free wherever C_dim = K, that is wherever k is odd or e = 1, since every element of V_2 has coefficient degree at most K while the candidate z^(C_dim - deg g_dim + 1) g_dim has degree C_dim + 1; the open rows are exactly k even with e >= 3, 448 of 1199 over odd dim = 5..2401 and 29116 of 99999 over odd dim = 5..200001 (witness: spectra.md, The tent identity, What it buys, census by lab/py/smith-window)
2026-09-14 Conjecture At the rows with k even and e >= 3, the family element of coefficient degree C_dim + 1 has mod-4 obstruction outside the image of the mod-2 symbol on the coefficient box, for a deficit of exactly K - C_dim = 2 J(e-1); this is the whole of what remains of the upper half of Law E (witness: spectra.md, The tent identity, What it buys)
2026-09-14 Refuted The parity-failing rows of Lemma 8 are not the family dim = 4^m + 3: dim = 11 fails the parity and is not of that form, and 9 of the 19 failing rows below dim = 2000001 lie outside the family; the true set is dim = 2^j + 3 for j >= 2 (witness: spectra.md, The tent identity, Lemma 8 and What it buys)

The Apollonian gasket

2026-09-14 Proved The Descartes reflection needs no square root and acts on all three coordinates: in the coordinates (k, k x, k y), a line being k = 0 with (k x, k y) its outward normal, the fourth circle tangent to three given ones is v' = 2(v_1 + v_2 + v_3) - v, by Vieta, so an integral root quadruple grows an integral packing; six identities ride along under B(u, v) = (sum u_i)(sum v_i) - 2 sum u_i v_i, B(k, k) = B(k, kx) = B(k, ky) = B(kx, ky) = 0 and B(kx, kx) = B(ky, ky) = -4, true on both roots and preserved by the reflection, which lies in the orthogonal group of B. Witness: lab/rs/apollonian, verbs strip and census, all six rechecked on 575969269 quadruples, 0 broken, an arithmetic check only.
2026-09-14 Proved The circles of the strip packing (0, 0, 2, 2) tangent to the line y = 0 are exactly the Ford circles, one over every reduced a/b, no interval assumed, of curvature 2 b^2: tangency is k y = 1 at positive curvature; (0, 2 b^2, 2 d^2, k) has square discriminant 64 b^2 d^2 and roots 2(b + d)^2 and 2(b - d)^2, the mediant and the Stern-Brocot parent; two are tangent exactly at (a d - b c)^2 = 1; the walk covers (0, 1), the root pair 0/1 and 1/1 the ends, and the period-1 translation the rest; conversely Dirichlet forces an overlap at irrational p and nesting equality at rational p. Witness: lab/rs/apollonian, verb ford, 4863601 mediants to denominator 4000 against sum_{b <= 4000} phi(b) - 1, 0 broken, 0 misses.
2026-09-14 Verified The Ford identification holds in both directions on the grown packing, not only on the Stern-Brocot walk: one period of the strip packing grown to curvature 2097152 gives 20770674 circles of which 318963 carry k y = 1, every one passing the Ford test that k/2 is a square b^2 and k x = 2 a b with gcd(a, b) = 1, 0 off-Ford, and 318963 is sum_{b <= 1024} phi(b) - 1; the same count returns at Q = 32 and Q = 181 as 323 and 10059, the far line carries 318963 by the strip's reflection symmetry, and no circle leaves the open period, 0 outside 0 < k x < k. Distinctness is controlled at T = 2048, 2448 circles and 2448 distinct. Witness: lab/rs/apollonian, verb strip.
2026-09-14 Proved The stack's brightness reads off the packing's curvature: the Farey stack lights the node a/b exactly floor(Q/b) times at depth Q, the one circle resting on that node has curvature k = 2 b^2, so the brightness is floor(Q sqrt(2/k)), and the nodes lit at depth Q are exactly the tangency points of the line-tangent circles of curvature at most 2 Q^2; summing over the half-open period [0, 1) gives sum_{b <= Q} phi(b) floor(Q/b) = sum_{n <= Q} sum over b dividing n of phi(b) = Q(Q + 1)/2, the walk carrying (0, 1) and the node 0/1 adding its Q. Witness: lab/rs/apollonian, verb ford, brightness 1275, 20100, 500500, 8002000 at Q = 50, 200, 1000, 4000 against Q(Q + 1)/2.
2026-09-14 Verified The curvature census grows like a power of T whose local exponent, read as the ratio log(N(T_2)/N(T_1))/log(T_2/T_1) and never as a fit, lands at 1.305, the fourth place set by the grid: the bounded packing (-1, 2, 2, 3) gives N(T) = 5, 165, 3325, 67163, 1359167, 27463391, 555198593, ratios ending 1.3055, 1.3057; the strip's one period gives 2, 48, 950, 19298, 390478, 7899138 on the decades, ratios ending 1.3061, 1.3060, and 20770674 at T = 2097152, octave ratios ending 1.3050, 1.3056. N(T) excludes the root quadruple, four circles bounded, one per strip period. Witness: lab/rs/apollonian, verbs census and strip, 67163 distinct against 67163 counted on the T <= 10^4 control.
2026-09-14 Verified The residues are the arithmetic the census can see: the bounded packing (-1, 2, 2, 3) uses exactly the eight classes 2, 3, 6, 11, 14, 15, 18, 23 mod 24 over all 555198593 circles of curvature at most 10^7, at counts 83211520, 55422929, 55455852, 83378348, 83354132, 55617906, 55576284 and 83181622, while the imprimitive strip packing uses exactly the four classes 0, 2, 8, 18 at 4144636, 6223160, 6241134 and 4161744 of its 20770674 circles; which integers inside those classes occur is closed by others and this tree makes no claim on it. Witness: lab/rs/apollonian, verbs census and strip.
2026-09-14 Verified The residual dimension is 1.3056867280498771846..., rigorous to 128 places by an effective Ruelle-Bowen computation on a Chebyshev-Lagrange approximation of the transfer operator, Theorem 1.1 of Vytnova and Wormell 2024 reading 1.3056867280 4987718464 5986206851 0408911060 ... +- 10^(-129); the counting asymptotic c T^alpha is Kontorovich and Oh 2011 with alpha ~ 1.30568(8), McMullen 1998 reads 1.305688. The census's bounded 1.3057 is alpha correctly rounded to four places, off 1.3e-5; the strip's 1.3056 and 1.3060 agree to three, off 8.7e-5 and 3.1e-4. Witness: REFS.md, read at source; lab/rs/apollonian, verbs census and strip.
2026-09-14 Verified No design carries the gasket's dimension inside the window the tree can see: a design is the attractor of similarities of one ratio 1/base under the open set condition so its dimension is log N/log base for an integer cell count N, equal to alpha only if base^alpha is an integer, and over 2 <= base <= 100 the nearest approach is 52^alpha = 174.005426001 at gap 0.005426001, then 68, 89, 49, 23, 20 at gaps 0.008182, 0.011684, 0.015094, 0.022279, 0.026750, worst 0.488110 at base = 47; the table refutes equality and nothing weaker, the nearest design dimension being log 351/log 89 = 1.305694144, off alpha by 7.4e-6. Witness: lab/rs/apollonian, verb design.
2026-09-14 Verified The packing grower is now in the publishable crate: mrlynum::apollonian takes a named integral root, grows it by the square-root-free reflection in exact i64 triples (k, k x, k y) and rechecks all six invariants of B on every quadruple, reproducing the generator's numbers from the crate: 2448 circles on one period of the strip to curvature 2048 and 950 to curvature 1000, the root excluded, 0 broken and 0 circles outside the open period, and 323 circles carrying k y = 1 below 2048, every one passing the Ford test k = 2 b^2, k x = 2 a b, gcd(a, b) = 1. Witness: mrlynum::apollonian::grow and is_ford, test every_line_tangent_circle_is_the_ford_circle_over_its_own_fraction, 2448 and 323 of 323.
2026-09-14 Verified The stack and the packing's tangency points agree fraction by fraction and not only in count: at depth 32 the 323 nodes the Farey stack lights inside the open period and the 323 tangency points of the line-tangent circles of curvature at most 2 Q^2 = 2048 are the same set of reduced fractions with 0 missed either way and 0 off-Ford, the brightness of the period summing to 528 against Q(Q + 1)/2, and at depth 16 the same reading gives 79 against 79 with 0 missed and brightness 136; the agreement is checked at every depth from 2 to 64 against mrlynum::lattice::farey. Witness: mrlynum::apollonian::shadow, test the_stack_is_the_shadow_of_the_line_tangent_circles, 0 missed at both depths.
2026-09-14 Verified Two further bounded roots sit in the integer coordinates with all six invariants exact and hand back new censuses: (-2, 3, 6, 7) placed as (-2, -1, 0), (3, 1, 0), (6, 5, 0), (7, 5, 2) and (-3, 4, 12, 13) placed as (-3, -1, 0), (4, 1, 0), (12, 7, 0), (13, 7, 2), each carrying a double Descartes root because k_1 k_2 + k_2 k_3 + k_3 k_1 = 0, giving N(1000) = 1297 and N(1000) = 741 beside 3325 for (-1, 2, 2, 3) and 950 for the strip period, the root quadruple excluded throughout. Witness: mrlynum::apollonian::root and grow, test the_growth_lands_on_the_counts_the_generator_prints, 1297 and 741.

The arithmetic pole

2026-09-06 Proved The comb zeta of base 3 digits {0,1} has a genuine pole at every s_(0,k) = log_3 2 + 2 pi i k/log 3, k = 1..10: the residue lambda_(0,1) lies in [0.231891517689918, 0.231891517689919] + i [-0.501067414481069, -0.501067414481068] by interval arithmetic on Burnol's Proposition 5.1 with every truncation bounded by a proved tail, and Res = s_(0,k) c_k ties it to the k-th Fourier coefficient of the log-periodic profile of A(x); the profile control from exact counts and a second certified enclosure by the functional-equation route both meet the first. Witness: lab/py/burnol-residue, dimensions.md THE ARITHMETIC POLE.

The circle count

2026-09-06 Proved A design's corner disc count has a self-similar main term with a log-periodic multiplier: with F the base-3 digit-restricted set of a design, fill its digit count and N(r) the filled cells whose centre lies in the closed Euclidean ball of radius r about the lattice corner, N is independent of the level, M(r) = fill^level mu(B_(r 3^(-level))) = r^(log(fill)/log 3) G(log_3 r) with G positive and 1-periodic, and |N(r) - M(r)| <= C(r), the crossing count, which is O(r^(dim-1)) in every dim because a cell meeting the sphere lies in the shell | |y| - r | <= sqrt(dim), of orthant volume 2^(-dim) omega_dim ((r + sqrt dim)^dim - max(r - sqrt dim, 0)^dim), so N(r) = r^(log(fill)/log 3) G(log_3 r) + O(r^(dim-1)), an unconditional saving r^0.8927892607 on the carpet and r^0.7268330279 on the sponge; the density-times-volume main term fails outright at the grid centre, whose middle block is empty at every level and where the relative error is exactly 1. Witness: lab/rs/circle-crop (22028 asserted rows, 6802 with a live error band, 41 mrlymath::shape::census cross-checks, the crossing bounds C_full <= 3r + 5 at dim = 2 and C_full <= pi sqrt 3 (r^2 + 1) at dim = 3 asserted at every radius), crop.md THE CIRCLE COUNT.
2026-09-06 Verified The crossing exponent of the corner disc count reads min 0.871371 / mean 0.898741 / max 0.969141 per triadic step over r = 27..19682 on the carpet and 1.704391 / 1.733764 / 1.757218 over r = 27..728 on the sponge, both bands containing the dimension minus one; the defect exponent sits in [0.220478, 1.015046] (carpet) and [0.645285, 1.730726] (sponge); the powers of the base carry no resonance, ranking inside [0.1200, 0.1296] in the crossing profile at k = 3..8 on the carpet while the defect's eight ranks average 0.4788 and reach 0.8333; mu(B_1) is certified in [0.750767350, 0.751113415] on the carpet, excluding 3/4, and [0.475928750, 0.485478125] on the sponge. Witness: lab/rs/circle-crop.
2026-09-06 Refuted That the corner disc count resonates at r = 3^n, the heuristic the page's own transform identity suggests (hat mu(3t) = (P(t)/m) hat mu(t), equality on integer t): |delta(3^n)|, delta(r) = N(3r) - m N(r), ranks 0.5000, 0.8333, 0.1111, 0.0185, 0.6605, 0.5514, 0.7167, 0.4390 inside its own triadic window on the carpet (the fraction of the window with |delta(r)| <= |delta(3^n)|) with no trend and delta(27) = 0 exactly; what is periodic is the crossing profile, whose maximum sits at 2.9671 times each window's start and triples exactly from r = 721; and square-root cancellation over the crossing cells fails, every central defect estimate sitting above half the crossing exponent. Witness: lab/rs/circle-crop.
2026-09-07 Proved The crossing shell is a tree: the shell at level j is the whole grid's shell at real radius r/3^j, so it is a rooted tree of depth level with 2r+1 leaves and 2*floor(r/3^j)+1 boxes per level, and C(r) counts the leaves whose path never takes the centre seat; mean branching is 3 + (2k-2)/(2Q+1), exactly 3 at every level where floor(r/3^j) is 1 mod 3. Witness: crossing cells brute-forced from the cell definition at every level of every r <= 150 and at 18 large and boundary radii with no fault, the live leaves equal the corner ball's Cut column at every radius of every depth 1..5 for codes 7, 11 and 15, no box lacks a crossed parent at any r <= 242, and C(100) = 134 and C(242) = 296 read twice by paths sharing no code (lab/rs/circle-crop, crates/mrlydemo/tests/shell.rs, crop.md THE CIRCLE COUNT).
2026-09-07 Proved The crossing ladder is an exact ratio of integer counts: Psi(r) = prod_k g_k(r) with g_k = u_(k+1)/(1 - p_(level-1-k)), u_k = T_k/T_(k-1) and k = level-1-j the depth from the top; g_0 = 1 identically, and g_1 = 1 exactly whenever the level-(level-1) or level-(level-2) centre box is uncrossed, so the profile carries at most level-1 informative ranks and often level-2. Witness: lab/rs/circle-crop ladder lines, the product asserted against Psi computed directly to 1e-12 at r = 80, 242, 1000, 6560.
2026-09-07 Proved The crossing shell's transfer operator is the tripling map on the offset: with R_j = r/3^j and y_j(x) = sqrt(R_j^2 - x^2), the offset a_j(i) = frac(y_j(i)) satisfies a_(j-1)(3i) = frac(3 a_j(i)) at every level and column, because y_(j-1)(3x) = 3 y_j(x) is an identity of reals and needs no hypothesis; the 9-bit box pattern is mask(floor(u - k sigma)) for k = 0..3 clipped to [0,2], with sigma the scale-free slope. Witness: lab/rs/circle-crop derivation pass, the derived pattern law reproducing the shell with 2 faults of 2188 boxes at level 0 and none at levels 1..5 on the shallow arc at r = 6560, the steep half following by the shell's own symmetry.
2026-09-07 Proved A 9-bit box pattern is realisable by a straight line exactly when max_(k<l) (v_k - v_l - 1)/(l - k) < min_(l<k) (v_l - v_k + 1)/(k - l) with the upper end positive; the test returns exactly thirty masks, which are character for character the crossing shell's thirty, and it forbids the thirty-first, a zero step beside a step of two, so the extra pattern seen only at the top of the tree is a curvature state and not a line state. Witness: lab/rs/circle-crop derivation pass, stable at search radius 12 and 15 and asserted at r = 6560, 19682, 12345.
2026-09-07 Verified The Perron root of the memory-one pattern matrix differs from 3 at r = 6560, 19682, 12345, brackets [3.000861, 3.000862], [3.000948, 3.000949] and [2.997616, 2.997617], but this does not test the Markov property: the matrix row sums are exactly popcount(s), so the model returns each level mass exactly (161, 485, 1457, 4373, 13121 at r = 6560), the tree's own pooled branching is 3.037736, 3.012422, 3.004124, 3.001373, 3.000457 and never the exact 3, the certified distance from 3 falls monotonically 0.323840, 0.002623, 0.002385, 0.000086 as the parent count rises, and over 18 unused radii the sign of rho - 3 is positive 9 times and negative 9 times at sizes 0.000336 to 0.005951. Witness: lab/rs/circle-crop operator, pair and sweep_radii lines.
2026-09-07 Verified The thirty-pattern alphabet of the crossing shell's transfer operator belongs to the truncation levels 1..level-3 and not to the tree: over levels 1..level the alphabet is 31 at r = 6560, 19682 and 12345, the extra being the root's own pattern, and 228 of the 16683 radii r = 3000..19682 leave the thirty over levels 1..level-3 and 1966 over levels 1..level-1, none reading fewer. Witness: lab/rs/circle-crop every_level_states and scan lines, with r = 1395, 1739, 6570, 15122, 3182 pinned at four truncations.
2026-09-07 Verified Folding N(r)/r^(log(fill)/log 3) at 16 offsets of log_3 r mod 1 and comparing consecutive triadic windows measures the collapse instead of assuming it: the carpet's largest gaps run 0.353553, 0.222183, 0.110138, 0.042663, 0.015114 from R = 1 to R = 243 with shares of the window level 0.534078, 0.305035, 0.145250, 0.055448, 0.019546, so R = 81 and R = 243 agree to two percent and the ladder falls like 1/r; the deepest window opens at N(243)/243^(log(fill)/log 3) = 0.751038, inside the certified mu(B_1) bracket [0.750767350, 0.751113415], while the sponge reads 0.229755 at R = 9 against R = 27 and the grid centre does not collapse at all, its share reading 3.081886. Witness: mrlydemo crop_collapse, crates/mrlydemo/tests/crop.rs, site/check.ts.
2026-09-07 Proved At dim = 2, with level the least level with r < 3^level, the crossing shell's digit rate is pinned to 1/9 with an unconditional power saving and the index ind(r) = prod_(j<level) (1 - p_j(r)) (9/8)^level is bounded: the shell is a monotone lattice path, so a level-j box carries leaves(X) = w + h - 1 cells with sum_X w = sum_X h = floor(r/3^j) + r + 1 and sum_X leaves = 2r + 1, the coordinate swap makes the seat class's two marginals equal, and p_j(r) (2r+1) = 2 sum_(seat X) w(X) - #seats exactly; that turns both counts into sums of floor of one circle arc over 3^j residue classes mod 3^(j+1), where van der Corput's Satz 5 on closed subintervals [a, b] with 3^(j+1) b + c < r and a residue split of the monotone increments y(x) - y(x+1) give |p_j(r) - 1/9| <= 13.60 R^(-1/3) + 305.08 R^(-1/2) + 9.84 R^(-1) at R = r/3^j, hence sum_(j<level) |p_j - 1/9| <= 781, and with 1 - p_j >= 3^j/(2r+1) from box column 0, |log ind(r)| <= 1191 for every r >= 1; Psi is untouched, so Phi does not follow, and the staircase is a plane fact so the sponge has none of it. Witness: lab/rs/circle-crop index lines, every identity asserted in exact integers at every level of r = 80, 242, 1000, 2186, 6560, 12345, 19682 and the bound asserted live at j = 0 for r = 212957, 531441, 2000000 where the cap clears the trivial 8/9; crop.md THE CIRCLE COUNT.
2026-09-07 Proved At dim = 2, with level the least level with r < 3^level, no level-j column block [3^j i, 3^j (i+1)] of the crossing shell of radius r tracks a slope window of width eps unless 3^j < 2 eps r, so a leaf's tracked run is at most floor(log_3(2 eps r)) + 1 levels, and no level-n column block is a line's staircase once n > log_3(1 + 2 sqrt(3r)): a block's slope span t(3^j (i+1)) - t(3^j i) is at least 3^j/r because t(u) = u/sqrt(r^2 - u^2) has t' = r^2 (r^2 - u^2)^(-3/2) >= 1/r, and tracking is inherited downwards so the tracked levels are a run from the bottom and never a gap; at the Dirichlet width eps = 1/b^2 no level-j block tracks a denominator b >= sqrt(2 r/3^j), so no block at rank k from the top of the Psi ladder tracks a rational with b >= sqrt(6) 3^(k/2) and ranks 0 to 4 are held to b <= 2, 4, 7, 12, 22; the block cap is sharp, since a >= 1, ab + 1 <= b^2 and 8 * 3^(2j) b^4 < r^2 force a level-j block to track; and if the staircase floor(sqrt(r^2 - X^2)) agrees with a line's at three columns U, U + m, U + 2m in [0, r) then the second difference reads at least -1 from the line and at most 2 - m^2/r from the arc, so m^2 <= 3r, which gives (3^n - 1)^2 <= 12 r for a block but only w(X), h(X) <= 2 + 2 sqrt(3r) for a box and so excludes no box the arc enters and leaves through one side; hence every rank k < level - 1 - log_3(1 + 2 sqrt(3r)), which is level/2 - 2.14 ranks to leading order and never half of them, carries no frozen-slope resonance across a block, halving the exponent the refuted frozen route produced without bounding Psi, which stays Conjecture. Witness: lab/rs/circle-crop track, budget, secant, blind, boxes and boxline lines, the run asserted between the two exact integer counts on all 1116 slope rows of F_30 at r = 3^level - 1 for level = 6, 7, 8, 9, run equal to the cap at 102, 91, 93, 98 of 279 slopes and to the floor at 26, 72, 66, 63, the three-point condition attained at exactly the largest level the block cap allows at each radius, and at r = 19682 level 6 exactly 3 of the 53 boxes with their whole content equal to a line's staircase, (16, 20), (19, 19) and (20, 16), the witness that the box statement fails; crop.md THE CIRCLE COUNT.
2026-09-07 Refuted That a spectral gap of the frozen-slope transfer operator bounds Psi: at slope 1/3, where the pattern law is exact, the straight-line ladder gives log Psi rising by 0.024224 a level over level = 4..12 identically at three line offsets, so Psi ~ 1.024520^level with survival rate 0.910342 above 8/9, while at slope 1/7 it falls by 0.009633 a level; the hole is a full triadic cylinder only at sigma = 0, where the digits are independent and Psi = 1 identically. Witness: lab/rs/circle-crop derivation pass; the circle escapes the frozen model only because it tracks a resonance for k-2 levels at width 3^-k.
2026-09-19 Proved At dim = 2, with R = r/3^j, the three constants of the crossing shell's digit-rate bound are explicit: over the 3^j classes each column sawtooth sum is at most 4.5310 r R^(-1/3) + 101.0364 r R^(-1/2) + 2 r R^(-1) + 1 and each box sawtooth sum at most 4.5310 R^(2/3) + 101.0364 R^(1/2) + 3 from 3 pi 3^(-2/3) and 175 * 3^(-1/2) rounded up, the residue split costs (7/9) sqrt(2r) <= 1.1000 r^(1/2) and (10/3) sqrt R + (4/3) sqrt(2R) <= 5.2190 R^(1/2), the counting terms are exact to 2 * 3^j and 2/3, the main terms cancel to (2r + 1)/9 plus a residue below 1/9, and the errors add to 27.1860 r R^(-1/3) + 610.1581 r R^(-1/2) + 9.3334 r R^(-1) + 10.3334, which 2r + 1 >= 2r and R <= r fold into 13.60 R^(-1/3) + 305.08 R^(-1/2) + 9.84 R^(-1). Witness: van der Corput 1923 Satz 5 as restated in Laugesen and Liu Theorem 18, with lab/rs/circle-crop asserting the bound per level at r = 80, 242, 1000, 2186, 6560, 12345, 19682 and live at j = 0 for r = 212957, 531441, 2000000.
2026-09-19 Proved The geometric sum behind the drift cap is explicit: sum_(j < level) R_j^(-delta) <= 3^delta/(3^delta - 1) reads 3.2612, 2.3661 and 1.5 at delta = 1/3, 1/2, 1, which is what carries sum_(j < level) abs(p_j(r) - 1/9) <= 781, and a Huxley-type delta = 77/208 in place of van der Corput's 1/3 moves only the first of the three, to 2.9927. Witness: arithmetic on the closed form, with the drift sums 0.223603, 0.245132, 0.328170, 0.256778, 0.260804, 0.446659, 0.267260 printed against 781 by lab/rs/circle-crop.
2026-09-19 Proved The certificate abs(log ind(r)) <= 1191 splits into two explicit blocks: where R_j clears 23157375, at which the digit-rate bound first falls under 1/9 with the crossing at 23157374.055, 1 - p_j >= 7/9 and the mean value theorem gives abs(log(1 - p_j) - log(8/9)) <= (9/7) abs(p_j - 1/9), so that block costs at most (9/7) 781 <= 1004.15, while at most 16 levels fall below it, since R_j < 23157375 asks j > log_3 r - 15.44 and log_3 r >= level - 1, each costing log(3 R_j) < (i + 1) log 3 at the i-th from the top, a tail of at most log 3 * n(n + 3)/2 over the top n levels and so at most 167.0 at n = 16. Witness: lab/rs/circle-crop index lines, abs(log ind) asserted against 1191 at every level of r = 80, 242, 1000, 2186, 6560, 12345, 19682.
2026-09-19 Verified Over the 277 slopes of F_30 meeting the sharpness hypothesis a >= 1 and a b + 1 <= b^2, 1108 rows at r = 3^level - 1 for level = 6, 7, 8, 9, the proved cap #{j : 3^j b^2 < 2r} less the proved floor #{j : 8 * 3^(2j) b^4 < r^2} is never above 2 and attains it, so the two exact integer counts pin those tracked runs to within two levels at every radius swept; the remaining 8 rows are 0/1 and 1/1, where the floor reads 0 by that hypothesis. Witness: lab/rs/circle-crop track lines.

The cocycle's joint spectral radius

2026-09-06 Proved The component cocycle of mixed Kronecker words is simultaneously triangularizable over Z and its joint spectral radius is the largest fill: in the frame (gamma, h, v, phi) = ((1,1,1,1), (1,1,2,2), (1,2,1,2), (1,2,2,4)) of components, horizontal runs, vertical runs and fill, every class matrix is nonnegative, integer and lower triangular with column sums (comp(A_c), r(c), s(c), fill_c) and diagonals (0,0,0,1), (0,1,0,2), (0,0,1,2), (0,0,0,2), (1,1,1,3), (1,2,2,4), so the cross-polytope P = conv{+/- gamma, +/- h, +/- v, +/- phi} satisfies M_c P subset fill_c P with exact integer residuals and its gauge is an extremal norm; hence JSR(F) = max fill and LSR(F) = min fill on all 2^15 - 1 subfamilies, the finiteness property holds with a one-letter spectrum maximizing product, and every word over {3, 6} has spectral radius exactly 2^level. Witness: connectivity.md THE JOINT SPECTRAL RADIUS OF THE COCYCLE, lab/py/jsr-schedules.
2026-09-06 Refuted That the joint spectral radius of a component-cocycle pair is a nontrivial invariant of the pair: the JSR depends on the alphabet only through its largest fill and the LSR only through its smallest, both attained by one-letter words, so on the 78 of 105 letter pairs whose fills differ the JSR rate log max_c k_c strictly exceeds chi at every interior frequency; an exhaustive scan of all 2^level words to level = 16 on {3, 6} and {3, 7} never improves on the one-letter rates 2 and 3, and the Blondel-Nesterov lifting equals (fill_1^k + fill_2^k)^(1/k) exactly, terminating at no finite k. Witness: connectivity.md THE JOINT SPECTRAL RADIUS OF THE COCYCLE, lab/py/jsr-schedules.

The codebook

2026-09-06 Proved Under a minimum-description-length score with a catalog of A = 667 atoms and at most P = 59049 positions, the name charge log2 A + log2 P kills every atom below 31 cells outright, 131 of the 667, while every surviving atom still pays its share of log2 A; the arithmetic of the score, not a measurement. Witness: lab/py/codebook.
2026-09-06 Verified The rearranged SVD of a level-5 carpet render returns its level-1 code 495 exactly at four block splits (sigma_2/sigma_1 <= 3.3e-15) and peels the five-letter magic word carpet(3), void(3), net(3), htree(3), vtree(3) into 495, 341, 186, 455, 365; under bit flips at 51 rates and 40 seeds each the code survives 40/40 to p = 0.30, 12/40 at 0.31, 0/40 from 0.32, against a block-mean baseline whose closed-form failure f/(1 + 2f) = 0.277638, f = (8/9)^4, is derived from the fill law and not from the sweep. Witness: lab/py/codebook.
2026-09-06 Verified The three-class codebook (Kronecker tiles, magic designs, life frames) under the MDL score loses to raw deflate on the tree's own render by 17212 bits, on text by 31375 and on a halftone by 31217 at Kronecker level 2, edges it by 29 bits on random bytes where deflate itself expands the stream by 47, and crosses deflate on the render only at catalog level 3; the uniform-bit control takes zero placements and saves -18 bits at every catalog depth 2 to 5, and every corpus reconstructs bit for bit from placements plus residual. Witness: lab/py/codebook.

The Collatz carry

2026-09-14 Proved In base 2 read least significant digit first, 3n + 1 = n + 2n + 1 has digit i equal to n_i + n_(i-1) + c_i over GF(2) with n_(-1) = 0, c_0 = 1 and c_(i+1) = MAJ(n_i, n_(i-1), c_i), and the substitution M = 2n + 1 clears the constant from the skeleton: M xor 2M = 2 (n xor 2n xor 1) + 1 for every n, so the carry-free step in the M coordinate is M -> M xor 2M, elementary rule 60 with no boundary term, mirroring to rule 102 in the most significant digit first render (witness: lab/rs/carry-skeleton README THE METHOD, 0 mismatches on all three identities over every n < 2^18).
2026-09-14 Proved The Collatz carry is a four-state Mealy transducer on the state (n_(i-1), c_i) reading least significant digit first, and its dependence radius is unbounded: with u_0 = 1 and u_j = 1 - (j mod 2) for 1 <= j < level, every majority from c_1 on propagates, so c_i = u_0 for 1 <= i <= level and flipping digit 0 alone changes every digit of 3n + 1 from 2 to level. A MrlyMath automaton is a triple (dim, mask, kind) with a finite offset mask (automata.md section 7), so the step is no rung of that ladder at any level; the dial grades acceptors and not transducers, so what it grades here is the step's zero-carry set and not the step (witness: lab/rs/carry-skeleton README THE CONTROLS, the radius witness at every level = 2..40).
2026-09-14 Proved For odd m the zero-carry set of m n + 1 in base 2 is exactly the even integers accepted by the width-(deg m + 1) rule W of beneath.md's memory dial forbidding two 1 digits at a distance in the difference set {abs(j - j') : j, j' in supp m}: digit 0 of the sum reads n_0 + 1 and digit i >= 1 reads sum_(j in supp m) n_(i-j), so the set depends on that difference set alone and m = 11 and m = 15 share one rule (witness: lab/rs/carry-skeleton README READS, set equality with 0 mismatches below 2^16 at m = 3, 5, 7, 9, 11, 15).
2026-09-14 Proved The Collatz carry fires with density exactly 1/2: on uniform independent digits the state (n_(i-1), c_i) is an irreducible aperiodic Markov chain on four states with stationary vector (1/3, 1/6, 1/6, 1/3) on (0,0), (0,1), (1,0), (1,1), whose carry-on mass is 1/6 + 1/3 = 1/2 (witness: lab/rs/carry-skeleton README READS, exact integer balance residuals all 0, and the mean of d_loc over every n < 2^level reading level/2 + 1/3 - (-1)^level/(3 * 2^level) as an exact integer identity at every level = 8..22).
2026-09-14 Proved The carry-free map T_free(n) = n/2 for even n and (n xor 2n xor 1)/2 for odd n never raises the base-2 digit count, and has exactly one cycle on the positive integers, {1}, which every orbit reaches: an odd n read as f in GF(2)[x] with f(0) = 1 has T_free equal to A(f) = ((1 + x) f + 1)/x, affine with a = (1 + x)/x and a + 1 = 1/x, so A^k(f) + 1 = a^k (f + 1) and A^k(f) = f forces a^k = 1 or f = 1, and (1 + x)^k = x^k fails at every k >= 1 by the constant term. The carry-free drift is therefore exactly zero and all of the growth of a Collatz step is carry (witness: lab/rs/carry-skeleton README READS, 0 digit-count increases and 0 values failing to reach 1 over every n < 2^20).
2026-09-14 Proved The carry-on block of the Collatz carry's transfer matrix is [[0, 1], [1, 1]], characteristic polynomial x^2 - x - 1, so a carry-on run survives one further digit at rate phi/2, and the worst-case carry run over level digits is level + 1, attained at n = 2^level - 1 (witness: lab/rs/carry-skeleton README READS, worst case pinned at level = 1, 2, 4, 8, 16, 32, 40).
2026-09-14 Verified The zero-carry rules of m n + 1 carry codes 7, 95, 23, 22015, 279 at m = 3, 5, 7, 9, 11 and widths 2, 3, 3, 4, 4 under mrlynum::memory::Rule::new(1, k, code) with Rule::allowed, with card W of 3, 6, 4, 12, 5, rho of 1.618034, 1.618034, 1.465571, 1.618034, 1.380278 and kappa of 0.098239, 0.167412, 0.115204, 0.201999, 0.115524; m = 15 repeats the m = 11 row (witness: lab/rs/carry-skeleton README READS, every code rebuilt from the difference set and rho, kappa taken by mrlynum::memory::perron and mrlynum::memory::kappa).
2026-09-14 Verified The golden rule, code 7 at width 2, and the supergolden rule, code 23 at width 3, are the zero-carry rules of 3n + 1 and 7n + 1, and mrlynum::memory::kappa reads 0.098239 and 0.115204 on them, reproducing beneath.md's two published couplings from the Collatz side with no shared code (witness: lab/rs/carry-skeleton README THE CONTROLS, the two values asserted to 5e-7).
2026-09-14 Conjecture The longest carry run of a Collatz step over level uniform digits grows like log_(2/phi) level, base 2/phi = sqrt 5 - 1 = 1.236068. The mechanism is Proved, the carry-on block being [[0, 1], [1, 1]] with Perron root phi; the constant is not established, sampled mean depths 6.338, 12.116, 18.462, 24.967, 31.481, 38.029 at level = 16, 64, 256, 1024, 4096, 16384 over 200000 uniform strings each, standard error at most 0.014 per mean, giving increments per quadrupling 5.778, 6.346, 6.504, 6.514, 6.548 against the predicted 6.541119, rising towards it from below with the offset settling near -7.76 (witness: lab/rs/carry-skeleton README READS, the depth table, no exponent fitted).
2026-09-14 Refuted The local carry count d_loc(n) = popcount((3n + 1) xor (n xor 2n xor 1)) is no function of popcount, of the longest run of 1 digits, of v_2, of the digit count, nor of all four read at once: the least clashing pairs are 1, 2 at d_loc 2, 0 for popcount and for the longest run, 1, 3 at 2, 3 for v_2, 2, 3 at 0, 3 for the digit count, 3, 5 at 3, 4 for (popcount, v_2) and 19, 25 at 3, 4 for all four, with 256217518, 417177932, 659301399, 662864636, 88157572, 9331881 unordered pairs below 2^16 sharing the statistic and disagreeing on d_loc; the last row subsumes every pair, so no pair is a summary either (witness: lab/rs/carry-skeleton README READS, the refutation table, six witnesses pinned as integer pairs).

The component exponent

2026-08-31 Proved The component count of a two-letter magic word has an exact closed form on 59 of the 105 letter pairs at base 2, dim = 2, namely 9 of the 15 pairs of distinct symmetry classes and all 6 pairs inside one class: 2^(k-m) with k the last unit place and m the unit count for a unit against a domino (16 pairs), 2^(number of diagonal letters) for a unit against a diagonal (8), 2^(level-r) with r the terminal run for two dominoes of unlike orientation (4), 2^k with k the last diagonal place for a domino against a diagonal (8), 2^(n-j) with n the number of full letters and j their terminal run for a domino against the full tile (4), 1 for the gasket class against the full tile (4), and 1 inside a class except 2^level inside the diagonal class (15); proved by the zero-contact cut on the first four families, again by a rank-1 telescope on the domino against the diagonal, by the row-block argument on the domino against the full tile, and by the contact split with the fill >= 3 line on the rest; exhaustive per pair on all 32766 words of length at most 14 against the representation and all 254 words of length at most 7 against the drawn cells, zero mismatches. The other 46 pairs are closed by the suffix recursion below. Witness: lab/rs/magic-words, connectivity.md.
2026-08-31 Proved The zero-contact cut: since contacts multiply, a suffix has h = v = 0 as soon as it holds one letter with h = 0 and one with v = 0, adjacent copies of that suffix tile can never merge, and comp(A_w) = fill(A_prefix) * comp(A_suffix) at the last such suffix; 49420 of all 54240 words of length at most 4 admit the cut with zero mismatches. Its scope is part of the law: the other 4820 words carry nonzero contact in one direction at every suffix, and every word over a domino and the full tile is among them, so that family needs the row-block argument instead. Witness: lab/rs/magic-words, connectivity.md.
2026-08-31 Proved The row-block law, and the one non-trivial exponent found: over a domino and the full tile the filled set is a product of a row set with the whole column range, |R| = 2^n at n full letters, each row of R is a full line, and two rows of R are adjacent exactly inside a block of 2^j rows at j the terminal run of full letters, so comp = 2^(n-j); at equal letter frequencies the exponent is (log 2)/2 against per-letter values 0, 0 and a fill ceiling of (3/2) log 2, strictly between them, which no other named pair achieves. Witness: lab/rs/magic-words, connectivity.md.
2026-08-31 Proved On every one of those 59 pairs, along any word in which both letters occur with positive frequency, the component growth rate exists, is a function of the letter frequency vector alone, and is read off the closed form; existence is earned from the formula and not assumed. The hypothesis is load-bearing and the same statement without it is false: over {3, 6} at frequency vector (1, 0) the constant word 3^level has rate 0, the word carrying the diagonal letter at the square places has rate log 2, and the word carrying it at the powers of 2 has upper rate log 2, lower rate (log 2)/2 and no limit, the last two having orbit closures that are uniquely ergodic but not minimal. Witness: lab/rs/magic-words, connectivity.md.
2026-08-31 Proved The constant word is a degenerate probe for the component exponent: comp(A_(c^level)) = comp(A_c)^level with comp(A_c) = 2 exactly on the diagonal class, so the one linear functional exact on constant words is Phi(f) = (f_6 + f_9) log 2, and it fails at every interior frequency on 7 of the 9 named class pairs, holding only for a unit against a diagonal and for the gasket class against the full tile; the failure is a wrong shape rather than a wrong coefficient, since the true exponent extends to no vertex. The caveat travels with the result: on five of the seven, 28 of the 32 letter pairs, the exponent equals the fill exponent and so saturates the trivial ceiling comp <= fill, carrying nothing the order-blind fill law did not already give. Witness: lab/rs/magic-words, connectivity.md.
2026-08-31 Verified The Thue-Morse run structure that makes the exponent exact: no three equal letters in a row, so every terminal run has length at most 2 and every prefix rate is within 2/level of its limit; exactly level/2 of each letter at every even length, not merely in the limit; over all lengths to 2^20 the terminal run of one letter takes the value 0 on 524288 prefixes, 1 on 349526 and 2 on 174762; the first 2^20 letters hold 349525 complete runs of length 1 and 349525 of length 2 with one unfinished run at the cut; and the run-boundary word t_n xor t_(n+1) is the period-doubling word on all 1048575 terms. Witness: lab/rs/magic-words.
2026-08-31 Proved The remaining 46 letter pairs carry exact closed forms too, so all 105 are solved, and the mechanism is a suffix recursion rather than anything spectral: a heavy suffix letter, meaning one of the five connected codes 7, 11, 13, 14, 15 that carry both contacts, leaves the count unchanged because the block graph of A_w (x) A_c is isomorphic to the cell graph; a zero-contact suffix letter, meaning one of 1, 2, 4, 8, 6, 9, leaves isolated cells and collapses the count to a fill; and a domino suffix letter turns the count into the number of maximal runs, whose recursion H(A_wq) = H(A_w) + fill(A_w) at a gasket letter telescopes. Hence comp(A_w) = fill(A_(w_1..p)) at p the last zero-contact place on the 30 pairs of a heavy letter against a light one, giving 3^(g-j), 2^d 3^(g-j), 4^(F-j) and 2^d 4^(F-j) with j the terminal heavy run, and comp(A_w) = 1 + sum of fill(A_(w_1..i-1)) over the gasket places i at or before the last domino place m on the 16 gasket-against-domino pairs, (3,7) and (5,7) among them, the eight column-domino pairs being the transpose of the eight row-domino ones; exhaustive on all 753572 words of length at most 13 against the representation and all 11684 of length at most 7 against the drawn cells, zero mismatches. Witness: lab/rs/magic-words, connectivity.md.
2026-08-31 Proved At every interior letter frequency the component exponent exists, is order-blind and equals the fill exponent f_1 log fill_1 + f_2 log fill_2 over the two letters on all 46 of those pairs; existence and order-blindness hence hold on all 105, saturation on 89 of them; the interior hypothesis is used exactly once, to force the terminal run of the heavy letter to be o(level), since a run of eps level would freeze the other letter's count on that block and contradict its positive frequency, and the gasket-against-domino case runs through the sandwich T < comp(A_w) <= 1 + (3/2) T at T = fill(A_(w_1..i*-1)) and i* the last gasket place at or before the last domino place, measured in [1.0004, 2.0000] on six named words with no violation. Nothing is claimed for a word whose letter frequencies fail to exist. Witness: lab/rs/magic-words, connectivity.md.
2026-08-31 Proved The two rates the study could not identify are identified exactly: along the Thue-Morse word over any of the 16 gasket-against-domino pairs, (3,7) and (5,7) included, the component exponent is (1/2) log 6 under either letter reading, with the two-sided certificate |log comp(A_(w_1..level)) - (level/2) log 6| <= log 108 + (1/2) log(3/2) < 4.885 at every level >= 4, because the word is cube-free, which caps the sandwich suffix at fill/T in [6, 108], and balanced, which pins the gasket-letter count to within 1/2 of level/2; the study prints the value as 0.895879734614027 nats, a float labelled as such. Measured to level = 2^14 on all 16 pairs and both readings the largest deviation is 4.273459 nats against the certificate's 4.884864. Witness: lab/rs/magic-words, connectivity.md.
2026-08-31 Proved The frequency functional Phi(f) = (f_6 + f_9) log 2 is refuted at every interior frequency on 78 of the 105 letter pairs and exact on 27, the 46 new pairs being refutations to a pair; and the exponent saturates the trivial ceiling comp <= fill on 89 of the 105 and falls short on 16, so the domino against the full tile is the unique class pair on the whole alphabet, and not merely among the named 59, whose exponent sits strictly between the constant-word values and the fill ceiling. The comparison is against the value of Phi and never against a periodic word. Witness: lab/rs/magic-words, connectivity.md.
2026-08-31 Proved The interior hypothesis is sharp on a pair carrying no diagonal letter, so the pathology is not a property of the diagonal class: over (3, 7) at frequency (1, 0) the constant word 3^level has rate 0, the gasket at the square places gives log 2, and the gasket at the powers of 2 has upper rate log 2, lower rate (log 2)/2 and no limit, with comp pinned to fill(A_(w_1..i*-1)) at i* = 2^k throughout (2^k, 2^(k+1)], so the accumulation set of the prefix rate is the whole interval [(1/2) log 2, log 2] and not its two endpoints. Witness: lab/rs/magic-words, connectivity.md.
2026-08-31 Proved A common invariant cone exists and the gasket-against-domino pair is primitive in it, and none of it is needed: phi = (1,2,2,4)^T is a common right eigenvector with M_c phi = k_c phi, so it normalises the row orbit and only that orbit, and in the resulting chart comp/fill = 1 - b - c with the letters acting by N_gasket(b,c) = ((1+b)/3, (1+c)/3) and N_domino(b,c) = ((1+b)/2, 0); the set {0 <= b <= 1, 0 <= c <= 1/2, b + c <= 1} is invariant under both and the length-3 word gasket-domino-gasket maps it strictly inside, vertex images (5/9,1/3), (11/18,1/3), (5/9,1/3), (7/12,1/3) with b + c at most 17/18, three being minimal since domino-gasket sends (1,0) to (2/3,1/3) on the face. Witness: lab/rs/magic-words, connectivity.md.
2026-08-31 Verified A by-product and a smoothness split: comp(A_((7,3)^k)) = (6^k + 4)/5, reading 2, 8, 44, 260, 1556, 9332 and checked to k = 8 against both the closed form and the representation, a stationary control whose per-letter rate is (1/2) log 6 again; and since every closed form on the other 89 pairs gives a count of the form 2^i 3^j, the gasket against a domino is the only family whose counts are not smooth, the largest at level = 8 over (3, 7) being 1094 = 2 x 547. Witness: lab/rs/magic-words, connectivity.md.
2026-08-31 Conjecture Which words with no letter frequencies carry a component exponent at all, now that all 105 letter pairs have closed forms and interior frequency settles the rate: over (3, 7) the tripling word W_(k+1) = W_k 7^|W_k| 3^|W_k| keeps both letters at lower density 1/4 and still has its prefix rate range over [0.4792, 1.4379] in log 2 units on 1024 <= level <= 4096 with no narrowing, so positive lower density is the wrong hypothesis, but that is measurement and not a proof that the limit fails. Witness: lab/rs/magic-words, connectivity.md.
2026-08-31 Conjecture The accumulation set of the saturation comp/fill along Thue-Morse over a gasket-against-domino pair, plausibly the attractor of the two affine chart maps read along the word; the sampled value 0.2325367033 at level = 4096 is a term of an oscillation and is neither a limit nor a maximum, the exact maxima at level >= 5 being 43397/186624 and 151/648 under the two readings. Witness: lab/rs/magic-words, connectivity.md.
2026-08-31 Conjecture Whether any alphabet of three or more letters, or any other order-sensitive observable, makes the component-style exponent depend on more than the letter frequencies, which is what a non-stationary result would need; at interior frequency on two letters it provably does not, on any of the 105 pairs. Witness: magic.md, connectivity.md.
2026-08-31 Conjecture Whether the same suffix recursion that closes the 46 reproduces the 59 forms proved by other means, which would collapse the whole table to one lemma, and whether the Euler, boundary and holes series of Hankel ranks 4, 8 and 11 answer to it as well. Witness: connectivity.md.
2026-08-31 Refuted That a difference between the component growth rate and the frequency-average prediction is a non-stationary result - on the whole settled class the exponent is a function of the letter frequencies alone, so Thue-Morse returns exactly what a periodic word of the same frequencies and almost every Bernoulli word return, and the tree's own stationary controls miss the prediction by the same amount; the difference refutes the frequency functional, not stationarity, and an aperiodic word witnesses nothing here. Witness: connectivity.md, lab/rs/magic-words.
2026-08-31 Refuted That the component exponent is a function of the letter frequencies for every word whose frequencies exist - over {3, 6} the frequency vector (1, 0) carries the constant word at rate 0, the diagonal-at-squares word at rate log 2, and the diagonal-at-powers-of-2 word with no rate at all; the statement holds only where both letters have positive frequency, and the same correction restores every family's closed-form rate. Witness: connectivity.md, lab/rs/magic-words.
2026-08-31 Refuted That the top Lyapunov or matrix-norm exponent of the cocycle is the component exponent, and with it every route to chi through a norm theorem, a joint spectral radius or a projective contraction of forward orbits - along 3^inf the largest entry of M_3^level is exactly 2^(level+2) - 2, reading 6, 14, 62, 1022, 262142, 17179869182 at level = 1, 2, 4, 8, 16, 32, so the norm exponent is log 2, while comp(A_(3^level)) = 1 at every level and the component exponent is 0; the observation functional gamma is a fixed vector of both heavy matrices and so sits on the boundary of the dual cone, which is the geometric form of the same obstruction. Witness: connectivity.md, lab/rs/magic-words.
2026-08-31 Refuted That entrywise positivity in the standard basis decides whether the letter matrices preserve a common cone - neither M_3 nor M_7 is a non-negative matrix, so the test never applied, and none of the 8190 products of length at most 12 is entrywise positive, yet a common invariant cone does exist in the chart normalised by the right eigenvector phi = (1,2,2,4)^T and the pair is primitive in it at length 3. Witness: connectivity.md, lab/rs/magic-words.
2026-08-31 Refuted That the Thue-Morse word is a named word along which the component exponent fails to exist - it converges, exactly, to (1/2) log 6 on every gasket-against-domino pair with a two-sided certificate; the genuine non-existence witness is the gasket at the powers of 2 at the boundary frequency (1, 0), whose orbit closure is countable and not minimal, so no uniquely ergodic minimal word is implicated. Witness: connectivity.md, lab/rs/magic-words.

The design Mobius meter

2026-09-07 Verified The design Mobius meter oscillates at the zeta ordinates and not at the design's pole lattice. Read M_F(x)/x^(alpha/2) uniformly in log x, Hann-windowed, against a local-median floor and a null of rigid shifts of each candidate list: at base 10 with the digit 9 missing all six strongest peaks sit within one bin of a nontrivial zeta zero, offsets 0.068 to 0.216, with the full-set control at the same depth reading ten of ten, offsets 0.018 to 0.196. Thirteen zeta ordinates are reachable in the band 4 < gamma < 60, so a peak lands within one bin of one by chance with probability 0.159 and six of six is P = 1.6e-5. The pole lattice 2 pi j/log base scores -0.592, -0.640, -0.734 at base 3 {0,1}, base 3 {0,2} and base 5 {0,1}, below its own null, while the counting function over the identical elements scores 3.602, 3.764 and 3.973, so the pipeline would have seen a lattice and there is none. Witness: lab/py/design-meter verb spectrum, lab/rs/mobius-designs, A084237.
2026-09-07 Proved The identity M_F(x) = sum_(n <= x) mu(n) A_F(n)/n + R_F(x) defines R_F at every base and digit set, and the echo's size splits at alpha = 1/2. Partial summation gives sum_(n <= x) mu(n) A_F(n)/n = A_F(x) H(x) - sum_(m in S_F, m <= x) H(m-1) with H(y) = sum_(n <= y) mu(n)/n, which is O(y^(-1/2 + eps)) under RH, so the echo is O(x^(alpha - 1/2 + eps)) when alpha > 1/2, while for alpha < 1/2 the second sum converges absolutely and the echo tends to the constant sum_n mu(n) A_F(n)/n, which is nonzero: base 16 {0,1} reads -0.0937, -0.1330, -0.1242, -0.1051, -0.1099 at 10^3 to 10^7 against x^(alpha - 1/2) falling 0.1778 to 0.0178, and base 10 {0,1} reads -0.0500 at 10^7 against 0.0405. Against the square-root bar x^(alpha/2) the echo dies at x^(-min(alpha, 1 - alpha)/2), equal to 1 only at alpha = 1, so the zeta zeros neither obstruct nor help the square-root conjecture, which is a statement about R_F alone. Verified separately at two designs: at base 10 missing 9 the echo carries six of six top peaks at zeta zeros and the residual none of the two it has, the echo being 0.1342 of the meter in root mean square against 0.6476, and the echo's share of the meter falls 0.356028, 0.242495, 0.207229 there and 0.208549, 0.099001, 0.047902 at base 3 {0,1}, share over prediction reading 1.0000, 0.7387, 0.6846 and 1.0000, 0.9219, 0.8663, each design decaying at least as fast as its own rate. Witness: lab/py/design-meter verb spectrum, mobius.md.
2026-09-07 Refuted A family law for the design meter's frequency set: it is not a function of (base, alpha). Base 3 {0,1} and base 9 {0,1,2,3} share alpha = 0.630930, element count 1048575 and log range to within 1.4%, and their meters split ten peaks against none, where support-matched random-sign meters reach 0 to 4 peaks on the first support and 0 to 2 on the second over eight draws each, so the ten sit above their own null and the none does not; base 9 {0,1,2,3} and base 9 {0,1,3,4} share base and alpha and split the same way, the second being base 3 {0,1} element for element since its digits are the base-3 pairs 00, 01, 10, 11. The scaled pair base 3 {0,1} and {0,2} shares eight of ten peaks, so the scaling transfer of mobius.md carries into the spectrum while no (base, alpha) law does. Witness: lab/py/design-meter verb family, mobius.md.
2026-09-19 Refuted The primitive quadratic Dirichlet L-zeros of conductor 3, 4 or 5, the quadratic conductor each base carries, are no frequency family of the design meter: over eleven designs the largest score is 0.372 against a null of 0.341 at base 4 {0,1,2} and the widest gap over a null is 0.371 against 0.290 at base 3 {1,2}, while the zeta ordinates on the same meters reach 1.130 against 0.392 at base 10 missing 9, so the pipeline would have seen an arc family; the wider prediction over arcs of denominator base^j is untestable by this spectrum and is not claimed. Witness: lab/py/design-meter verb spectrum.

The diagonal cut

2026-08-28 Conjecture Cook's (2011) Menger-slice code cuts along the normal (1, 1, 0.5), not the centroid diagonal, so the base-3 diagonal cut is not Cook's slice; the reference row scoring the cut as Cook's is kept at yes with that correction recorded beside it, and the row once scored yes in error now reads no.

The digit-restricted Mobius exponent

2026-09-01 Conjecture theta(F) = 1/2 for every digit set with 2 <= |F| <= base - 1 and squarefree digit gcd - square-root cancellation against the set's own counting function: the 47 running-maximum exponents across base = 3, 4, 5, 10 read 0.4465..0.5358 with last-five-level drifts 0.0157..0.1056, while the full-set controls, whose limiting exponent is 1/2 under RH and at least 1/2 unconditionally, read 0.4413..0.4517 at the same depths; the finite tables are consistent and decide nothing, single-cut exponents scattering 0.22..0.53 on the same data. Witness: lab/rs/mobius-designs, mobius.md.
2026-09-19 Conjecture An unconditional Mertens-shape bound on the dense column: for F omitting exactly one digit, base >= 92317 and x = base^level with level past a point depending on base alone, abs(M_F(x)) <= C(base) A_F(x) exp(-c(base) sqrt(log x)) with C(base) and c(base) > 0 effective, through a Dirichlet-approximation dissection whose region A is the ladder rung b = 4/5 and whose one load-bearing minor-arc input is unread. Witness: lab/rs/mertens-numerology for the wall 92317, Maynard 2022 for the imported lemmas.
2026-09-19 Proved That dissection is dead at fixed digit count: {0,1} at base 3 has l^1 exponent log 2 / log 3 = 0.630929, above every bar the dissection sets, region B's 1/4 and region A's own ask included, the l^1 floor again in arc-local form. Witness: exact arithmetic in the sentence that prints it.
2026-09-19 Proved Vaughan's identity is circular here at power strength: the mu_{<=U} * mu_{<=U} * 1 piece carries the main term fill^level M_1(U)^2, so bounding the pieces one by one at power strength forces M_1(U) << U^(-delta), which continues 1/zeta into sigma > 1 - delta. Witness: the identity, carried out in the sentence that prints it.

The digit-restricted Mobius meter

2026-09-01 Proved Carry-free scaling ties the digit designs' Mobius meters together: for digit sets F = a F' inside {0..base-1}, m -> a m is a digit-length-preserving bijection S_F' -> S_F (each scaled digit stays below base, so no carry occurs), giving M_F(base^level) = sum mu(a m); a square factor in a kills the meter identically (F = {0,4} at base = 5: zero at all 21 levels), and prime a = p gives M_(pF')(base^level) = -sum_(p not | m) mu(m), the {0,2} column at base = 3 reading as the {0,1} column twisted by the Thue-Morse sign of the binary index; asserted at every level on all eight scaled census families. Witness: mobius.md, lab/rs/mobius-designs.
2026-09-01 Proved The base-4 anti-symmetry M_{0,2}(4^level) = -M_{0,1}(4^level): 4 | base forces every element of S_{0,1} to 0 or 1 mod 4, so even elements carry mu = 0 and M_{0,2}(x) = -M_{0,1}(x/2) at every real x, running maxima included since S_{0,1} is empty strictly between (4^level - 1)/3 and 4^level; exact at all 22 levels, -110/110 at level = 15, 34/-34 and shared Mmax = 1553 at level = 22. Witness: mobius.md, lab/rs/mobius-designs.
2026-09-01 Proved No Euler product for a digit design: S_F is not multiplicatively closed, witness 4 = 11_3 and 13 = 111_3 in S_{0,1} at base 3 with 4 x 13 = 52 = 1221_3 outside, so M_F is not the coefficient sum of an inverse Dirichlet series; the series itself is built literature (abscissa Kohler and Spilker 2009, continuation and poles Burnol 2026) and carries no Mobius sum anywhere. Witness: mobius.md, REFS.md.
2026-09-01 Verified The digit-restricted Mobius census: exact M_F(base^level) and running maxima max |M_F(x)| for all 38 digit sets with 2 <= fill <= base - 1 at base = 3, 4, 5 (depths 24, 22, 14, 21, 13, 11 by class), the ten base-10 one-digit-excluded columns to 10^8, and full-set controls to 3^17, 4^13, 5^11, 10^8; factorization and sieve agree on the base = 3 {1,2} family at every level to level = 16, the base-10 control reproduces A084237, and an independent second-language recompute matched 99 sampled rows exactly. Witness: lab/rs/mobius-designs, mobius.md, A084237.
2026-09-03 Proved A power saving for the Mobius meter on the dense digit columns, under GRH: assume L(s, chi) has no zero in sigma > 1/2 for every Dirichlet character chi, let F omit exactly one digit e_0, and let base >= 1499, or base >= 1032 when e_0 is 0 or base - 1; then for every eps > 0 and all x >= 2, |M_F(x)| <<_{base,eps} x^(3/4 + c'_base(e_0) + eps) with c'_base(e_0) = log PB'_base(1, e_0)/log base, PB_base(1) = 1 + Phi_base/base and Phi_base = (4/pi) base + (2q/pi) H(ceil((base-2)/2)) + (1 - 2/pi)(base-2) + 0.727, and 3/4 + c_base < alpha_base = log(base-1)/log base, so |M_F(x)| <<_{base,eps} A_F(x)^(1 - delta_base + eps) with delta_base = (alpha_base - 3/4 - c_base)/alpha_base > 0, every fixed delta' < delta_base delivered and the endpoint never; orthogonality mod base^level, the shifted-grid l^1 recursion c_level <= B_base(F) c_{level-1}, the kernel bound B_base(F) <= base PB_base(1) from sin(pi v) <= 4v(1-v) and 1/sin x <= 1/x + 1 - 2/pi with Parseval exact on the excluded digit, and the assembly with its geometric sum are derived, and the uniform max_theta |sum_{n <= x} mu(n) e(n theta)| <<_eps x^(3/4 + eps) of Baker and Harman 1991 is quoted at source; the corollary at m excluded digits runs whenever PB_base(m) < (base-m) base^(-3/4), which holds at m <= 6, 78, 451 at base = 10^4, 10^5, 10^6 and asymptotically for m <= base^(1/2)(1-o(1)), and c_base -> 0 gives delta_base -> 1/4. The attempt to break it drives the chain below the wall, where the failure is quantified rather than hidden (c_base = 0.28087 against alpha_base = 0.999855 at base = 1000), checks the exponent test against the constant-space certificate gap_base(m) = (base-m) base^(-3/4) - PB_base(m) > 0 at every 3 <= base < 20000, the cancellation-reduced and direct forms of delta_base against each other to 10^-9 relative at every printed base, and Phi_base against the exact shifted-grid kernel sum on a 4001-point grid at base = 50, 101, 200, where it is loose by under 20%. Witness: mobius.md a power saving under GRH at large base, lab/rs/mertens-numerology.
2026-09-03 Proved The ladder above that theorem, and its floor: for 1/2 <= a < 1, if L(s, chi) has no zero in sigma > a for every Dirichlet character then the same five steps give |M_F(x)| <<_{base,eps} A_F(x)^(1 - delta_base(a) + eps) with delta_base(a) = (alpha_base - b(a) - c_base)/alpha_base > 0 at every base >= base_0(a), b(a) the smaller of the Baker and Harman 1991 table and Zhang 2024 Theorem 1.1 (Zhang strictly smaller inside (1/2, 4/7) and equal at both ends, by the factorisations -5(a - 1/2)(a - 2/5)/(4 - 2a) and -7(a - 4/7)(a - 4/5)/(4 - 2a), with b(a) >= 3/4 throughout), so every common zero-free half plane buys the saving and GRH is only its first rung, the price of a weaker hypothesis being paid entirely in the base; the wall base_0(a) exists and is a true least base at every a, since PB_{base+1}(1) - PB_base(1) < 1.291/(base-2) for base >= 40 while the mass term gains (1-b)(base+1)^(-b) per step, so the gap steps up at every base >= Q(b), the least base with (1-b)(base-2)(base+1)^(-b) >= 1.291, and below that it is negative: exhaustively on 3 <= base < 3690, and on [3690, Q(b)] by a majorant with one interior minimum whose endpoint values are both negative. The attempt to break it looks for a rung the floor misses and finds none: at every b in [3/4, 1), printed rung or not, minimality of Q(b) gives gap_{Q(b)}(b, 1) < -1.56 and a majorant below -0.95 at both ends, with any b < 1417/1850 forcing Q(b) <= 1486 and an empty range, the constants reproduced on a b-grid across the whole interval. Witness: mobius.md a power saving under GRH at large base, lab/rs/mertens-numerology.
2026-09-03 Verified The rungs of that ladder: (a, b(a), source, base_0(a), Q(b)) reads (1/2, 3/4, both, 3690, 723), (13/25, 1417/1850, Zhang, 8578, 1486), (11/20, 913/1160, Zhang, 33547, 4754), (4/7, 4/5, both, 92317, 11221), (3/5, 4/5, BH, 92317, 11221), (2/3, 5/6, BH, 3107080, 216023), (3/4, 7/8, BH, 6939524168, 129458304), then (4/5, 9/10, BH, <= 3.09358e13, 128606353005), (9/10, 19/20, BH, <= 3.23663e34, <= 1.73431e28) and (19/20, 39/40, BH, <= 9.24614e83, <= 3.30712e68), a wall printing as an exact integer only below 2^53 with both neighbouring gaps above 1024 ulps and otherwise as an upper bound on the least base; the GRH rung reproduces the wall 3690 and the margin there is delta_base <= -2.395807653 * 10^-6 at base = 3689 against delta_base >= 5.863425182 * 10^-6 at base = 3690, with gap_base(1) <= -1.533059397 * 10^-4 and >= 3.752213034 * 10^-4; the m-budget at base = 10^7 falls 1971, 1002, 365, 176, 176, 8 along the rungs below that base. The attempt to break them reproduces every wall under 4 * 10^6 by an exhaustive scan from base = 3 against the bisection, requires Q(b) < q_0(a) at every rung, sweeps 3 <= base < 3690 for an early close at every rung and finds none, and pins each rendered row as a string. Witness: lab/rs/mertens-numerology, mobius.md a power saving under GRH at large base.
2026-09-03 Proved The l^1 floor is a wall on the method, not on the problem: sum_{r mod base} |g_F((t+r)/base)|^2 = base fill exactly, so sum_{r mod base} |g_F((t+r)/base)| >= base fill / max_r |g_F| >= base for every t, the shifted-grid recursion never contracts, B_base(F) >= base and c_base >= 0 at every base and every digit set; hence the decomposition needs alpha_base > 3/4, that is fill > base^(3/4), and every fixed-fill column, F = {0,1} at base = 3 included, is beyond it with or without GRH, so it never meets the census or the exponent conjecture. The same floor kills the two neighbouring routes: Davenport's unconditional x (log x)^(-A) in the quoted step exceeds A_F(x) by the power x^(1 - alpha_base), so no unconditional saving follows inside this decomposition without an input of zero-free-strip strength, and Cauchy-Schwarz with Parseval on both factors gives exponent (1 + alpha_base)/2 > alpha_base, worse than trivial. The attempt to break it hunts a negative c_base over 3 <= base < 5000 and a PB_base(1) below 1 and finds neither, Parseval forbidding both. Witness: mobius.md a power saving under GRH at large base, lab/rs/mertens-numerology.
2026-09-03 Verified The cost-out of that saving against a hypothetical Type I defect: with the saving delta_base set beside the defect exponent m/(2(base-m) ln base) carried by a level-x^(alpha_base/2) distribution bound for the digit strings, a bound no page here states, the saving is below the defect at the wall (5.86342e-6 against 1.65022e-5 at base = 3690, a factor above 2.8) and above it from base = 3692 on, the least such base in a scan of 3690..10^5 in which the difference rises at all 96310 steps, monotonicity beyond the scan unproved; at base = 10^9 it is 1.16951e-1 against 2.41275e-11, and the tightest corollary row base = 10^6, m = 451 reads 3.14081e-5 against 1.63296e-5. The attempt to break it checks the crossover for a premature crossing at base = 3690, 3691 and for a single down-step in the scan and finds none, and holds the yardsticks apart: delta_base is normalised to the mass, so as a power of x the saving is x^(alpha_base delta_base) with alpha_base >= 0.99993 on every row compared, while the defect multiplies fill^level. Witness: lab/rs/mertens-numerology, mobius.md a power saving under GRH at large base.
2026-09-03 Conjecture That the defect x^(m/(2(base-m) ln base)) of a level-x^(alpha_base/2) distribution bound for the digit strings, a bound no page here states, is absorbed by the GRH saving at all: the cost-out sets two exponents from two unrelated statements on two yardsticks side by side and no derivation joins them, so it is neither a necessary condition nor a proof that a Type I estimate for M_F follows, the string-to-interval bookkeeping and the bilinear half of any such argument being untouched; the comparison is decided at the wall and nowhere else, lost there by a factor under 3 and won two steps later, so any sharper constant that moves q_0 must be re-costed rather than inherited. Witness: lab/rs/mertens-numerology, mobius.md a power saving under GRH at large base.
2026-09-06 Verified The coefficient sequence a real Vaughan decomposition hands the bilinear sum is not the sequence that beats the method's diagonal floor: at the eight swept boxes with both sides above x^(2/5), the boxes the identity produces, the Type II coefficient sum_{d | l, d <= x^(2/5)} mu(d) takes values in {-1, 0, 1} at seven of the eight and its full quadratic form sits in [0.6929, 1.2045] of its own diagonal, where a sign vector engineered against the column reads 0.2043 on such a box; over all eighty coefficient cells of the census, sixteen boxes, four families, two depths, two cuts and five real sequences, the form over the diagonal stays in [0.3138, 52.6676] with none below 0.1 and every departure from the swept band upward. Witness: lab/rs/rho-decoupling section menergy signed vaughan.
2026-09-06 Proved The large-values refinement of the moment route is the l^2 route itself: splitting the grid at |hat F_level(a/base^level)| >= fill^level x^(-eta), bounding the large set by its l^2 mass under the fourth moment and Parseval and the rest by the threshold, all against Parseval on the bilinear side, gives the exponent max(min(alpha + 1/2 - eta, (1 + alpha)/2), min((1 + alpha)/2, alpha + (nu_4 + 2 eta)/2)) = (1 + alpha)/2 identically at every eta >= 0 and every digit set with alpha < 1, and the large-sieve constant of any grid subset for base^level consecutive frequencies is base^level exactly, so no spacing enters. Witness: lab/rs/rho-decoupling section riesz large values chain, 66 rows with c = -(1 - alpha)/2.
2026-09-06 Verified The large frequencies are adjacent or isolated grid points, 407 in 331 runs at {0,1} base 3 level = 12 eta = eta_4 against the fourth-moment count 4096, least gap 1/base^level, large-sieve constant on D_level in 1.06009e5..2.13280e5 nearly at its l^2 floor 1.05611e5; at the eight dense census cells the Type II sum at a_m = b_l = 1 on the box M = N = floor(x^(1/2)/2) is the representation count, 0.26 to 0.41 of fill^level, and the balanced sum at a_m = 1_(base | m), b_l = 1 a fixed share of fill^level, so no bound uniform over bounded coefficients holds there; the sparse cell {0,1} base 100 level = 3 is void at the box. Witness: lab/rs/rho-decoupling sections riesz large values and riesz large values witness.
2026-09-06 Proved The second-largest grid value of the digit transform is max_(a != 0) |hat F_level(a/base^level)| = fill^(level-1) max_(b != 0 mod base) |g_F(b/base)|, equal to fill^(level-1) at one excluded digit and at {0,1} base 3, so the large set is the zero frequency alone exactly below eta_1(level) = log(1/gamma_1)/(level log base), a threshold that vanishes with depth. Witness: lab/rs/rho-decoupling section riesz large values cells, eight cells at 1/fill.
2026-09-06 Conjecture Whether a Vaughan decomposition's coefficient sequence, a convolution and not a free sign vector, can be steered near the engineered sign vector that beats the Cauchy-Schwarz diagonal floor by a factor thirty-eight at a top box; and whether the arc regime M, N >= x^(2/5) carries a dyadic box with R = x^(alpha - o(1)), the middle-divisor question on which the balanced route's refutation for alpha < 2/5 is conditional. Witness: lab/rs/rho-decoupling, sections menergy signed engineered and menergy type II.
2026-09-06 Refuted The adversarial pass on the census: an independent linear-sieve recompute in a second language rebuilt 99 rows - nine families, four controls, one excluded-digit column, meters, counts and running maxima - and first DISAGREED on eleven {0,1}-family rows, traced to the recompute itself double-counting the boundary base^l its length filter had already caught; fixed, it agrees on all 99. Two generator runs differ in zero of 784 shared rows, and a first-draft page table assembled by hand was wrong in multiple cells before every page table was switched to script extraction from the generator's printed rows. Witness: lab/rs/mobius-designs.
2026-09-06 Refuted That the Mobius signs cancel the digit column's off-diagonal multiplicative correlation better than an unstructured sign vector on the same support: over sixteen boxes |Sigma_mu| is 0.0913 to 0.7178 of the random-sign root mean square against 0.0359 to 1.5048 for the support-matched controls, the split against those controls is 3, 9, 4 at chi-square 0.375 against the uniform-rank null, and the fifteen-of-sixteen advantage over Liouville is the support of the Mobius function; a sign vector engineered against a known column drives the same Cauchy-Schwarz bound to 0.0265 of its diagonal floor, so the census refutes the arithmetic of the coefficients and not the method. Witness: lab/rs/rho-decoupling, sections menergy signed, menergy signed summary and menergy signed engineered.
2026-09-07 Proved The major-arc input for mu on a digit design is effective. Let F be a digit set with fill >= 2 every prime of whose digit-difference gcd divides base, condition (E) in one dimension, and let x = base^level. Every real primitive Dirichlet character whose modulus has all its primes dividing base has conductor dividing 8 rad(base), so the possible exceptional zeros run over a set of size bounded in base and Siegel's theorem is never invoked; with that, x^(-1) Sum_{a in M} hat F_level(a/x) S_mu(-a/x) is at most fill^level exp(-c sqrt(log x)) with c effectively computable, over the arcs |a/x - b/d| <= (log x)^C/x with d <= (log x)^C, and there is no main term at any arc. Witness: mobius.md The pair route.
2026-09-07 Proved Under condition (E) in one dimension and the large sieve Sum_{d <= Q} Sum_{gcd(b,d)=1} |hat F_m(b/d)| << fill^m (Q^(2 alpha_1) + Q^2 base^(-m(1 - alpha_1))) at every scale m <= level, with alpha_1 < 1/2 the sup-over-shift l^1 exponent, a digit design's level of distribution survives restriction to an initial segment: Sum_{d <= Q, gcd(d,base)=1} max_{y <= x} |#{n in D_level : n <= y, d | n, gcd(n,base)=1} - (1/d) #{n in D_level : n <= y, gcd(n,base)=1}| << fill^level (log x)^(-B) at Q <= x^(1 - alpha_1)(log x)^(-C), the same level as the full-range statement and one power of log x less saving, because the transform's error is uniform in the target residue and the segment splits into at most fill blocks per scale. Witness: mobius.md The pair route.
2026-09-07 Proved The hybrid bound that carries the digit-restricted bilinear estimate holds at every base with the digit set's own dimension as its exponent. Let F be a digit set with fill = abs(F) >= 2, alpha = log(fill)/log(base), sup-over-shift l^1 exponent alpha_1, and assume the shifted and perturbed large sieve it supplies by Farey spacing, sup over shifts of Sum_{a <= d} sup_{abs(eta) <= delta} F_Y(a/d + shift + eta) << (1 + delta d)(d^(alpha_1) + d Y^(-(1 - alpha_1))) at every scale. For D, E, Y, Q_1 powers of base with D E << Y, Q_2 >= 1, q_1 ~ Q_1 coprime to base and d ~ D all of whose primes divide base, the sum of F_Y(a/(d q_1 q_2) + eta) over q_2 ~ Q_2 coprime to base, over a < d q_1 q_2 coprime to d q_1 q_2, and over abs(eta) <= E/Y with (eta + a/(d q_1 q_2)) Y an integer is << (D E)^(alpha_1) (Q_1 Q_2^2)^(1 - alpha) + E^(alpha_1 + alpha/2) D^(1 + alpha/2) Q_1 Q_2^2 Y^(-alpha/2). Both exponents come from Parseval on a window base^r, where int F^2 = base^(-r alpha) exactly when 0 is in F and ≍ otherwise, so the base-10 values 1/21 and 10/21 are 1 - alpha rounded up and alpha/2 rounded down. Since alpha + alpha_1 >= 1 at every design, this never loses to the plain l^1 bound in the modulus aspect. Witness: mobius.md The pair route.
2026-09-07 Proved The lattice half of the digit-restricted bilinear estimate transfers to every base, and the five inequalities it asks are free below 1/3. With x = base^level, the window N K >= x^(1 - 2 beta), delta >= N/x and Q <= x^(1/2), the sum of F_x(a_1/x) F_x(a_2/x) over pairs whose large contribution comes from a rank-2 lattice is << (log x)^5 (Q + E)^(-eps/4) x/(N K), the source's own log power, whenever 2 alpha_1 < alpha, (2 - alpha) 2 beta < 1 - alpha_1, 2 beta (alpha_1 (3 - u) + u - 1) < u alpha/2 for some u in (0, min(1, 2 alpha_1/alpha)], 5 beta < 1 + alpha/2 and 2 beta < (1 - alpha_1)(1 - alpha_1 + alpha/2). The source writes a numerical check for the second and the fourth only; the first, third and fifth are read off steps it performs silently. All five are monotone in the three exponents, so the corner alpha = 1 - alpha_1, beta = 1/4 decides them, and every one holds under alpha_1 < 1/3, beta <= 1/4 and the l^1 floor alpha + alpha_1 >= 1, with 1/3 sharp since three become equalities there. The floor and the threshold on beta alone do not suffice, as alpha_1 = 0.40, alpha = 0.60, beta = 1/4 shows. Base 10 clears all five as published. Witness: mobius.md The pair route.
2026-09-07 Proved The pair route's eight inequalities are two. Write alpha = log(fill)/log(base) for a digit set's dimension, alpha_1 for the sup-over-shift l^1 exponent of its transform and beta for the exceptional-set threshold. Of the eight inequalities the route asks, one is a ceiling on alpha_1 alone, 2 alpha_1 < alpha, and seven are caps on beta at fixed (alpha, alpha_1); four of those fall in alpha_1 and two are constant in it, so each takes its minimum over the region at the wall alpha_1 = alpha/2. At that wall the lattice cap (2 - alpha) 2 beta < 1 - alpha_1 and the geometric-mean condition read exactly 1/4, the last lattice cap reads (2 - alpha)/4 and the fourth (1 + alpha/2)/5, all identities in alpha, so none of them ever cuts below the window threshold 1/4 inside the wall. For alpha in (1/2, 1) the region is therefore exactly alpha_1 < alpha/2 and beta <= min(1/4, (2/5)(1 - alpha_1)), the greedy cap cutting from alpha_1 = 3/8 and from nowhere else. A sweep of 66000 cells, 264000 cap tests, finds no exception, and the two wall equalities hold at each of 330 rational alpha. Witness: lab/py/mobius-region verb boundary.
2026-09-07 Proved The exceptional-set threshold obeys the same Parseval floor as the l^1 exponent, and the pair route reaches only fill >= base^(3/4). The normalised transform is at most 1 pointwise, so the moment exponent m_t is non-increasing in t; and m_2 = 1 - alpha exactly, since two length-level digit strings congruent modulo base^level are equal. Hence m_t >= 1 - alpha for every t <= 2, and since 2 - t <= 1 for t >= 1 the threshold beta = inf over t in [1,2) of m_t/(2 - t) is at least 1 - alpha at every base and every digit set, the same floor alpha + alpha_1 >= 1 puts on the l^1 exponent. The route's window condition beta <= 1/4 alone then forces alpha >= 3/4, that is fill >= base^(3/4), with equality only when the l^1 floor is also an equality. That window condition is a convenience rather than a necessity, and dropping it does not widen the route: on the single-window branch, which carries the greedy step under alpha_1 <= 1 - (13/4) beta, the same two floors give alpha >= 13/17 = 0.764705..706, so that branch reaches only fill >= base^(13/17) and the gate rises. The weaker reading fill > sqrt(base), which follows from alpha_1 < 1/2 alone, stays true and is simply not sharp, so no earlier row is contradicted. Witness: lab/py/mobius-region verb check.
2026-09-07 Verified The census of the pair criterion over 49 digit designs, and a second machine at base 21. Over the 38 proper digit sets of base = 3, 4, 5, the ten base-10 one-missing-digit columns and base 21 missing 0, one design clears the criterion, 47 are refuted and one is open, the open cell being base = 5 with F = {0,1,3,4}, where the transform vanishes inside a window cell and the infimum matrix loses a row. A pass is decided at the pessimistic corner and a failure at the optimistic one, every cap being monotone in each parameter. A second implementation of the window method returns alpha_1 in [0.2499715, 0.2499822] for base 21 missing 0 at five window digits and sub-scan 8, against the five-digit [0.2499715, 0.2499821] already certified, agreeing on the lower bound to all seven printed digits and differing by one unit in the last on the upper; both run the same method at the same depth, so the agreement witnesses transcription and the upper-bound gap is the only independent information. The same machine reproduces base 10 missing 5 at alpha_1 in [0.3505101, 0.3506471], m_(235/154) <= 0.1362891 and beta <= 0.2875140 against the three published values 27/77, 59/433 and 23/80. Witness: lab/py/mobius-region verbs criterion and params.
2026-09-07 Proved The line half of the digit-restricted bilinear estimate and its two bookkeeping steps, at every base. With x = base^level, a threshold beta admissible and at most 2/5, which with the Parseval floor beta >= 1 - alpha forces alpha >= 3/5 on the design, delta >= N/x, N K >= x^(1 - 2 beta), K above the absolute constant of the pair dichotomy, and N >= x^(eps + max((5/4) beta, (5 beta - 1/2)/3)), the pair sum over the pairs whose large contribution lies on a line is << (log x)^(O(1)) x^(-eps') x/(N K) for x past a point depending on base, fill and eps, with eps' a function of eps and the implied constant depending on those three alone. The statement asks nothing of the l^1 exponent and asks of the dimension only what the admissibility of beta already encodes, so the whole l^1 content of the route sits in the lattice half and the greedy step. Two write-outs complete it. For coefficients bounded by the j-fold divisor function, orthogonality on the grid with tau_j^2 <= tau_(j^2) gives #{a mod x : the exponential sum is at least x/C} <<_j C^2 (log x)^(j^2 - 1), so a Heath-Brown decomposition costs a log power where a 1-bounded sequence costs none. And Cauchy-Schwarz in the long variable turns the bilinear sum into x/N times the pair sum of the transform against the sum over l_1, l_2 <= N of min(x/N, the inverse distance from (a_1 l_1 - a_2 l_2)/x to the nearest integer), which is the exact step at which all four coefficient factors leave by the triangle inequality; the dyadic split into level sets and pair-mass classes costs two more log powers. Witness: mobius.md The pair route.
2026-09-07 Conjecture A digit set satisfying condition (E) in one dimension whose sup-over-shift l^1 exponent obeys alpha_1 < 1/4 has Sum_{level in S_F, level <= x, gcd(level,base) = 1} mu(level) = O(A_F(x) (log x)^(-B)) for every B. The program is named: two Proved steps for mu, the rest set-only or coefficient-free, and the lattice branch of the source's Section 14 in general parameters owed. Base 10 fails on two independent numbers, 27/77 against 1/3 and 23/80 against 1/4. Witness: mobius.md The pair route. Superseded by the criterion row that names five lattice conditions and the threshold beta <= 1/4, under the same subsection in OPEN.
2026-09-07 Conjecture , whose owed list and whose base-10 diagnosis are both superseded. A digit set satisfying condition (E) in one dimension whose sup-over-shift l^1 exponent obeys alpha_1 < 1/4 has Sum_{level in S_F, level <= x, gcd(level,base) = 1} mu(level) = O(A_F(x) (log x)^(-B)) for every B. The program is named: the major-arc lemma and the level of distribution on an initial segment are Proved for mu, and the lattice branch is Proved in general parameters. Three things are owed and none is a new idea: the line branch at general base, whose two lemmas are set-free and coefficient-free but whose own conditions m_t < (2 - t) beta and N >= x^max((5/4) beta, (5 beta - 1/2)/3) are gathered into no statement yet; and the write-out at general base of two bookkeeping steps, the Parseval count of large frequencies for the Heath-Brown pieces and the dyadic reduction of the bilinear sum to the pair sum, both stated at source for arbitrary 1-bounded sequences. Base 10 now fails on one number only, the sharp threshold beta = inf_t m_t/(2 - t), at 23/80 against 1/4. Witness: mobius.md The pair route.
2026-09-07 Refuted The pair criterion cannot be met at base 10 at any excluded digit. An upper bound on a moment exponent bounds the threshold above and can never show the criterion fails, so the published miss of 3/80 prices a gap and refutes nothing. Two monotonicities close it: on a cell [t_0, t_1] every t has m_t/(2 - t) >= m_(t_1)/(2 - t_0), and above a cut the Parseval value 1 - alpha alone forces the ratio past 1/4. With the moment bounded below by the infimum window matrix, adaptive chains of 25 to 53 cells certify beta > 1/4 at all ten one-missing-digit sets of base 10, the certified lower bounds running 0.2502716 to 0.2541480, so no admissible threshold clears the window condition there and the route is dead at base 10 at every digit rather than merely unreached. The refuting certificates do not order the columns, their brackets [0.2510933, 0.2625620] at the digit 9 and [0.2515026, 0.2875159] at the digit 4 overlapping; run at the target 0.2626 the same chain certifies beta >= 0.2632014 at each of the eight non-extreme digits, up to 0.2645208 at the digit 7, above both extreme upper bounds, while the digits 0 and 9 come back undecided as they must, and that settles the two extreme digits as strictly the cheapest columns. The miss is at most 0.0125620 at the cheapest column and at least 0.0139557 at the digit 4; the factor 2.99 between the two printed upper bounds is a ratio of upper bounds and not of misses. Witness: lab/py/mobius-region verbs threshold and threshold 0.2626.
2026-09-14 Proved The one-step constant of the l^1 recursion is exact and cheap at one excluded digit. Let F = {0..base-1} less {e_0}, phi_r = (t+r)/base, A_r = (-1)^r sin(pi t)/sin(pi phi_r) and c = e_0 - (base-1)/2. Then abs(g_F(phi_r)) = abs(A_r - e(c phi_r)) = sqrt(A_r^2 + 1 - 2 A_r cos(2 pi c phi_r)) for every t not in Z, which is where A_r is defined, since D_base(phi_r) = e((base-1)phi_r/2)(-1)^r sin(pi t)/sin(pi phi_r) and the unimodular factor divides out, so B_base(F) = sup_t sum_(r mod base) abs(g_F((t+r)/base)) is a sup of base real square roots and costs O(base) per t; the reduction reproduces the direct sum over F to 12 digits and reproduces the grid sups 4.0000000000 at base 3 {0,1} and 19.8885438199 at base 10 missing 9, the floored readings of split's 4.000000000 and 19.888543820. Witness: lab/py/mrly-pairing, verb onestep.
2026-09-14 Proved Two exact symmetries of that constant: B_base(F) is unchanged by e_0 -> base-1-e_0, because the digit reflection multiplies g_F by a unimodular factor, and the shifted-grid sum is symmetric in t about 1/2, because r -> base-1-r carries t to 1-t with sign(A_r) cos(2 pi c phi_r) fixed; so the scan for the sup runs on t in [0, 1/2] and on e_0 <= (base-1)/2. Witness: lab/py/mrly-pairing, verb onestep.
2026-09-14 Proved The phase identity behind the one-step constant: for every base, every e_0 and every t in (0,1), sum_(r mod base) (1 + sign(A_r) cos(2 pi c phi_r)) = base + cos(2 pi c (t - 1/2)/base)/cos(pi c/base), by summing the geometric series sum_r (-1)^r e(c r/base) = e(-c/(2q))/cos(pi c/base), which is where e(c) = (-1)^(base-1) collapses the numerator to 2; the right side is at least base + 1 at every t, since abs(2 pi c (t-1/2)/base) <= abs(pi c/base) < pi/2, and it reaches base + 1/sin(pi/(2q)) at t = 1/2 and e_0 in {0, base-1}. Witness: lab/py/mrly-pairing, verb onestep.
2026-09-14 Proved The triangle split abs(g_F) <= abs(D_base) + abs(g_E) of the shifted-grid step can be sharpened by a fixed share of base at one excluded digit, with no new input. For base >= 17 and m = 1, B_base(F) <= (4/pi) base + Psi_base + base/2 - sec(pi (e_0 - (base-1)/2)/base)/2, where Psi_base = (2 base/pi) H(ceil((base-2)/2)) + (1 - 2/pi) base with H(n) = ln n + gamma + 1/(2n), the desk convention of mobius.md, is the step 3 kernel constant less its two-point part, against the step 3 bound base PB_base(1) = base + Phi_base = (4/pi) base + Psi_base + base + 0.00023954, the constant being 0.727 - 2(1 - 2/pi) exactly. The proof is abs(a - e(psi))^2 = (a+1)^2 - 2a(1 + cos psi) with sqrt(1-X) <= 1 - X/2, then abs(A_r) >= sin(pi t) = s and s/(1+s) >= s/2, then the phase identity, then the t-dependent kernel bound sum_r abs(D_base(phi_r)) <= (4/pi) base + s Psi_base that step 3's own two-point and pairing estimates give, and finally h(tau) = cos(pi tau)(Psi_base - base/2) - cos(pi tau) cos(2 beta tau)/(2 cos beta) with beta = pi (e_0 - (base-1)/2)/base has h' <= 0 on [0, 1/2] once Psi_base >= (1 + pi) base/2, first true at base = 17, by sin(pi tau) >= 2 tau, sin x <= x and sec beta <= base. Witness: lab/py/mrly-pairing, verb onestep.
2026-09-14 Proved That sharpening lowers the base of the conditional power saving with no new idea and no change to any other step: the least base with (base-1) base^(-b(a)) > B_base(F)/base falls from 3690 to 2446 at every excluded digit and to 1812 at e_0 in {0, base-1} at the GRH rung b = 3/4, from 8578 to 5700 and 4242 at b = 1417/1850, and from 33547 to 22416 and 16816 at b = 913/1160, each an exhaustive scan from base = 17 in the generator, whose held column prints 3997555 = 4000000 - 2446 + 1 and the five like counts, so every wall is an up-set over its whole scan and not a first crossing, while the three step 3 baselines are quoted from mobius.md and not rescanned. Witness: lab/py/mrly-pairing, verb onestep.
2026-09-14 Proved The l^1 floor is higher than base at one excluded digit: letting t -> 0 in the shifted-grid sum gives abs(g_F(0)) = base-1 and abs(g_F(r/base)) = 1 at every r != 0, so B_base(F) >= 2(base-1) and c_base >= log(2 - 2/base)/log(base) > 0 for every base >= 3, and that endpoint is the seat at base = 3 by hand, the three terms collapsing to 4 cos u on [0, pi/6) and 4 cos(u - pi/3) on [pi/6, pi/3), both at most 4. Hence no exact constant can push the method below the base where (base-1) base^(-3/4) > 2 - 2/base, which is base^(1/4) > 2 and so base >= 17, and the sup-times-l^1 method needs fill > (2 - 2/base) base^(3/4) and not fill > base^(3/4); the floor is too weak to give fill > 2 base^(3/4), since at base = 17 the base fill = 16 lies between (2 - 2/base) base^(3/4) = 15.759 and 2 base^(3/4) = 16.744. Witness: lab/py/mrly-pairing, verb onestep.
2026-09-14 Verified The exact one-step constant falls with the excluded digit, while the step 3 bound is one number for all of them: at base = 3690, B_base(F)/base >= 5.750052 at e_0 = 0 and >= 6.392410 at e_0 = 1844, against the exact kernel sup K_base/base >= 6.191324, the proved kernel bound Phi_base/base <= 6.791445 and 1 + Phi_base/base = 7.791445; the split defect (K_base + base) - B_base(F) reads 1.441272 base at e_0 = 0, flat to 1.3e-5 across base = 100, 1000, 2234, 3690 and agreeing to six digits with 1 + (2/pi) ln 2 = 1.4412712, which nothing here proves is its limit, and 0.798914 base at the middle digit, where those same four base read 0.808644, 0.799643, 0.799091 and 0.798914, a spread of 9.8e-3, so that branch is stable only to 1e-2; Phi_base - K_base reads at most 0.600121 base there, so the two losses are the same order and the excluded digit's position is worth 0.64 base. Each number is a grid scan on t in [0, 1/2] at cut 1/4000 with the sup seated at t = 1/2. Witness: lab/py/mrly-pairing, verb onestep.
2026-09-14 Verified The ceiling of the exact-constant lever, and what it is not: the measured B_base(F) itself would put the GRH base at 927 at every excluded digit, the last failure being base = 926 at e_0 = 462 on a downward scan of 17..2000 over every digit, and at 304 at e_0 in {0, base-1}, last failure 303 on 17..8000, against the proved 2446 and 1812, so a further 0.9 base of slack is left in the sharpened bound, of which 0.6 base is the gap between Phi_base and the exact kernel sup. The middle digit is not the maximiser at odd base and reading it alone reports the crossing 232 bases early: at base = 695 the worst digit is e_0 = 463 with B_base(F)/base = 5.327344 against (base-1) base^(-3/4) = 5.127089, a failure, while the middle digit passes by 2.1e-5. Both bases are readings of Sigma(1/2), which is the sup on the cut at every (base, e_0) checked in the range but is not proved to be the sup, and neither crossing is proved monotone in base, so they bound nothing and never enter a statement. Witness: lab/py/mrly-pairing, verb onestep.
2026-09-14 Verified Nothing measured contradicts the sharpened bound: over every excluded digit at base = 17..60 the worst ratio of B_base(F) to the bound is 0.807189 at base = 60, e_0 = 29, at the seats base = 100, 1000, 2234, 3690 it is 0.876716 at base = 3690, e_0 = 1844, and 4000 draws at seed 1009 over base in {17, 23, 60, 101, 333, 1000, 3690} with e_0 and t uniform give worst ratio 0.872146 at base = 3690, e_0 = 1701, t = 0.499866 and no violation. Witness: lab/py/mrly-pairing, verb onestep.
2026-09-14 Proved The pair route's gate is fill >= base^(3/4), and dropping its window condition narrows the route rather than widening it. F_x <= 1 pointwise makes m_t non-increasing and m_2 = 1 - alpha is exact by Parseval on the grid, so beta >= 1 - alpha at every base and every digit set, and the window condition beta <= 1/4 forces alpha >= 3/4. The branch that drops it asks alpha_1 <= 1 - (13/4) beta, and with beta <= m_1 <= alpha_1, the t = 1 term of the infimum against the grid sum being one shift of the supremum, that reads beta <= 4/17 = 0.235294 and alpha >= 13/17 = 0.764705, so the gate rises to fill >= base^(13/17). The step beta <= alpha_1 is load-bearing (lab/py/mobius-region, verb boundary prints the branch): without it alpha = 0.9, alpha_1 = 0.155, beta = 0.26 clears both floors, 2 alpha_1 < alpha, the greedy condition and all five lattice conditions with beta > 1/4. The weaker reading fill > sqrt(base) stays true and unsharp. Witness: mobius.md The pair route, lab/py/mobius-region verb check, which prints the two floors at four designs and beta <= alpha_1 at t = 1 at the base-21 recompute.
2026-09-14 Proved The region the pair route asks for is exactly two inequalities. Of the eight, one is a ceiling on alpha_1 alone, 2 alpha_1 < alpha, and seven are caps on beta at fixed (alpha, alpha_1); four of the seven fall in alpha_1 and three are constant in it, so each takes its minimum at the wall alpha_1 = alpha/2. There (2 - alpha) 2 beta < 1 - alpha_1 reads 1/4, 2 beta < (1 - alpha_1)(1 - alpha_1 + alpha/2) reads (2 - alpha)/4 and 5 beta < 1 + alpha/2 reads (1 + alpha/2)/5, three identities in alpha with the last two strictly above 1/4 on (1/2, 1); the u-condition has wall coefficient (3 alpha/2 - 1) + u (1 - alpha/2), so it reads exactly 1/4 for alpha >= 2/3 and is vacuous for alpha in (1/2, 2/3), where that coefficient is negative at small admissible u. Vacuous or 1/4, none of the four cuts, so for alpha in (1/2, 1) the region is alpha_1 < alpha/2 with beta <= min(1/4, (2/5)(1 - alpha_1)), the greedy cap cutting below 1/4 exactly from alpha_1 = 3/8, a threshold free of alpha. Witness: mobius.md The pair route, lab/py/mobius-region verbs region and boundary.
2026-09-14 Verified The pair route and the Mobius census cannot break each other, and no proved conditional exponent sits below a measured one. The criterion's conclusion is a log saving, so it caps theta(F) at 1 in A_F units, far above every measured running-maximum exponent of the census, 0.4465 to 0.5358. The chain that does print an exponent is the GRH one, theta(F) <= 1 - (1/4 - alpha_1)/alpha, and it reads above 1 at every base-10 one-missing-digit column, 1.1054746 at the digit 4, and 0.9999819 at base = 21 missing 0, a saving under 2 * 10^(-5) in the exponent against the trivial bound; so neither chain is falsified by the census at any design of it, and a design whose measured exponent rose above its own proved ceiling would refute one of them. Witness: lab/py/mobius-region verb criterion, mobius.md The pair route.
2026-09-14 Proved The chord 1/sin x <= 1/x + (2/pi)(1 - 2/pi) x holds on (0, pi/2], where csc x - 1/x has an all-positive Taylor series and so lies under its own chord; pairing r with base-1-r puts every shifted-grid argument inside (0, pi/2] at t in (0, 1/2], and K(t) = K(1-t) carries the rest, so K(t) = sin(pi t) sum_(r mod base) 1/sin(pi (t+r)/base) <= (4/pi) base + sin(pi t) Psi'_base with Psi'_base = (base/pi)(2 H(P-1) - 1 + 1/P) + (1 - 2/pi) base/2 at even base and (base/pi)(2 H(P-1) - 1 + 2/P) + (1 - 2/pi)(base/2 + 1/(2 base)) at odd base, P = floor(base/2), H(n) = ln n + gamma + 1/(2n) the desk convention of mobius.md; the paired argument sum is base^2/4 at even base and P(P+1) + t at odd base, the source of the odd 1/(2 base), and at even base the constant is Psi_base - base/2 + 2/pi. Witness: lab/py/mrly-pairing, verb onestep, the chord kernel block.
2026-09-14 Verified The chord kernel bound cuts the gap between the proved kernel constant and the exact kernel sup K_base by a factor 5.98: at base = 3690 the up-rounded gap columns give Psi_base - (K_base - (4/pi) base) <= 0.600121 base and Psi'_base - (K_base - (4/pi) base) <= 0.100293 base, the same quantity the lemma-slack column floors to 0.100292 base; the chord column reads 0.100290, 0.100293, 0.100293 at base = 100, 1000, 2234, so the slack is flat to 1e-5, and it is attained at the seat t = 1/2. Witness: lab/py/mrly-pairing, verb onestep, the chord kernel block.
2026-09-14 Proved At one excluded digit and base >= 36 the one-step constant of the shifted-grid l^1 recursion obeys B_base(F) <= (4/pi) base + Psi'_base + base/2 - sec(pi (e_0 - (base-1)/2)/base)/2, the phase identity and the sqrt(1-X) <= 1 - X/2 chain of the earlier sharpening run against the chord kernel bound; the monotone step needs Psi'_base >= (1 + pi) base/2, which first holds at base = 36 with H(n) = ln n + gamma + 1/(2n), the desk convention of mobius.md, and at base = 37 with the harmonic number itself, the over-estimate safe elsewhere since it only raises an upper bound; the hypothesis is sufficient and not necessary, the max of h(tau) sitting at tau = 0 at every e_0 from base = 8 up on the exhaustive scan 4..79. Witness: lab/py/mrly-pairing, verb onestep, the chord wall block; lab/rs/mertens-numerology, the_chord_floor_carries_its_harmonic_convention.
2026-09-14 Verified The chord bound moves the GRH wall base_0(1/2) from 3690 at step 3 and 2446 at the phase sharpening to 1499 at every excluded digit, and from 1812 to 1032 at e_0 in {0, base-1}, each an up-set over the exhaustive scan 36..4000000; the rungs b = 1417/1850 and b = 913/1160 move from 5700 and 22416 to 3525 and 14078, and from 4242 and 16816 to 2459 and 10013, up-sets over 36..8000000 and 36..40000000. Witness: lab/py/mrly-pairing, verb onestep, the chord wall block.
2026-09-14 Verified Every chord wall costs out against the level-x^(alpha/2) defect within five steps of itself: the saving delta_base first exceeds 1/(2(base-1) ln base) at base = 1502 for the wall 1499 and at base = 1036 for the wall 1032, against 2450 for 2446, 1815 for 1812 and 3692 for 3690, every crossing an up-set to 100000, each row scanned from its own floor, base >= 3 at step 3, 17 at the phase sharpening and 36 at the chord, and no wall inherited. Witness: lab/rs/mertens-numerology, sharpened cost-out block, sharpened_cost_out_is_pinned.
2026-09-14 Verified Nothing measured contradicts the chord bound: the worst ratio of the exact B_base(F) to it is 0.902124 over every e_0 at base = 36..60, 0.941239 at the larger seats and 0.936333 over 4000 seeded draws of (base, e_0, t), against 0.807189, 0.876716 and 0.872146 for the phase sharpening alone. Witness: lab/py/mrly-pairing, verb onestep, the chord falsification block.
2026-09-14 Verified The weight the chord bound leaves behind is not free: at the seat t = 1/2 the kept weight w_r = abs(A_r)/(abs(A_r) + 1) >= s/(1 + s) >= s/2 is worth base/2, while dropping the singular terms by (1 + sign(A_r) cos) <= 2 costs 2 sum_(r mod base) 1/(abs(A_r) + 1) = (2 - 4/pi) base, measured 0.726761 base at base = 1000, 3690, 20000 against its exact limit 2(1 - 2/pi) = 0.726761, a net -0.226761 base, so the route loses more than it wins. Witness: lab/py/mrly-pairing, verb onestep, the weight route block.
2026-09-14 Proved Above each rung's step 3 wall the sharpened m = 1 certificate needs no scan: base PB_base(1) - base PB_base(1, e_0) = base/2 + sec(pi (e_0 - (base-1)/2)/base)/2 + 0.727 - 2(1 - 2/pi) with 0.727 - 2(1 - 2/pi) = +2.3954 * 10^(-4) and abs(pi (e_0 - (base-1)/2)/base) < pi/2, so PB_base(1) - PB_base(1, e_0) > 1/2 at every base >= 17 and every excluded digit, and the step 3 certificate (base-1) base^(-b(a)) - PB_base(1) > 0, proved positive from q_0(a) = 3690, 8578, 33547 on by the monotone floor at Q(b) = 723, 1486, 4754, carries the sharpened certificate over [q_0(a), infinity) unscanned. Witness: mobius.md step 3 sharpened and step 5, on lab/py/mrly-pairing verb onestep.
2026-09-14 Verified Below each rung's step 3 wall the sharpened m = 1 certificate (base-1) base^(-b(a)) - PB_base(1, e_0) > 0 is exhaustive and not a first crossing, so with the proved half above q_0(a) each sharpened wall is a half line and not a window: the scans 17..4 * 10^6, 17..8 * 10^6 and 17..4 * 10^7 each run past their own q_0(a) and hold at every base from 2446 and 1812 at b = 3/4, 5700 and 4242 at b = 1417/1850, 22416 and 16816 at b = 913/1160, the held counts printing 3997555, 3998189, 7994301, 7995759, 39977585, 39983185, each equal to hi - w + 1. Witness: lab/py/mrly-pairing, verb onestep.
2026-09-14 Proved The per-denominator Mobius input states, at the exact frequencies, and buys no exponent. Group the grid a/base^level of the orthogonality step by base-power level j, so a = a' base^(level-j) with base not dividing a' and hat F_level(a'/base^j) = fill^(level-j) hat F_j(a'/base^j) because g_F at an integer is fill; with c_j the primitive level-j sum sum abs(hat F_j(a'/base^j)), c_0 = 1 and C_level = sum_j fill^(level-j) c_j, Baker-Harman's PROPOSITION p.194 eq. 6 under its hypothesis (4), that L(s, chi) is zero-free in sigma > a for EVERY Dirichlet character, taken at (r,Q) the frequency itself, where its second factor is 1, gives abs(M_F(base^level)) <<_(base,eps) x^eps base^(-level) sum_(j <= level) fill^(level-j) c_j min(x^(b(a)), x^a base^(j/2)) at x = base^level, the reduced denominator of a'/base^j dividing base^j. That sum lies in [m/base, 1] times the uniform base^(-level) C_level x^(b(a)): Parseval on the shifted grid is exact, sum_(s mod base) abs(g_F((t+s)/base))^2 = qk, and abs(g_F) <= fill, so sum_s abs(g_F((t+s)/base)) >= base, hence C_j >= base C_(j-1) at every j and the top level carries c_level = C_level - fill C_(level-1) >= (m/base) C_level, while b(a) <= a + 1/2 at every rung keeps the uniform constant x^(b(a)) on that level. So the exponent stays b(a) + c_base, the saving is at most -log(c_level/C_level)/(level log base) <= log(base/m)/(level log base) and vanishes with level, and the whole lever is worth one bounded factor base/m. The bracket is proved for this corollary and for it alone, the level charge being an upper bound and not the pointwise truth: at composite base a top-level a' = 5^level u has true denominator 2^level = x^0.301. Witness: lab/py/mrly-pairing verb perden, the exponent block, level(den - unif) reading -0.657068 at base = 3 one digit off and -0.292383 at base = 10 missing 9, constant in level, the largest term sitting at argmax j = level at every printed row by measurement and not by proof.
2026-09-14 Verified The same tool at full strength buys no exponent either. Letting any reduced r/Q serve any frequency, Q(1 + x abs(a/base^level - r/Q)) = Q + abs(aQ - r base^level), so the per-frequency constant is min(x^(b(a)), x^a nu(a)^(1/2)) with nu(a) = min_Q (Q + norm(aQ)_(base^level)), and the honest ratio to the uniform input rises with level while its exponent gain decays faster than 1/level. Witness: lab/py/mrly-pairing verb perden, the full minor-arc block, nu checked against a full search over every reduced r/Q at base^level = 81 with 0 mismatches, ratio 0.817368, 0.835986, 0.851049, 0.861910 at base = 3 level = 6, 8, 10, 12 and 0.684136, 0.705013, 0.737968 at base = 10 level = 4, 5, 6, with level times the gain falling from -0.183564 to -0.135265 and from -0.164858 to -0.131963; that the ratio is bounded below in level is measured over these seven rows and not proved.
2026-09-14 Proved The l^1 mass of a digit transform decays geometrically downward from the top denominator. From C_j >= base C_(j-1) the top level's share is c_j/C_j >= m/base at every j, and the levels below J carry sum_(j <= J) fill^(level-j) c_j = fill^(level-J) C_J <= (fill/base)^(level-J) C_level, so the mass on the levels of reduced denominator at most x^(1/2), the tie at j = level/2 included, is at most (1 - m/base)^(ceil(level/2)) of the whole and falls geometrically in level. Witness: lab/py/mrly-pairing verb perden, the level-profile block, which asserts the decomposition identity, the floor and the cap at every printed row.
2026-09-14 Verified The measured level profile at one excluded digit. Top-level shares 0.485846, 0.510055, 0.573574 and 0.676152 at base = 3 level 12, base = 10 level 6, base = 101 level 3 and base = 1499 level 2, against the proved floor m/base = 0.333333, 0.100000, 0.009901 and 0.000667, the last two short rows where C_j/C_(j-1) is still moving, 244.658399 then 234.507307 at base = 101, and so not converged constants; the levels of reduced denominator at most x^(1/2) carry 0.018474, 0.117603, 0.174294 and 0.323848 against the proved cap 0.087791, 0.729000, 0.980296 and 0.999333; the one-step ratios C_j/C_(j-1) read 3.889889, 18.369403, 234.507307 and 4625.632148, each above the proved floor base. Witness: lab/py/mrly-pairing verb perden, level-profile block, reproducing verb split's 18.369402635 and its top share 0.510055, and matched by brute force over the digit strings at C_j = 234.856179, 913.566768 and 331.978584 with top shares 0.485833, 0.485848 and 0.512017 for base = 3 j = 4, 5 and base = 10 j = 2.
2026-09-14 Proved Baker-Harman's PROPOSITION is a d-form minor-arc bound, and that is where it pays, inside eq. 6's printed range on Q, which the desk has not read. On a Dirichlet arc abs(theta - l/d) <= 1/d^2 with (l,d) = 1 it reads S_mu(theta) << x^(a+eps) d^(1/2) (1 + x/d^2)^(1/2) = x^(a+eps) (d + x/d)^(1/2) <= x^(a+eps) (d^(1/2) + x^(1/2) d^(-1/2)), which at a = 1/2 is x^eps ((x d)^(1/2) + x d^(-1/2)) under the generalized Riemann hypothesis. Unlike the rows above, where the min caps the PROPOSITION by the uniform THEOREM at the level that decides, this one applies eq. 6 at an arbitrary arc denominator d and so carries that unread range. Witness: lab/py/mrly-pairing verb perden, the arc block, derived from the statement quoted at source in REFS.md.
2026-09-14 Proved The chord kernel constant beats the step 3 one at every base with no scan: Psi'_base < Psi_base at every base >= 5, by Psi'_base = Psi_base - base/2 + 2/pi at even base and by Psi_base - Psi'_base = (2 base/pi)(H(P) - H(P-1)) + (base/pi)(1 - 2/P) + (1 - 2/pi)(base/2 - 1/(2q)) at odd base, P = floor(base/2), positive since H(P) - H(P-1) = ln(P/(P-1)) - 1/(2P(P-1)) and ln(P/(P-1)) > 1/(P - 1/2) > 1/(2P(P-1)) at P >= 2. Witness: cargo test -p mertens-numerology the_chord_constant_saves_half_a_base, psi_chord(base) < psi(base) at every base in 36..3999, 34 passed; 0 failed.
2026-09-14 Proved Hence each chord wall is a half line and not a window: base PB_base(1) - base PB'_base(1, e_0) = (Psi_base - Psi'_base) + base/2 + sec(pi (e_0 - (base-1)/2)/base)/2 + (0.727 - 2(1 - 2/pi)) > 0 at every base >= 36, so the step 3 certificate, positive from base_0(a) on by the monotone floor, carries the chord certificate over [base_0(a), infinity) unscanned. Witness: mobius.md step 5, on lab/py/mrly-pairing verb onestep, chord walls 1499 and 1032 holding 3998502 and 3998969 of the scan 36..4 * 10^6.
2026-09-14 Conjecture The criterion, with the owed list it now carries. A digit set satisfying (E1) whose l^1 exponent obeys alpha_1 < 1/4 has sum_(level in S_F, level <= x, (level,base) = 1) mu(level) = O_B(A_F(x) (log x)^(-B)) for every B. The two steps that read the sequence rather than the set are proved here, the major arcs for mu at base-smooth moduli and the level of distribution on an initial segment; the lattice half, the line half and both bookkeeping steps are now written out at general base, so the three write-outs the program listed as owed are written and this supersedes the owed list carried by the earlier criterion rows in OPEN. One arithmetic item is left, the level of distribution at base-divisible moduli, needed only to drop (level, base) = 1: the level carries by the divisor split but the main term does not, being a digit-string count times 1/d_2 and not 1/d. Of the source reading, the line branch's close is read once and owes a second reading. Witness: mobius.md The pair route.
2026-09-14 Refuted The seat of B_base(F) is not t = 1/2 at every base >= 10: at base = 11, e_0 = 0 the sup on the cut is 22.5094271855 at t = 0.47275 against Sigma(1/2) = 22.4703926508, and at base = 13, e_0 = 0 it is 27.9876970872 at t = 0.47875 against 27.9570308138, so the seat is interior at both. t = 1/2 is the seat at every family printed from base = 100 up, and is where the constant is read and not where it is proved to sit; the floor 2(base-1) and the sup scan are untouched, while any wall read off Sigma(1/2) alone is a reading and not a bound. Witness: lab/py/mrly-pairing, verb onestep.
2026-09-14 Refuted Base 10 is refuted at every excluded digit, on both branches, rather than merely unreached. A published or certified moment exponent is an upper bound on beta and can never show the criterion fails, so the published beta <= 23/80 prices a gap of 3/80 and refutes nothing. From below, F_x <= 1 makes m_t non-increasing and m_2 = 1 - alpha is exact, so adaptive chains of 25 to 53 cells certify beta > 1/4 at all ten one-missing-digit sets of base 10, the certified lower bounds running 0.2502716 to 0.2541480. One such certificate kills both branches: the merged window asks beta <= 1/4, and the single-window branch asks alpha_1 <= 1 - (13/4) beta, which with beta <= alpha_1 reads beta <= 4/17 < 1/4. At the target 0.2626 the same chain gives beta >= 0.2632014 at each of the eight non-extreme digits, up to 0.2645208 at the digit 7, with the digits 0 and 9 undecided, so the two extreme digits are strictly the cheapest columns of the base. Witness: mobius.md The pair route, lab/py/mobius-region verbs threshold and threshold 0.2626.
2026-09-14 Refuted The per-denominator weighting lowers neither the exponent cost c_base nor the base q_0 of the conditional power saving on the dense digit columns. At the top level j = level the charge is x^(a + 1/2) and the uniform constant x^(b(a)), and a + 1/2 - b(a) is 1/4, 47/185, 61/232, 19/70, 3/10 and 1/3 as exact rationals at a = 1/2, 13/25, 11/20, 4/7, 3/5, 2/3, so the top level is strictly worse at EVERY rung of both exponent tables and the crossing 2(b(a) - a) never exceeds 1/2; the certificate (base-1) base^(-b(a)) - PB_base(1, e_0) > 0 takes no per-denominator quantity at all, so the GRH walls of the chord certificate stand untouched. Beating x^(3/4) on the top level means bounding the Mobius exponential sum at denominator base^level = x, which is the conjectural x^(1/2 + eps) at essentially every frequency and lies past the method's ceiling. Witness: lab/py/mrly-pairing verb perden, the rung block, every rung printing no in its beats uniform at j = level column.
2026-09-19 Proved Above the supremum the L^p norms of the digit transform buy nothing: max(fill^p, base fill^(p/2)) <= Lambda(p) <= base fill^(p-1) forces Lambda(p)^(1/p)/fill to 1, reading 1.224744, 1.029883, 1.004288, 1.000653, 1.000107 at p = 2, 4, 6, 8, 10 for {0,1} at base 3. Witness: lab/rs/rho-decoupling, riesz higher moments.
2026-09-19 Proved The unbalanced kernel carries no Type II estimate uniform over bounded coefficients at any digit set containing 0: a_m = b_l = 1 makes the sum the box representation count and some admissible box carries R >= x^(alpha - o(1)), while on the balanced sum the bound-to-trivial ratio rises through 1, 1.0134 at level 12 and 1.0730 at level 14 at {0,1} base 3, the margin of alpha over the achieved exponent falling 0.035675 to 0.027009, and inside the arc regime the digit column is worse for the method than a random column of the same density at every cell. Witness: lab/rs/rho-decoupling, menergy type II.
2026-09-19 Verified Two box witnesses floor every coefficient-free route at x^alpha over the nine dense cells: a_m = b_l = 1 reads 0.19 to 0.41 of fill^level and a_m = 1_(base divides m), b_l = 1 reads 0.0024 to 0.104, the second carried by the frequencies a'/base^j at bounded j. Witness: lab/rs/rho-decoupling.
2026-09-19 Conjecture That the same floor holds at every digit set, which rests on R >= fill^level (log x)^(-C) and is void at alpha = 0.15 where the box is empty. Witness: lab/rs/rho-decoupling.
2026-09-19 Verified The cost-out of the GRH saving against the level-x^(alpha/2) defect runs at the rung b = 3/4 and at no rung above it, so no rung past a = 1/2 is set against the defect anywhere. Witness: lab/rs/mertens-numerology, the cost-out block.

The Eisenstein stack

2026-09-09 Proved The Eisenstein spun stack is the hexagonal twin of the Gaussian one: layers are the nonzero associate classes of Z[omega], layer z the lattice z^-1 Z[omega]; the hexagonal circle count is the floor sum h(t) = sum_j (floor(t/(3j+1)) - floor(t/(3j+2))), the twin of the Gaussian floor(t/(4j+1)) - floor(t/(4j+3)), equal to direct enumeration of classes for every t from 0 to 400; a node of reduced denominator class [d] has brightness h(floor(N/N(d))), checked by literal stacking of all 31 layers at norm bound 50 in exact rational coordinates, 630 nodes and 0 mismatches, the origin at 31; the lit set has sum_{[d], N(d) <= N} Phi(d) points with Phi(d) = N(d) prod_{p | d} (1 - 1/N(p)), reading 630, 9606, 151020, 337026, 945486 and 2419950 at norm bounds 50, 200, 800, 1200, 2000 and 3200. Witness: lab/py/eisenstein-stack hex_classes_closed, hex_classes_direct, literal_stack, closed_brightness, totient_sum.
2026-09-09 Proved A rotation keeps two hexagonal layers coincident iff it lies in Q(sqrt -3), iff its cosine is rational and its sine a rational multiple of sqrt 3, and those rotations are exactly w/conj(w) for nonzero w in Z[omega] by Hilbert 90 (all 58 rational rotations of denominator at most 60 are reached from the box of side 20); at whole degrees exactly the six multiples of 60 survive, against four of 360 on the square lattice, by cyclotomic reduction modulo Phi_360 and twelve minimal-polynomial spot checks; the hexagonal coincidence series is (1 + 3^-s)^-1 zeta_K(s)/zeta(2s) = prod_{p = 1 mod 3} (1 + p^-s)/(1 - p^-s), read at source in Pleasants, Baake and Roth and re-derived coefficient by coefficient to bound 100. Witness: lab/py/eisenstein-stack rotation_hits, field_rotation_degrees, spot_check_degrees, csl_zeta_ratio, csl_euler_product, REFS.md.

The hexagonal slice

2026-08-28 Verified The carpet face-count law V(i) = 2 * 20^i + 4 * 8^i, visible faces 6, 72, 1056, 18048, 336384, is A332705 verbatim, the surface area of the stage-i Menger sponge, with the same closed form on the entry. Witness: mrlymath::formulas::surface::surface, A332705.
2026-08-28 Proved The carpet slice census 6, 42, 306, 2250, 16578 is A299916(level+1): sectioning the sponge at level level on x + y + z = 1.5 * 3^level, a surviving cube cuts a hexagon of 6 mesh triangles or a triangle of 1, refinement triples the plane offset, and the 20 surviving subcubes split by coordinate sum as 1, 3, 3, 6, 3, 3, 1, so H_(level+1) = 6 H_level + T_level and T_(level+1) = 6 H_level + 3 T_level, the hexagon-triangle substitution proved by exhaustion; the ledger 54 = 6*6 + 6*1 + 12 punches exactly one 12-triangle hexagram per hexagon and none per triangle, so hexagram holes of the n-th size number A299916(n) and the mesh census is one index up; the recurrence a(n) = 9 a(n-1) - 12 a(n-2) gives the slice dimension log((9 + sqrt(33))/2)/log(3) = 1.818410; the empty area at level = 0..5 resolves into 1, 6, 42 components of descending size, each with six radial maxima, sixfold symmetry to 0.001 and max/min radius 1.68 against the hexagram's sqrt(3). Witness: slice-recurrence-order, mrlymath::six::topology test the_carpet_slice_percolates_at_base_three, A299916.
2026-08-28 Verified The slice vertex count 12k^2 - 6k + 1 is A154105 at n = k - 1 and the centered hexagonal number A003215 at index 2k - 1 (3m(m+1) + 1 at m = 2k - 1), so a prime vertex count is a cuban prime, A002407; at k = 1..20 ten values are prime, 7, 37, 271, 397, 547, 919, 1657, 1951, 2269, 4219, and ten composite, 91, 169, 721, 1141, 1387, 2611, 2977, 3367, 3781, 4681. Witness: mrlymath::formulas::six::solid_slice_vertices, A154105, A003215, A002407.

The layers as a dilation system

2026-09-09 Proved The stack's layers are a dilation system: the square wave s(x) = (-1)^floor(x) is the odd 2-periodic extension of the constant 1, so the layers s(nx) are the dilates of one function in the sense of Hedenmalm, Lindqvist and Seip, with sine coefficients a_n = 2 sqrt 2/(pi n) on odd n and 0 on even, and the symbol S(s) = (2 sqrt 2/pi)(1 - 2^(-1-s)) zeta(1 + s); the moire correlation law is exactly that system's Gram matrix sum_j a_(j n') a_(j m'), its both-odd hypothesis being the symbol's support on the odd integers, the lcm-grid integral and the symbol sum agreeing at all 78 pairs m <= n <= 12 (1/15 at (3,5), 1/3 at (3,9), 0 at (2,3)); the Gram vanishes unless v_2(m) = v_2(n) and every block is the odd Gram, so the full system's Gram operator is a direct sum of copies of the odd one and the half-period form is one block over 4. Witness: lab/py/stack-dilations check_dilation_shift, check_gram_two_ways, check_blocks.
2026-09-09 Proved The parity layers are complete and minimal in L^2(0,1) but neither a Riesz basis nor a frame: the Riesz criterion needs the symbol bounded on Re s > 0 and S is unbounded at the pole of zeta(1 + s) (Hedenmalm-Lindqvist-Seip Theorems 5.2 and 3.1), the normalised coefficients a_n/a_1 being totally multiplicative with divergent prime sum (Corollary 5.3), completeness and minimality following from Corollary 5.8 with the biorthogonal system built from the Dirichlet inverse mu(n)/n on odd n and not itself a dilation system; the sharp frame bounds are the supremum and infimum of |S| on the half-plane, infinity and zero; the parity stack is the boundary case tau = 1 of the source's example zeta(tau + s), Riesz iff tau > 1. Witness: lab/py/stack-dilations check_inverse, REFS.md.
2026-09-09 Proved No weighting of the scales moves the stack's symbol off the line Re s = 1: a weight multiplies the symbol by its own Dirichlet series at the same s, and the Mobius weight gives a_1^2/S, the reciprocal up to a_1^2 = 8/pi^2; over the first K odd scales lambda_max rises 2.01467 to 2.47224 and lambda_min falls 0.4393 to 0.3570 for K = 25 to 200, condition number 4.586 to 6.926, the determinant exact against the Smith product at K = 1..13. Witness: lab/py/stack-dilations check_inverse, spectrum, check_determinant.
2026-09-09 Refuted The pole heuristic that the largest Gram eigenvalue grows like (log N)^2: the spectral norm of a gcd matrix at exponent one over k distinct integers is of order (log log k)^2 by Lewko and Radziwill 2014 Theorem 2, which settles the exponent-one case that Gal 1949 bounded for the gcd sum, and the window agrees, the (log N)^2 ratio falling 1.93 across K = 25..200 against 1.41 for (log log N)^2. Witness: REFS.md, lab/py/stack-dilations spectrum.

The leaning stack

2026-09-09 Proved The leaning stack shifts layer n of the line stack by a drift t_n, its lines sitting where n x - theta_n is an integer with theta_n = n t_n mod 1. The linear lean theta_n = n delta is a translation, lit set F_Q + delta and brightness floor(N/b), 278 nodes and 0 mismatches at N = 30. The quadratic lean theta_n = n^2 c/d lights a/b iff n (a d - b c n) = 0 mod b d, the lit layers a union of 2^w residue classes modulo the period lcm(b, d*) with d* the least k with d | k^2 and w the number of primes with v_p(b) <= v_p(d) < 2 v_p(b); 0 mismatches against literal stacking on 10240 point-drift pairs at N = 60, b <= 20, d <= 16, and 0 solution-set mismatches; lcm(b, d*) is a period and not always the least, the least being lcm(b, d*)/2 exactly when v_2(b) >= 1 and v_2(d) = 2 v_2(b) - 1, 417 of 16384 tuples with b, d <= 20 halving and 0 breaches of the rule. Witness: lab/py/leaning-stack linear_lean, quadratic_lean, brightness_form, minimal_period_law, adversarial.
2026-09-09 Proved The leaning stack's lit set reads the numerator and its brightest node leaves the origin: at delta = 1/4 the point 1/4 is lit by n = 0, 1 mod 4 and 3/4 by n = 0, 3 mod 4, B_61 = 31 against 30, 48 such pairs over b <= 12 and d in 2, 4, 8, 9; the origin's brightness at drift c/d is floor(N/d*), so at delta = 1/2 the point 1/2 reads 60 at N = 60 and the origin 30, the origin beaten at five of seven printed drifts; the origin's density 1/d* is A019554, multiplicative with a(p^e) = p^ceil(e/2), its Dirichlet series zeta(2s+1) zeta(s+1)/zeta(2s+2) recovered as 1.826902 against 1.826907 at s = 1; the phase census at prime drift denominator is a Legendre symbol, n^2 c/p taking (p+1)/2 values with multiplicity 1 + (j c^-1/p), the centred twist S(c, p) = (c/p) S(1, p) holding coefficient by coefficient and the solution count being the Fourier sum of quadratic Gauss sums, 0 breaches at p = 5, 7, 11, 13. Witness: lab/py/leaning-stack lit_set, origin_law, gauss_sums.
2026-09-09 Proved Layers m != n of the quadratic lean share a lit point iff lcm(m, n)(m - n) delta is an integer, 0 criterion failures against literal intersection on 5280 pairs over m < n <= 12 and every reduced drift with d <= 16, so at irrational drift no two layers ever coincide and brightness is at most 1 everywhere at every N, the translation twin of the dead-spin theorem; the 325 lit points of layers 1..25 are distinct at sqrt 2 - 1 and phi - 1, the closest approach to N = 120 being 9.202e-09, and the near-coincidences follow Weyl's equidistribution of n^2 delta, star discrepancy 0.019450, 0.006421 and 0.022253, 0.009391 at N = 1000, 10000 against 1/sqrt N, four points and no exponent. Witness: lab/py/leaning-stack sharing_law, adversarial, irrational_lean.
2026-09-09 Refuted The lean is a weight on the scales: no weight w reproduces it, its brightness density 2^w/lcm(b, d*) per unit N depends on the numerator while every weighted stack reads sum_{k <= N/b} w(kb), denominator-only, and no weight makes a node brighter than the integer, which x = 1/2 at delta = 1/2 is; the lean is the first operation on this tree outside the Dirichlet group with a closed form. Witness: lab/py/leaning-stack quadratic_lean, lit_set.

The memory dial

2026-09-13 Proved A width-k rule W over the 2^dim digit vectors has N_W(level) = 1^T A_W^(level-k+1) 1 for level >= k - 1, where the states of A_W are the 2^(dim(k-1)) windows of width k - 1 and A_W[s][t] = 1 iff s and t overlap in k - 2 digits and the k-window they form is allowed, with the k = 1 case reading one state and A_W = [#W]; an accepted word of length level is exactly a path of level - k + 1 steps (witness: lab/py/memory-census README THE TRANSFER MATRIX).
2026-09-13 Proved The memory number kappa(W) = log_2(#W)/k - log_2 rho(W) is nonnegative for every width-k rule, since an accepted word of length mk splits into m disjoint allowed windows and so N_W(mk) <= #W^m, giving rho^k <= #W; it is zero on every product rule W = F^k with F non-empty, where N_W(level) = #F^level, while the empty rule has #W = 0 and rho = 0 and carries no kappa at all (witness: lab/py/memory-census README THE MEMORY NUMBER).
2026-09-13 Proved Every element of G_(dim,k), the signed permutations B_dim applied diagonally to the k digits of a window together with window reversal, preserves N_W(level) for every level: a diagonal B_dim element conjugates A_W by a permutation matrix and reversal transposes it, and A and A^T share a characteristic polynomial (witness: lab/py/memory-census README THE GROUP).
2026-09-13 Proved At dim = 1 the classes of width-k rules under diagonal B_1 alone number 2^(2^k - 1) + 2^(2^(k-1) - 1), because the digit flip acts on the 2^k windows as w -> 2^k - 1 - w in 2^(k-1) two-cycles (witness: lab/py/memory-census README THE CENSUS, Burnside).
2026-09-13 Proved The set of Perron roots occurring at width k is contained in the set occurring at width k + 1: the rule W' = {(d_1..d_(k+1)) : (d_1..d_k) in W and (d_2..d_(k+1)) in W} accepts the same words of length k + 1 and above, which is all rho needs, while at level = k exactly it has no window and accepts every word (witness: lab/py/memory-census README THE PERRON ROOTS).
2026-09-13 Proved G_(1,4) < G_(2,2) < B_4 as permutation groups of the 4-cube: flipping all four bits is the diagonal B_2 element flipping both axes, and the width-4 window reversal is the (2,2) block swap composed with the diagonal axis swap (witness: lab/py/memory-census README THE GROUP).
2026-09-13 Verified Width-k rules at base 2 in dimension 1 fall into 3, 9, 88, 16960 classes under G_(1,k) for k = 1..4, out of 4, 16, 256, 65536 rules, the orbit walk agreeing with an independent Burnside average on every row (witness: lab/py/memory-census census.csv, orbit walk and Burnside).
2026-09-13 Verified Under diagonal B_dim alone with no window reversal the counts are 3, 10, 136, 32896 at dim = 1 and k = 1..4 and 6, 8548 at dim = 2 and k = 1, 2, against 3, 9, 88, 16960 and 6, 4660 with reversal (witness: lab/py/memory-census census.csv, orbit walk and Burnside).
2026-09-13 Verified The memory dial at k = 1 is the plain design census: 3 classes at dim = 1 and 6 at dim = 2, A000616 at 1 and 2, with the two groups agreeing because reversal is trivial (witness: lab/py/memory-census census.csv, the (dim,1) rows).
2026-09-13 Verified A width-k rule in dimension dim is a subset of the k dim-cube counted with a smaller group, so the class counts meet or exceed A000616 at k dim, by factors 1, 3/2, 4, 42.19 at dim = 1 and k = 1..4 and 1, 11.59 at dim = 2, equal at k = 1 where reversal is trivial and the two censuses coincide (witness: lab/py/memory-census census.csv column a000616).
2026-09-13 Verified Code 7 at dim = 1 and k = 2, the rule forbidding the window 11, has Perron root the golden ratio, minimal polynomial x^2 - x - 1 and rho = 1.618033988749 (witness: lab/py/memory-census classes.csv, PARI factor and polrootsreal).
2026-09-13 Verified At dim = 1 and k = 3 the four named roots land on the four expected codes, each the least code of its class: code 127 gives tribonacci x^3 - x^2 - x - 1 at 1.839286755214, code 55 golden, code 23 supergolden x^3 - x^2 - 1 at 1.465571231876, code 54 plastic x^3 - x - 1 at 1.324717957244 (witness: lab/py/memory-census classes.csv, PARI).
2026-09-13 Verified The same four named polynomials occur in dimension 2 at width 2, on least codes 327 tribonacci, 19 golden, 323 supergolden and 326 plastic, carrying 121, 588, 54, 48 classes, so the named roots are not a dimension-one accident (witness: lab/py/memory-census classes.csv).
2026-09-13 Verified Across the whole census kappa(W) = 0 holds on exactly the non-empty product classes, 2 at every (1,k) and 5 at every (2,k), with zero counterexamples over 19563 live classes; the test is exact, kappa = 0 iff the minimal polynomial of rho divides x^k - #W (witness: lab/py/memory-census census.csv columns classes_kappa0 and kappa0_nonproduct).
2026-09-13 Verified For k >= 2 the largest memory number in the census is attained at rho = 1, by the largest rule of zero entropy, on a tie of 1, 3, 4, 3 classes whose least codes are log_2(3)/2 = 0.792481 on code 11 at (1,2), log_2(6)/3 = 0.861654 on code 175 at (1,3), log_2(13)/4 = 0.925110 on code 49071 at (1,4) and log_2(10)/2 = 1.660964 on code 36079 at (2,2); at k = 1 every live rule is a product, kappa is identically 0 and the maximum is attained on every live class, the full rule at rho = 2^dim included (witness: lab/py/memory-census census.csv columns kappa_max, kappa_max_code and kappa_max_ties).
2026-09-13 Verified The window budget of zero entropy, the largest number of windows a live class with rho = 1 allows, is 1, 3, 6, 13 at dim = 1 and k = 1..4 and 1, 10 at dim = 2: a rule may allow that many windows and still accept subexponentially many words, code 11 at (1,2), which allows 00, 01, 11, accepting every 0^a 1^b with N_W(level) = level + 1 (witness: lab/py/memory-census census.csv column rho1_windows).
2026-09-13 Verified The distinct characteristic polynomials number 3, 6, 23, 431 at dim = 1 and k = 1..4 and 5, 333 at dim = 2 and k = 1, 2, and the distinct minimal polynomials of rho number 3, 4, 10, 177 and 5, 185 (witness: lab/py/memory-census census.csv, exact Faddeev-LeVerrier and PARI factor).
2026-09-13 Verified The Perron roots that fail to dominate their conjugates strictly number 12 of 177 at (1,4) and 5 of 185 at (2,2), and every one of them is p(x^m) for some m >= 2 with p the minimal polynomial of a strict root already in the census, for instance x^4 - x^2 - 1 and x^6 - x^3 - 1 for the golden ratio; strictness is decided numerically, PARI complex roots against a 1e-20 gap, and the strict column is left empty on the dead row x so that the live count reads 12 and 5 (witness: lab/py/memory-census census.csv columns weak_perron_polys and weak_are_radicals).
2026-09-13 Verified Burnside extends the class counts past the orbit walk at no cost: under G_(1,k) for k = 1..8 they are 3, 9, 88, 16960, 1074036736, 4611686053860868096, 85070591730234617055658644612208132096, 28948022309329048855892746252171977006958709724020498949042189405102555529216 (witness: lab/py/memory-census memory.py Burnside extension, 0.01s).
2026-09-13 Verified None of 3, 9, 88, 16960, 6, 4660, 3, 4, 10, 177 or 3, 6, 23, 431 appears in the local OEIS dump; 3, 10, 136, 32896, 2147516416 greps three hits, A055708, A056006 and A191363, each a list of integers with a sigma property agreeing only through the closed form 2^(m-1)(2^m + 1) at m = 2^(k-1) (witness: grep of the local dump for each comma-delimited string, names read at source).
2026-09-13 Verified The census, the crate and the demo read a corner the same way: mrlymath::bang::universe::corners(dim) emits the corner vector row first and corner_index folds it most significant first, so a crate design's corner integer at dim = 2 is c = x + 2y, bit 0 the column and bit 1 the row, and no code label moves between the three (witness: crates/mrlydemo/tests/memory.rs::width_one_is_the_plane_design_cell_for_cell, which pins codes 11 and 13, exchanged by the axis swap and drawn differently).
2026-09-13 Verified The width-one memory rule is the plane design of the same code cell for cell, for codes 1, 7, 9, 11, 13, 14 at every level one to six (witness: mrlydemo::memory::memory_sheet against mrlydemo::two::two_grid, test crates/mrlydemo/tests/memory.rs::width_one_is_the_plane_design_cell_for_cell)
2026-09-13 Verified The golden rule, dim = 1 width 2 code 7, which forbids the window 11, accepts 2, 3, 5, 8, 13, 21, 34, 55 words at levels one to eight, the Fibonacci numbers (witness: mrlynum::memory::counts, test the_golden_rule_counts_the_fibonacci_numbers)
2026-09-13 Verified The golden rule has Perron root 1.618034, growth exponent 0.694242 and memory number kappa = log_2(3)/2 - log_2(phi) = 0.098239 (witness: mrlynum::memory::perron, exponent, kappa, check row memory golden growth)
2026-09-13 Verified The supergolden rule, dim = 1 width 3 code 23, the sponge code read as a window rule, allows at most one 1 a window and accepts 2, 4, 4, 6, 9, 13, 19, 28 words at levels one to eight, the Narayana cow recurrence a(level) = a(level - 1) + a(level - 3) holding from level = 2k = 6 on and failing at level = 5, where the count is 9 against a(4) + a(2) = 10 (witness: mrlynum::memory::counts, test the_supergolden_rule_counts_the_narayana_cows)
2026-09-13 Verified The supergolden rule has Perron root 1.465571, the supergolden ratio, the real root of x^3 = x^2 + 1 (witness: mrlynum::memory::perron, check row memory cow root)
2026-09-13 Verified The rule that forbids a digit twice in a row, dim = 2 width 2 code 31710, accepts 4, 12, 36, 108 words at levels one to four, root exactly 3 since its transfer matrix is J - I on the four digits with minimal polynomial x - 3, and memory number kappa = log_2(12)/2 - log_2(3) = 0.207519 (witness: mrlydemo::memory::memory_read, check row memory no repeat, lab/py/memory-census classes.csv row x - 3 at (2,2))
2026-09-13 Verified The full rule accepts 2^(dim level) words at every dimension one to three and every width its span allows, and its growth exponent is the dimension (witness: mrlynum::memory::counts and exponent, test the_full_rule_counts_every_word)
2026-09-13 Verified The empty rule accepts every word shorter than its window and nothing at or past it, and its Perron root is exactly zero (witness: mrlynum::memory::counts and perron, test the_empty_rule_dies_past_its_window)
2026-09-13 Proved The memory number kappa(W) = log_2(card W) / k - log_2 rho, for card W the allowed windows, is the bits per digit a rule spends on memory and is zero on every memoryless design W = F^k with F non-empty, so every width-one rule reads zero (witness: mrlynum::memory::kappa and allowed_windows, test width_one_is_the_memoryless_design)
2026-09-13 Verified The plastic rule, dim = 1 width 3 code 54, accepts 2, 4, 4, 5, 7, 9, 12, 16 words at levels one to eight, Perron root 1.324717957 the plastic number, the real root of x^3 - x - 1, and memory number 0.260981 (witness: mrlydemo::memory::memory_read, test the_width_three_presets_name_the_plastic_and_tribonacci_roots, check row memory plastic root)
2026-09-13 Verified The tribonacci rule, dim = 1 width 3 code 127, which forbids only the window 111, accepts 2, 4, 7, 13, 24, 44, 81, 149 words at levels one to eight, Perron root 1.839286755 the tribonacci constant, the real root of x^3 - x^2 - x - 1, and memory number 0.056639 (witness: mrlydemo::memory::memory_read, test the_width_three_presets_name_the_plastic_and_tribonacci_roots, check row memory tribonacci root)
2026-09-13 Verified The supergolden rule has memory number kappa = 2/3 - log_2(1.465571232) = 0.115204 (witness: mrlynum::memory::kappa)
2026-09-13 Proved A width-k digit rule W in dimension dim at base base counts its accepted words by a path count: with states the base^(dim(k-1)) words of k-1 digit vectors and A[x,y] the number of allowed windows with prefix x and suffix y, N_W(level) = 1^T A^(level-k+1) 1 for every level >= k-1, and at k = 1 the matrix is [card W] so the count is card W^level, today's fill law. (witness: beneath.md, The transfer matrix)
2026-09-13 Proved A width-k rule in dimension dim is a subset of the corners of the k dim-cube, so the raw census 2^(base^(k dim)) is the design count at dimension k dim and transports unchanged, while the quotient does not: cube symmetry acts diagonally on the k windows, so the group is B_dim of order 2^dim dim! and not B_(k dim) of order 2^(k dim) (k dim)!. (witness: beneath.md, Width k in dimension dim is a subset of the k dim-cube)
2026-09-13 Proved The coupling kappa(W) = log(card W)/(k log base) - log(rho(A))/log(base) of a width-k rule is nonnegative, by cutting an accepted word of length mk into its m disjoint windows so that N_W(mk) <= card W^m, while rho^(level-k+1) <= N_W(level) by the entry sum of A^(level-k+1); and kappa = 0 on every product W = G^k with G non-empty, where N_W(level) = card G^level and rho = card G, so every memoryless design with a non-empty rule sits at coupling zero. (witness: beneath.md, The coupling)
2026-09-13 Proved Memory does not leave the lattice class: the counting series sum_level N_W(level) x^level of a width-k rule is rational with denominator det(I - x A), so at x = base^(-s) its poles sit on finitely many vertical lines Re s = log(abs(lambda))/log(base) over the nonzero eigenvalues lambda of A and the pole set is invariant under s -> s + 2 pi i / log base, the same period as the memoryless case; one line can carry a finer progression, as at dim = 1, k = 2, code 6, whose eigenvalues 1 and -1 put poles at gap pi / log base on Re s = 0. (witness: beneath.md, What the dial does not buy; lab/py/memory-census, the class of code 6)
2026-09-14 Proved The class count of width-k binary rules under G_(1,k) is a(2m) = 2^(2^(2m)-2) + 2^(2^(2m-1)-2) + 2^(2^(2m-1)+2^(m-1)-1) for m >= 1 and a(2m+1) = 2^(2^(2m+1)-2) + 2^(2^(2m)-1) + 2^(2^(2m)+2^m-2) for m >= 0, by Burnside over the order-4 group: the digit flip fixes no window, reversal fixes the 2^ceil(k/2) palindromes, and flip-reversal fixes the 2^(k/2) antipalindromes at even k and none at odd k; the form reproduces 3, 9, 88, 16960 and every Burnside extension term through k = 8 (witness: lab/py/memory-census verb burnside, the cycle index and the closed form agreeing at k = 1..11, and beneath.md, The memory dial).
2026-09-14 Proved At dim = 1 and k >= 2 every transfer matrix has determinant in {-1, 0, 1}, so the constant term of every characteristic polynomial is 0, 1 or -1: rows s and s + 2^(k-2) are both supported on the columns 2s and 2s+1 taken modulo 2^(k-1), those column pairs partition the columns as s runs over 0..2^(k-2)-1, and the matrix is therefore a row permutation of a block diagonal matrix with 2^(k-2) blocks of size 2 x 2 over {0,1}; checked over all 16, 256, 65536 rules at k = 2, 3, 4 (witness: lab/py/memory-census verb lemmas, a Bareiss determinant and the signed block product agreeing in {-1,0,1} on every rule at k = 2, 3, 4, and beneath.md, The memory dial).
2026-09-14 Proved Distinct minimal polynomials of the Perron root are distinct Perron roots: every conjugate of rho(W) is a root of the characteristic polynomial of A_W and so an eigenvalue of A_W, hence at most rho(W) in modulus, so two conjugate Perron roots are equal in modulus and, both being nonnegative, equal; the 3, 4, 10, 177 minimal polynomials at dim = 1 and k = 1..4 are therefore 3, 4, 10, 177 distinct growth rates (witness: lab/py/memory-census verb lemmas, no conjugate above rho on any of the 463 characteristic polynomials at k = 1..4, the 3, 4, 10, 177 minimal polynomials carrying 3, 4, 10, 177 distinct rho, and beneath.md, The memory dial).
2026-09-14 Verified Second generators reproduce the memory census whole: a cycle-index Burnside counter gives 3, 9, 88, 16960, 1074036736, 4611686053860868096 under G_(1,k) at k = 1..6 and 6, 4660, 1152921592116822016 under G_(2,k) at k = 1..3, with the remaining extension terms at k = 7, 8 and k = 4 agreeing as well, and an exact integer Faddeev-LeVerrier enumeration over all rules, factored in PARI, gives 3, 6, 23, 431 characteristic polynomials and 3, 4, 10, 177 minimal polynomials at dim = 1 and k = 1..4 (witness: lab/py/memory-census verb burnside at dim = 1, the census run's Burnside extension at dim = 2, verb lemmas for the polynomial counts, every cell reproduced).
2026-09-14 Verified None of the memory-census counts is in the local OEIS dump: 3, 9, 88, 16960 with every Burnside extension term through k = 8, 6, 4660 with its extension through k = 4, 3, 4, 10, 177 and 3, 6, 23, 431 each grep to zero hits as comma-delimited runs, while 3, 10, 136, 32896 hits A055708, A056006 and A191363 through the coincidence 2^(2^k-1) + 2^(2^(k-1)-1) = A007582(2^(k-1)), and 9 and 16960 recur inside A367526, a grid tiling count with different neighbours (witness: grep of the local OEIS dump on the runs printed by lab/py/memory-census verb burnside, each hit read at source).
2026-09-14 Verified The binary words of length n counted under the same group that the width-n rules are counted under, reversal together with bitwise complementation, are A005418 at n: 1, 2, 3, 6, 10, 20, 36, 72 at n = 1..8 (witness: lab/py/memory-census verb burnside, the word orbits printed beside the class count, agreeing with the entry in the local OEIS dump).
2026-09-14 Verified The Perron root of a transfer matrix is the largest root over the strongly connected components of its digraph, each component being irreducible with a simple root, so it is exact against that component's integer characteristic polynomial; at dim = 1, k = 3, code 5 gives 1, codes 62, 125 and 190 give the plastic number 1.324717957, code 91 gives 1.380277569, code 95 gives the golden ratio 1.618033989, and the coupling of codes 125 and 190 is log_2(6)/3 - log_2(1.324717957) = 0.455969; a stop comparing one scalar across two sweeps halts on a plateau of the I + A mass ratio and misses all six (witness: mrlynum::memory::perron and its test, lab/rs/memory-meter).
2026-09-14 Verified The memory meter's control column is the Mertens function: the width-1 rule of code 3 at base 2 accepts every integer, and its meter reads -1, 1, 2, -23, -48, 212, 1037, 1928 at 10^1..10^8, asserted inside the run, which is A084237 (witness: lab/rs/memory-meter, the control mertens line, and beneath.md, The memory meter).
2026-09-14 Verified The same linear sieve reproduces the memoryless base-3 design meters exactly, so the memoryless row of the dial is pinned against the existing census: digits {0,1} read (M, max abs M) = (11, 105) at level = 14, (149, 173) at level = 16 and (-30, 312) at level = 18, digits {1,2} read (-1461, 1582) at level = 18, each asserted; digits {0,2} at level = 20 wants 3^20, past the 2^30 sweep, and is printed unpinned at level = 14, 16, 18 as (-10, 67), (-124, 152) and (67, 249). The four pinned pairs are read at source in lab/py/design-meter, which computes them and cites lab/rs/mobius-designs as their census (witness: lab/rs/memory-meter, the control design lines, and beneath.md, The memory meter).
2026-09-14 Verified The window-profile recurrence, profile(n) being profile(n >> 1) unioned with the window n mod 2^k, reads the accepted integer set of every width-1, 2 and 3 rule at dim = 1, base 2: on all 276 rules its masses agree with a direct digit recount below 2^20, profile containment agrees with mrlynum::memory::Rule::accepts below 2^12, and at every one of the 89 phases the subset-sum transform of the per-profile mu sums equals the independently carried per-rule meter (witness: lab/rs/memory-meter, the control recount line, and beneath.md, The memory meter).
2026-09-14 Verified Code 7 at (dim, k) = (1, 2), the golden rule forbidding the window 11, opens exactly the fibbinary integers A003714 without its zero, and its mass below 2^level is a Fibonacci number, A = 2178309 below 2^30; code 11, forbidding 10, opens exactly the Mersenne numbers A000225 without its zero and holds 30 elements below 2^30, one per level (witness: lab/rs/memory-meter, the control code 7 and control code 11 lines, and beneath.md, The memory meter).
2026-09-14 Verified The Mobius meter of every width 1, 2, 3 rule at dim = 1, base 2, read to 2^30 at the 89 phases x = floor(2^(level + j/4)), level = 8..30, j = 0..3, no exponent fitted, each ratio at a named phase: at 30.00 the full line reads A = 1073741824, M = -10374, max abs M = 11173, ratios -0.316589 and 0.340973; the golden rule code 7 at k = 2, kappa = 0.098239, reads A = 2178309, M = 551, max abs M = 716, ratios 0.373329, 0.485125; the k = 3 least codes 23, 54, 127 read normalised peaks 0.731125, 0.677943, 1.239625 at kappa = 0.115204, 0.260981, 0.056639 (witness: lab/rs/memory-meter, the rule and row lines, and beneath.md, The memory meter).
2026-09-14 Verified A rule's rho and kappa are read off the transfer matrix of its word language while A and M are read off its integer set, and on a rule that is not zero-closed those are different objects: at k = 3 code 5 the word language grows on the self-loop 000 and carries rho = 1, while the integer set holds 3 elements below 2^30 (witness: lab/rs/memory-meter, the rule k=3 code=5 line, and beneath.md, The memory meter).
2026-09-14 Verified Over the census of 53 rules of all three widths holding at least 10^4 integers below 2^30, the floor fixed before any reading, the normalised peak max abs M_W/sqrt(A_W) spans [0.293624, 1.239625] at phase 30.00, least on k = 3 code 125 and largest on k = 3 code 127, and spans [0.500000, 2.169240] over all 89 phases, the top on k = 3 code 232 at phase 16.75; the last-phase leader and the sweep-wide leader are different rules, so no rule is the dial's widest excursion (witness: lab/rs/memory-meter, the span lines, and beneath.md, The memory meter).
2026-09-14 Verified Read against the full line at the same phase, which needs no band and no grid, the factor max abs M_W/sqrt(A_W) over max abs M/sqrt(x) runs [0.861136, 3.635552] at phase 30.00 over the 53 census rules and reaches 6.375774 on k = 3 code 190 at phase 12.75 over all phases (witness: lab/rs/memory-meter, the factor lines, and beneath.md, The memory meter).
2026-09-14 Verified Grouped by the coupling over the same 53 rules at phase 30.00, the mean normalised peak reads 0.340973 on kappa = 0 with 3 rules, then 0.622171 on 6, 0.581446 on 14, 0.644840 on 12 and 0.645966 on 18 for the bands [10^-9, 0.1), [0.1, 0.2), [0.2, 0.3) and [0.3, 0.5); every band of positive coupling sits above the kappa = 0 band, whose three rules are the full line under three codes, and among the positive bands the means are not monotone in kappa. Restricted to k = 3 the same bands read 0.340973, 0.698386, 0.581446, 0.644840, 0.645966 on populations 1, 4, 14, 12, 18 (witness: lab/rs/memory-meter, the kappaband lines, and beneath.md, The memory meter).
2026-09-14 Verified The full line's own normalised peak runs [0.272410, 0.500000] over the grid, the ceiling at phase 8.00, and that ceiling is a property of where the grid starts and not of the full line: below the grid the same ratio reads 1.000000 at x = 1, 0.894427 at 5, 0.832050 at 13, 0.718421 at 31 and 0.565685 at 200 (witness: lab/rs/memory-meter, the gridstart line, and beneath.md, The memory meter).
2026-09-14 Verified The falsification fires. Five of the 53 census rules never enter the full line's band at any phase where they hold 10^4 elements, all of them above it: k = 3 codes 159, 182, 190, 218 and 250, holding 211116, 13607, 31535, 59860 and 4126645 integers. The band's ceiling being grid-dependent, the same-phase factor is the instrument that carries the reading, and it is read rule by rule on the generator's factor lines (witness: lab/rs/memory-meter, the band outside lines, and beneath.md, The memory meter).
2026-09-14 Verified 16 of the 53 census rules attain their sweep-wide normalised peak in the last quarter of the phases, from 24.75 on, so most of the dial peaked earlier and is not growing at the end of the sweep. No rule at any width holding at least 1000 elements has M_W/sqrt(A_W) or max abs M_W/sqrt(A_W) rise at every one of the last eight phases, but that test asks max abs M_W to grow about 9% per quarter-level across two whole levels, so its empty answer carries little and the late-peak count is the informative statistic (witness: lab/rs/memory-meter, the latepeak and climbing lines, and beneath.md, The memory meter).
2026-09-14 Verified Leading zeros move most rules' integer sets. A rule is zero-closed when prepending one zero changes no membership below 2^20: the zero-closed codes number 3 of 4 at k = 1, 8 of 16 at k = 2 and 64 of 256 at k = 3, and are the codes allowing 0 with the empty code, those allowing 01, and those allowing both 010 and 011, asserted code for code. Under any number of zeros the word language agrees with the integer set on 2 of 4, 4 of 16 and 16 of 256 codes, tested on every word to length 14, so at k = 3 the two readings part company on 240 of 256 (witness: lab/rs/memory-meter, the zeroclosed and reading lines, and beneath.md, The memory meter).
2026-09-14 Verified Equal mass is not the same set: code 14 at k = 2, forbidding 00, and code 126 at k = 3, forbidding 000 and 111, each hold 28655 integers below 2^20 and each carry rho = 1.618033989, yet they share only 1077 of them, the symmetric difference is 55156, and 4 is the least integer the second holds and the first does not; below 2^30 both hold 3524576 integers while their meters read -466 and 435 and their normalised peaks 0.454355 and 0.594444 at phase 30.00 (witness: lab/rs/memory-meter, the pair line, and beneath.md, The memory meter).
2026-09-14 Verified The matrix ladder of a memory design runs in double precision in the public crate. mrlynum::automaton carries Automaton, zeta, cofactor, residue and denominator, every value beside its bound. The full rules, code 15 at k = 2 and 255 at k = 3, meet mrlynum::ladder on the base 2 full design to 7.18e-11 at s = 2 and 2.03e-14 at s = 0.3 + 40i, inside bounds; the product rule code 8 meets a direct Mersenne sum to 1.12e-16; the golden rule code 7 meets a direct fibbinary sum with its Fibonacci tail bound, and every pinned golden reading meets the arbitrary-precision control inside its own bound, the four zeta_W values to 3.0e-15 and the largest of the fourteen rows 1.134e-11 against 7.348e-11 (witness: mrlynum::automaton, lab/py/memory-zeta, beneath.md, The memory zeta).
2026-09-14 Proved The polynomial beneath.md names the string equation of a memory rule is also the denominator of the rule's Dirichlet series, strictly more than that page proves. beneath.md proves det(I - x A) denominates the counting series and names det(I - base^(-s) A) = 0 the Moran replacement; the peel carries it to zeta_W, each level of (I - base^(-w) T) G_P(w) = E_P(w) + sum_(l >= 1) binom(-w,l) base^(-w-l) Gamma_l G_P(w+l) dividing by det(I - base^(-w) T). So the poles of zeta_W lie in base^(-s) lambda_i = base^m, lambda_i a nonzero eigenvalue and m >= 0 whole, and that determinant is the m = 0 level's denominator, not the whole one (witness: mrlynum::automaton, beneath.md, The memory zeta).
2026-09-14 Proved Right of the abscissa the matrix ladder carries its bound as a nonnegative vector and needs no norm and no primitivity, which settles the Conjecture row the ladder unit left open. For nonnegative y, abs((I - base^(-w) T)^(-1)) y <= sum_(i >= 0) (base^(-Re w) T)^i y entrywise, since abs(base^(-w)) = base^(-Re w) and T is nonnegative; the remainder closes on the guide v = (I + T)^60 1, which meets T v <= mu v for the bracket's upper end mu, as sum_(i > N) A^i y <= (max_u y_u/v_u) theta^(N+1)/(1 - theta) v, theta = base^(-Re w) mu < 1. The seed is exact: sum_(j >= P) E_j(sigma) <= base^(-(P-1)sigma) (I - base^(-sigma) T)^(-1) c_P (witness: mrlynum::automaton, beneath.md, The memory zeta).
2026-09-14 Proved The residue of a memory zeta at a simple pole needs no eigenvector: the adjugate is the spectral projector in polynomial form. Faddeev-LeVerrier on T gives integer matrices M_k and integer coefficients c_k with adj(I - x T) = sum_(k < n) x^k M_k and det(I - x T) = sum_(k <= n) c_k x^k, so with x = base^(-w) and N the ladder numerator, Res_(w0) zeta_W = 1^T adj(I - x_0 T) N(w0) / (-x_0 log base det'(x_0)). The form is stable at the pole, where det(I - x_0 T) (I - x_0 T)^(-1) is not, and it dies exactly where the ladder unit said it would, at a multiple root, where det'(x_0) = 0 and the pole order exceeds one (witness: mrlynum::automaton, beneath.md, The memory zeta).
2026-09-14 Verified The golden rule carries two genuine pole combs interleaving at half a tooth. T of code 7 at k = 2 is [[1,1],[1,0]] and det(I - x T) = 1 - x - x^2, so one comb sits at Re s = log_2 phi = 0.6942419136306174 with Im s in 2 pi Z / log 2 and one at Re s = -log_2 phi with Im s in (2Z + 1) pi / log 2, the argument pi of the negative eigenvalue shifting it half a period. The six residues at m = 0 are under THE POLE COMB and none is zero. Comb two comes from the left-of-abscissa branch, so its path is the arbitrary-precision control and not the contour average: the six agree to 6e-16 and 4.3e-12 (witness: mrlynum::automaton, lab/py/memory-zeta, beneath.md, The memory zeta).
2026-09-14 Verified Burnol's Proposition 5.1 survives the memory dial verbatim: the residue at the abscissa is the limit of the level digit sums. For code 7 the direct sum of n^(-alpha) over the fibbinary integers of exactly level bits, over log 2, reads 0.946747043404283, 0.946743630023052 and 0.946743410426742 at level = 16, 20, 24 against the ladder's 0.946743395641970, the gap falling like 2^(-level) (witness: mrlynum::automaton, beneath.md, The memory zeta).
2026-09-14 Verified The matrix Lyndon cofactor is the determinant with its adjugate, and it reads on the whole m = 0 comb where the series is singular. Z_W(s) = det(I - base^(-s) T) zeta_W(s) is carried as det(I - base^(-s) T) D_(P-1)(s) + 1^T adj(I - base^(-s) T) N(s), so it never divides by the vanishing determinant; for code 7 it reads 0.991729890316722 at s = 3, 0.973380053858285 at s = 2 and 0.913335748872126 at s = 0.8, each to a bound near 1e-13, while zeta_W(0.8) = 9.536379694275015 is already climbing the pole at 0.6942419136306174 (witness: mrlynum::automaton, beneath.md, The memory zeta).
2026-09-14 Proved Left of the abscissa a matrix ladder has no free denominator bound, and the module buys one with the residual of its own inverse. abs(1 - k base^(-w)) has no matrix analogue and the Neumann majorant diverges once base^(-Re w) rho >= 1, so the level closes on the computed inverse C certified against R = I - (I - base^(-w) T) C: for nonnegative y, abs((I - base^(-w) T)^(-1)) y <= abs(C)(y + (max_u y_u) r/(1 - r) 1) with r the max row sum of abs(R), and the module raises when r >= 1. The second comb of code 7 is read only through that branch, at bounds near 4e-9 against 1.3e-13 on the first (witness: mrlynum::automaton, beneath.md, The memory zeta).
2026-09-14 Verified The matrix Lyndon cofactor Z_W(s) = det(I - 2^(-s) T) zeta_W(s) of the golden rule, code 7 at width 2, has exactly 20 zeros in the box -0.95 < Re s < 2, 0.02 < Im s < 43.1: the determinant strips both m = 0 combs in one factor, leaving Z_W meromorphic there with exactly 4 simple poles, the level-one teeth on Re s = -0.305758086, so each cell count is its argument-principle winding plus the level-one teeth the cell holds; all 20 are located with largest abs(Z_W) 9.694e-12, largest surviving phase step 0.999894 radians against a cap of one, largest propagated bound 1.474e-10, nothing within 0.02 of an outer box edge, and identical cell rows and zeros at contour seeds 0.1, 0.05 and 0.025, at 7298, 11777 and 21599 evaluations, and with 0 < Im s < 0.02, Im s > 43.1 and Re s < -0.95 uncounted, 20 is exact on the box (witness: lab/py/memory-zeta, verb census, and beneath.md, The memory zeta).
2026-09-14 Verified The 4 poles that census adds back are read and not assumed: a 48-point circle mean of Z_W at each level-one tooth of code 7 gives residues -1.990368154340-0.795661945868i, -0.350975872907-0.436714265460i, -3.135030562964-2.032376530959i and -1.028888122837+2.036528502776i, radius 0.05 against radius 0.02 agreeing to 7.3e-14, each simple to 5.1e-05 against (s - s_0) Z_W at 1e-5, while a blank point on the same line reads 4.6e-16, and the same read at code 23 gives four residues of modulus 2.231820, 3.226224, 2.438482 and 1.779516, the two radii agreeing to 5.1e-14, against a blank point at 6.3e-16 (witness: lab/py/memory-zeta, verb census, and beneath.md, The memory zeta).
2026-09-14 Proved Z_W has no zero in Re s >= 2 on code 7: the least element of S_W is 1 and the coefficients are nonnegative, so abs(zeta_W(s) - 1) <= zeta_W(2) - 1 < 1 there from zeta_W(2) = 1.415825532885 < 2, and det(I - 2^(-s) T) has no root right of the abscissa log_2 phi, which makes the census box's right edge a wall and not a choice (witness: lab/py/memory-zeta, verb census, and beneath.md, The memory zeta).
2026-09-14 Verified On code 7 the first comb carries a zero comb and the second carries none: at the design family census radius 0.45 all 4 teeth of the comb on Re s = log_2 phi below Im s = 43.1 carry a zero, at distances 0.317490225, 0.045406362, 0.143076282 and 0.070233751, while 0 of the 5 teeth on Re s = -log_2 phi do, least distance 0.666213518 and largest 0.758440773, and the emptiness holds over every point of every disc, the radius 0.45 disc reaching Re s = -1.144241913631, since the same census on -1.2 < Re s < 2, which admits no new pole line before -1.305758086369, returns the same 20 zeros, nineteen to twelve decimals and the twentieth to eleven, the same 4 of 4 and 0 of 5 and the same five distances (witness: lab/py/memory-zeta, verb census, and beneath.md, The memory zeta).
2026-09-14 Verified The first-order tooth law u_1 = -r/R holds on code 7's first comb to 0.000951131, 0.003342909, 0.020403749 and 0.062287774 and misses on the second by 0.214394685, 0.645682670, 0.408924841, 0.489190529 and 0.519744494, and it is a reading beside the census and not a second falsification: R is a circle mean of radius 0.3, three of the five second-comb predictions of abs(u_1), 0.501975708, 0.398920848 and 0.301764481, are read outside that disc, and on the first comb the one prediction past 0.3 carries the worst miss (witness: lab/py/memory-zeta, verb census, and beneath.md, The memory zeta).
2026-09-14 Verified The second comb's line carries zeros where its teeth do not: three of code 7's 20 zeros sit within 0.05 of Re s = -log_2 phi, at 0.000322593, 0.014258173 and 0.043369824 from that line, while their distances to the nearest tooth of the same comb are 4.104099275, 0.747618761 and 4.283371881, and stripping the 4 first-comb teeth leaves a second family of 16 with real parts in [-0.737611737911, 0.540957439322]; the excess is 4.4 times the 0.68 that 20 uniformly spread real parts would put in a window of width 0.1 on a box 2.95 wide, on a sample of 20 (witness: lab/py/memory-zeta, verb census, and beneath.md, The memory zeta).
2026-09-14 Verified Exactly 9 of the 88 width-3 rule classes under G_(1,k) carry two pole lines, none with a repeated eigenvalue, and the cofactor's zeros are a resolved census on one printed box for all nine, S_W read off the minimal base-2 string: -1.15 < Re s < 2, 0.02 < Im s < 20, contour seed 0.05, which holds every radius 0.45 occupancy disc of every second line, the deepest reaching Re s = -1.144241913631; the nine read 9 to 14 zeros in 20 to 36 cells, every zero located, largest residual 3.236e-11, largest surviving phase step 0.999909 radians against a cap of one, largest propagated bound 9.110e-09, and no pole of Z_W within 0.02 of any contour, the least clearance being exactly 0.02, from the cut Im s > 0.02 to the level-m pole on the real axis. Resolved and not certified: nothing bounds Z_W'/Z_W on the contour, and two zeros sit within 0.02 of the left contour, -1.134547677+3.580553251i on code 127 and -1.143621954+17.814806003i on code 63 (witness: lab/py/memory-zeta, verb teeth, and beneath.md, The memory zeta).
2026-09-14 Verified The occupancy reading is invariant across two boxes whose contours fail a 0.02 guard in disjoint ways. On -1.2 < Re s < 2 the pole lines Re s = -1.202842615688 of codes 54 and 62 and Re s = -1.188629537248 of code 223 sit 0.002842615688 and 0.011370462752 from the left contour, the first pair outside the box and so never added back; on -1.15 < Re s < 2 every pole clears 0.02 and two zeros do not. Both boxes read 50 teeth, 34 occupied at radius 0.45, 23 teeth predicting abs(u_1) < 0.3 occupied 22, 15 predicting at or above 0.45 occupied 4, 12 between occupied 8, and the same occupancy column on all nine rules; only the totals off the discs move, code 23 from 13 zeros to 11 and code 31 from 14 to 13 (witness: lab/py/memory-zeta, verb teeth, at --left -1.2 and --left -1.15, and beneath.md, The memory zeta).
2026-09-14 Proved With S_W read off the minimal base-2 string, so that a word shorter than the window holds no window and is accepted, the width-3 rule 55 accepts exactly the set of the width-2 rule 7 with the single integer 3 adjoined, hence zeta_55(s) = zeta_7(s) + 3^(-s). Code 55 forbids exactly the windows 011, 110 and 111, which is exactly the ban on an adjacent pair of ones inside a 3-window, and for length at least 3 the window starting at min(i, level-3) holds the pair at (i, i+1), while the word 11 carries no window and is accepted; checked over 1 .. 262143 with 3 the only difference either way. Padded to the window width instead the two sets are equal and the claim is empty (witness: lab/py/memory-zeta, verb bridge, and beneath.md, The memory zeta).
2026-09-14 Verified The width-3 ladder meets the controlled width-2 one across that gap, which is the control on a new rule: Z_55(s) - Z_7(s) - det(I - 2^(-s) T) 3^(-s) reads at most 1.168e-13 over seven points including three teeth and one located zero, every point inside the sum of its own two bounds, so the 4-state ladder, adjugate and peel meet the 2-state ones that carry the stored arbitrary-precision control, itself met to 1.134e-11 on 14 rows with none outside its bound (witness: lab/py/memory-zeta, verbs bridge and control, and beneath.md, The memory zeta).
2026-09-14 Verified One tooth of the 50 carries two zeros inside the occupancy radius, so 34 occupied teeth hold 35 zeros: code 54, second line, tooth Re s = -0.202842615688, Im s = 14.612532469, holds 0.213711738933+14.629308261174i at 0.416892021 and -0.597460129789+14.668266829134i at 0.398533940, and abs(u_1) = 0.631828588 there puts it in the bin the first-order law reads as empty. The other 49 teeth hold at most one, and the count is printed by the census itself as tooth zeros 6 against 5 occupied teeth on that rule (witness: lab/py/memory-zeta, verbs teeth and census --width 3 --code 54, and beneath.md, The memory zeta).
2026-09-14 Proved For a finite set F of positive integers disjoint from S_W, the set S_W + F = S_W u F has zeta_(W+F)(s) = zeta_W(s) + P_F(s) with P_F(s) = sum_(n in F) n^(-s) a Dirichlet polynomial and so entire, hence the two series carry the same poles, the same orders and the same residues at every point of the plane, and Z_(W+F)(s) = Z_W(s) + det(I - base^(-s) A) P_F(s); adding a finite set is a knob on the zero set alone (witness: lab/py/memory-zeta verb dial, the off-tooth identity Z_(W+F)(s) - Z_W(s) - det(I - 2^(-s) T) P_F(s) missing by at most 2.384e-15 at code 7 and 4.003e-16 at code 23 against an added part of up to 1.912203 and 1.708983).
2026-09-14 Proved The knob cannot move the cofactor at a tooth: at every m = 0 tooth t the determinant vanishes, so Z_(W+F)(t) = Z_W(t) exactly, and the principal part of zeta_W at t is fixed while the constant term becomes R + P_F(t), so the first-order zero position is u_1(F) = -r/(R + P_F(t)); every higher coefficient of the regular part moves too, the linear one by P_F'(t) = -log(n) n^(-t) (witness: lab/py/memory-zeta verb census, code 55 at width 3 reading r = 0.210170579-0.581938843i at Im s = 9.064720 digit for digit against code 7's and R = 1.313430833+1.119663028i against 1.714940435+0.882338583i, a difference of -0.401509602+0.237324445i which is 3^(-t) to nine decimals).
2026-09-14 Verified The residue and tooth probes of the dial cannot falsify the perturbation's entirety and the off-tooth identity can: a 48-point circle mean annihilates an entire addition and the determinant vanishes at a tooth, so the printed gaps are an aliasing floor and a determinant residual and not a measurement, while the identity read off the teeth agrees to fifteen decimals against an added part of order one (witness: lab/py/memory-zeta verb dial, largest residue gap over every probe 1.776e-15, largest tooth value gap 1.250e-13, largest off-tooth identity miss 2.384e-15 against an added part of up to 1.912203 at code 7).
2026-09-14 Verified The dial's disc probe reproduces the zero census it is read against: on code 7 at width 2 it reads 4 of 4 teeth occupied on the comb at Re s = log_2 phi and 0 of 5 on the comb at -log_2 phi, and on code 23 at width 3 it reads 3 of 4 on the first comb and 7 of 10 on the second line, each occupancy an argument-principle count on a circle of radius 0.45 about the tooth with the level-m poles inside added back (witness: lab/py/memory-zeta verb dial, edge guard splits 0 on both baselines, a guard that covers the baselines and not the perturbed grid).
2026-09-14 Verified Occupancy at radius 0.45 under the knob is undetermined wherever a zero sits within the 0.02 guard of the occupancy circle and the inner count is 0, which is 9 of the 110 cells of code 7's second comb, 10 of the 108 of code 23's abscissa comb and 9 of the 270 of code 23's second line, so a minimum taken over a tooth's candidate row has two readings and the reading must be named (witness: lab/py/memory-zeta verb dial, per-line rows occupancy undetermined 9, occupancy undetermined 10 and occupancy undetermined 9).
2026-09-14 Verified Every empty tooth of code 7's second comb is occupied by a single added integer from the 22 integers of 2 .. 40 outside S_W, so the smallest F that occupies a tooth of the empty comb has one element and that element is at most 11 under either reading of the seam, while the least singleton itself is radius-dependent at two of the five teeth: {3}, {6}, {7}, {11}, {11} on the inner reading against {3}, {6}, {3}, {11}, {6} on the outer, at Im s = 4.532360, 13.597080, 22.661801, 31.726521 and 40.791241 (witness: lab/py/memory-zeta verb dial, per-tooth rows smallest singleton and outer reading occupied ... smallest singleton).
2026-09-14 Verified Occupancy moves both ways on code 23 and the count is stable under either reading of the seam: four empty teeth are filled, Im s = 9.064720 by {7} or {6}, 20.807773 by {5}, 29.872493 by {7} and 38.937214 by {15} or {11}, and two occupied second-line teeth are emptied by a singleton off the seam, 2.678332 by {6} and 42.645269 by {6} and by {7}, so 6 of the 14 teeth of the box change occupancy under a one-element perturbation (witness: lab/py/memory-zeta verb dial, occupied teeth emptied by a singleton 2, both emptied teeth printing an empty seam list).
2026-09-14 Verified The dial meets the cross-width control exactly where one exists: S_7 + {3} is S_55, and the dial's grid at code 7 with the added element 3 reads the second comb's tooth at Im s = 4.532360 occupied and the tooth at 13.597080 empty, which is the 1 of 2 the width-3 four-state ladder prints for code 55 (witness: lab/py/memory-zeta verbs dial, bridge and census, bridge reading Z_55 - Z_7 - det(I - 2^(-s) T) 3^(-s) at most 1.168e-13 over seven points and the code 55 census reading zeros in the disc 1 at 4.532360 and 0 at 13.597080).
2026-09-14 Verified The exact minimum of abs(R + P_F(t)) over all 4158861 subsets F of size at most 16 of the 22 integers of 2 .. 40 outside S_W is 1.095277075, 0.784350607, 1.295268436 and 1.662786204 at code 7's four abscissa-comb teeth, the greedy chain attains every one of them, and the disc at each exact minimiser keeps its zero off the seam, 1/1, 2/2, 1/1 and 1/1, the tooth at Im s = 18.129441 gaining a second zero rather than losing its first (witness: lab/py/memory-zeta verb dial --deep 16, rows exact minimiser over the 4158861 subsets of size at most 16).
2026-09-14 Conjecture Every width-k rule at dim = 1, base 2, with rho > 1 has M_W(x) = O(A_W(x)^(1/2 + eps)) for every eps > 0: over all 89 phases every one of the 53 census rules has its sweep-wide maximum of max abs M_W/sqrt(A_W) inside [0.500000, 2.169240] and its same-phase factor against the full line inside [0.861136, 3.635552] at phase 30.00, the sweep-wide maximum being 6.375774 on code 190 at phase 12.75, with 16 of 53 peaking in the last quarter of the grid, against Mullner 2017, which gives M_W(x) = o(x) for an automatic set and no rate at all. A band at finite depth is not a rate and nothing here bounds the constant (witness: lab/rs/memory-meter, Mullner 2017, and beneath.md, The memory meter).
2026-09-14 Conjecture What selects an occupied tooth is the first-order quantity u_1 = -r/R, the residue of zeta_W at the tooth against the regular part of Z_W/det, and not the spectrum. Over the 50 teeth of the nine two-line classes at width 3, 34 teeth are occupied at radius 0.45, the 23 whose prediction abs(u_1) falls below 0.3, inside the radius 0.3 disc that builds R and so where the reading is self-consistent, are occupied 22 times, and the 15 with abs(u_1) at or above 0.45 are occupied 4 times; the one exception inside 0.3 is code 55's second-line tooth at Im s = 13.597080, abs(u_1) = 0.267301885 with the nearest zero at 0.497761908. Occupancy is a per-tooth Boolean and the law is a law on abs(u_1): the largest modulus miss abs(d - abs(u_1)) is 0.739013203 and the largest vector miss abs(z - t - u_1) is 1.287895060, both at code 63's second-line tooth at Im s = 13.597080, abs(u_1) = 0.520202113 against a nearest zero at 1.259215316 (witness: lab/py/memory-zeta, verb teeth, and beneath.md, The memory zeta).
2026-09-14 Conjecture The knob's strength at a tooth t is abs(n^(-t)) = n^(-Re t) and its direction the phase -Im(t) log n mod 2 pi, so on a line with Re t < 0 the strength grows with n and the largest candidate still reading empty rises with the tooth height, while on the abscissa comb, where Re t = log(rho)/log(base) > 0, the strength decays and the flippers are confined to a bounded range of n that the phase selects inside: code 7's second comb reads 11, 25, 28, 35 on the inner seam convention and 11, 24, 28, 35 on the outer, increasing under both, over a candidate range stopping at 40 (witness: lab/py/memory-zeta verb dial, per-tooth rows last candidate reading empty on both readings).
2026-09-14 Refuted The Euler wall of zeta.md stands over the memory dial and its construction does not. The conclusion transfers: S_W for code 7, the fibbinary integers, holds the coprime pair 5 and 9 whose product 45 = 101101 carries adjacent ones and leaves the set, so the indicator of S_W is not multiplicative and no Euler product over primes exists; 45 is the least such product over all coprime pairs of S_W below 2^16. The construction does not: zeta.md builds its witness from the repunits R_c and R_(c+1) of the least missing digit c, and a memory rule has no missing digit to take the least of (witness: mrlynum::automaton, beneath.md, The memory zeta).
2026-09-14 Refuted "A memory rule's second pole comb carries no zero comb": the supergolden rule, code 23 at width 3, has det(I - x T) = 1 - x - x^3, one comb on Re s = log_2 psi = 0.551463089746 and two interleaved on Re s = -0.275731544873 from the conjugate eigenvalue pair of modulus psi^(-1/2), and its census on -0.75 < Re s < 2, 0.02 < Im s < 43.1 reads 24 zeros in 70 cells, largest phase step 0.998514, of which at radius 0.45 the first comb holds 3 of 4 teeth and the second line holds 7 of 10, least distance 0.170257380 (witness: lab/py/memory-zeta, verb census, and beneath.md, The memory zeta).
2026-09-14 Refuted "Symmetric pole combs give a symmetric zero set": code 7's two combs sit symmetrically about Re s = 0 and code 23's about Re s = 0.137865772436, yet under reflection in that line no zero of either census has a partner other than itself within 0.05 in both coordinates, 0 of 20 and 0 of 24, there being no functional equation on either side; the exclusion bites once, code 7's zero -0.023033432741+33.122746617086i sitting 0.046066865482 from its own reflection and being the only self-match inside the tolerance on either census, code 23's nearest missing at 0.075316339787 (witness: lab/py/memory-zeta, verb census, and beneath.md, The memory zeta).
2026-09-14 Refuted No function of the spectrum selects an occupied comb, which is the falsification L7 named. Codes 55 and 63 at width 3 and code 7 at width 2 all carry det(I - x T) = 1 - x - x^2, so all three have the same two combs, Re s = log_2 phi at argument 0 and Re s = -log_2 phi at argument pi, and the same teeth; on the second line at radius 0.45 they read 1 of 2, 0 of 2 and 0 of 2 occupied, least tooth-to-zero distances 0.264392586, 1.259215316 and 0.702616482. Codes 54 and 62 share the whole spectrum, det(I - x T) = 1 - x^2 - x^3, and differ on both lines, 2 against 1 and 3 against 4. Two rules with one spectrum reading two occupancies kills every function of it, monotone, threshold or otherwise; the ratio abs(lambda_2)/rho is neither, code 223 at 0.430159709002 reading 0 of 4 while code 127 at the smaller 0.400890564601 reads 2 of 4 and code 62 at 0.655865618097 reads 4 of 4 against code 31 at 0.563624162161 reading 2 of 4 (witness: lab/py/memory-zeta, verb teeth, and beneath.md, The memory zeta).
2026-09-14 Refuted The pole data cannot select occupancy at all, the residue included, and one line proves it: zeta_55 - zeta_7 = 3^(-s) is entire, so codes 55 and 7 carry the same poles, the same orders and the same residues at every m >= 0, while their zero sets differ, code 55 having a zero at -0.442302243578+4.612546440182i where code 7 reads -0.097731686660-0.868473160333i and reading 1 of 2 against 0 of 2 on the second line. Read at m = 0 through the determinant the residues agree digit for digit, -0.259501222742937-0.592535006433179i at -log_2 phi + pi i/log 2 and 0.896350590641921+1.403072744223695i at -log_2 phi + 3 pi i/log 2 from both rules (witness: lab/py/memory-zeta, verb bridge, and beneath.md, The memory zeta).
2026-09-14 Refuted The first-order quantity u_1(F) = -r/(R + P_F(t)) selects occupancy under perturbation on the abscissa comb: at code 7's tooth Im s = 9.064720 the exact minimiser F = [3, 6, 11, 12, 19, 22, 23, 35, 38, 39] drives abs(R + P_F) to 1.095277075, below the emptying threshold abs(r)/rho = 1.374951382, so the law predicts abs(u_1) = 0.564905571 and an empty disc, and the disc reads 1/1 with no seam (witness: lab/py/memory-zeta verb dial --deep 16, row exact minimiser ... predicted abs(u1) 0.564905571 emptying threshold abs(r)/rho 1.374951382 zeros in the disc 1/1).
2026-09-14 Refuted The first-order quantity is a selector across the perturbed family on a subdominant line: on code 7's second comb the grid holds 110 cells of which 77 read occupied, the law calls 75 right and the constant occupied predictor 77, and on code 23's second line, 270 cells and 222 occupied, the law calls 218 against 222; the deficit only widens on the outer reading of the seam, 80 against 86 and 219 against 231 (witness: lab/py/memory-zeta verb dial, per-line rows first-order law agrees against the constant occupied predictor agrees on both readings).
2026-09-14 Refuted Only the smallest added integers flip a tooth of the abscissa comb: code 23's empty tooth at Im s = 9.064720 is occupied by {7}, {13} and {14} and by none of the smaller candidates 5, 6, 10, 11 and 12, so the flipping set is not an initial segment of the candidate list, and the two occupied teeth a singleton empties are emptied by {6} and {7} while the smaller candidate 5 occupies both (witness: lab/py/memory-zeta verb dial, code 23 tooth rows measured 001000110000000000000000000 occupied 3 of 27, measured 101111111111111111111111111 and measured 100111111111111111111111111).
2026-09-19 Proved G_(1,4) < G_(2,2) < B_4 as permutation groups of the 4-cube, flipping all four bits being the diagonal B_2 element that flips both axes and the width-4 window reversal being the (2,2) block swap composed with the diagonal axis swap, so one cube carries three nested groups and its class counts nest the other way, 16960 > 4660 > 402, the last A000616 at 4. Witness: lab/py/memory-census.

The moments of the digit transform

2026-09-06 Proved The even moments of the digit transform are additive energies counted by a carry DP: sum_{a mod base^level} |hat F_level(a/base^level)|^(2r) = base^level E_r(level), E_r the r-fold additive energy modulo base^level of the length-level strings, C-finite in level of order at most r(r+1)/2 with growth constant Lambda(2r) = base rho, rho the certified Perron root of the carry-pair transfer matrix; Lambda(4) = 18 at {0,1} base 3 (x - 6), 2(23 + sqrt 353) at {0,1,2} base 4, (275 + 5 sqrt 2369)/2 at {0,1,2,3} base 5, every value strictly inside [max(fill^4, base fill^2), base fill^3]; brute force at level <= 7, direct grid evaluation at level <= 6, the bounds and the recurrence asserted to level = 60. Witness: lab/rs/rho-decoupling (the riesz module, 19 tests), mobius.md THE METER AND ITS YARDSTICK.
2026-09-06 Proved The multiplicative energy E_x(level) = #{n_1 n_2 = n_3 n_4} of a digit-restricted column has exponent 2 alpha for every base and digit set: 2K^2 - K <= E_x(level) <= K^2 max_m r(m) with r(m) <= d(m), so E_x(level) = fill^(2L) x^(o(1)); the census reads 1, 15, 111, 655, 3179, 14211, ... to 58760487 at level = 1..12 for {0,1} base 3 with theta_x = 1.475642, 1.410978, 1.356938 at level = 4, 8, 12 falling toward 1.261860; the shift family (base^i u, base^j v, base^(i') u, base^(j') v), i + j = i' + j', i != i', counted in closed form in fill and level when 0 is a digit, is a floor on the excess over the two diagonals, 0.4418 of it at {0,1} base 3, level = 12 and 0.19 to 0.0003 at the other families. Witness: lab/rs/rho-decoupling (the menergy module, 27 tests), mobius.md THE METER AND ITS YARDSTICK.
2026-09-06 Refuted That a moment of the digit transform alone carries the Type II estimate: Holder with the 2r-th moment and Parseval on the bilinear side gives x^(theta_p/p + 1/2 - 1/p) >= x^(alpha + 1/4) for every even p >= 4 and every digit set, above the trivial x^alpha, so the route needs the bilinear sum on the minor arcs below its own root mean square, which random-sign coefficients defeat on the census (minor arcs carrying 0.79 to 0.86 of the l^2 mass, the supremum 2.8 to 3.2 times x^(1/2)). Witness: lab/rs/rho-decoupling (the arcs lines), mobius.md THE METER AND ITS YARDSTICK.
2026-09-06 Refuted The sparse large-sieve shape (fill^level + x^beta) x^(o(1)) for the digit set at the points r/base^j: the exact constant is fill^(level-j) base^j = x^(alpha + beta(1 - alpha)), above both x^alpha and x^beta for 0 < beta < 1 (the Gram eigenvalue at base = 3, {0,1}, level = 2, j = 1 is exactly 6). Witness: lab/rs/rho-decoupling.
2026-09-07 Refuted That a Type II estimate on a digit set is a statement about coefficients whose sums over residue classes mod base^j cancel for base^j up to x^(2 eta/alpha): the Type II coefficients are hypothesised 1-bounded and nothing more, the polytope being a support constraint that supplies a divisor in [X^(9/25), X^(17/40)], and the Cauchy-Schwarz in m spends even that bound, the triangle inequality dropping the coefficient product to 1; residue sums of the coefficient side occur only on the major arcs at base <= (log X)^C. Witness: Maynard 2019 Proposition 7.2, Lemma 13.1 and the reduction (13.2), both read at source and quoted verbatim, with an adversarial pass confirming the wording and the pagination.

The node stack

2026-09-09 Proved The corner stack of the odd parity carpet is the Farey field, odd-restricted: with the edge set of a layer the cell sides separating ink from paper and the corner set the vertices of inked cells, layer n's edges are the interior grid lines restricted to the odd rows and columns and its corners the full interior vertex grid, so the corner stack lights exactly the pairs (a/b, c/d) with b and d odd and lcm(b,d) <= N, at brightness the number of odd multiples of lcm(b,d) up to N, and the edge stack is the line stack in one coordinate times the 1D parity stack in the other; at N = 15 literal stacking gives 536 lit points against 536 predicted, none missed and none invented, and 2352 segment tests with no breach. Witness: lab/py/node-stack parity_corner_predicted, parity_segment_check.
2026-09-09 Proved The base-3 carpet's edge stack is a denominator-restricted Farey family: a period boundary k/n is never an edge line, every other residue mod 3^level is (J_1 = {1, 2}, J_2 = {1, ..., 8}), so the lit lines are the reduced a/b with 3 | b and b/3^min(v_3(b), level) <= N, at brightness floor(N/(b/3^min(v_3(b), level))) - floor(N/b); at level = 1 the lit set is exactly the multiples of 3 in F_3N, 106 lines at N = 12 and 100 at N = 6, level = 2, with no breach on 14628 and 10800 segment midpoints; the level-1 corner stack at N = 12 lights 5029 corners at brightness floor(N/(m/gcd(m, 3))) against the plain Farey pair bound 157609. Witness: lab/py/node-stack carpet_lines_reach, check_line_brightness_carpet, carpet_corner_stack.
2026-09-09 Refuted The Sierpinski carpet's edge family is a numerator-restricted Farey sequence: the edge condition a (3^level n/b) mod 3^level in J_level is vacuous on the numerator because J_level is every nonzero residue, so the family is restricted by a divisibility of the denominator and must not be conflated with the cutoff-restricted Farey sequence of the Farey page; sibling designs removing another digit vector have proper residue sets and are the open door. Witness: lab/py/node-stack edge_residues.

The parity fill

2026-09-09 Proved The parity blend of the odd carpet stack has an exact rational fill: with s_n = 1 - 2 C_n the signed layer, the XOR of the layers is (1 - prod_n s_n)/2, and expanding the product over subsets S of the scales and splitting prod_{n in S} C_n(u, v) into its two coordinates gives fill(N) = (1 - sum_S (-2)^|S| m_S^2)/2 with m_S the measure of the set of u where every floor(nu), n in S, is odd, a cell count on the grid of lcm(S); the fill reads 1/9, 53/225, 3524/11025, 36284/99225, 19619/51975, 117419647/289864575, 109067744/289864575, 17006699344/45107387325, 6812188030619/19244451701475, 1114185811873/2749207385925 at N = 3, 5, ..., 21, matching a literal 2D XOR count on the lcm cell grid at every N <= 9 and a 4096 raster at every N within 2.42e-04; coprime layers being independent, the exact fill equals the independent-Bernoulli fill (1 - prod_n (1 - 2 p_n))/2, p_n = ((n-1)/(2n))^2, at N = 3 and 5, and the vanishing triple mass m_{3,5,7} = 0 against the independent 2/35 breaks the agreement at N = 7. Witness: lab/py/parity-fill fill_exact, fill_literal, fill_independent, mass_table.
2026-09-09 Verified The parity fill is not monotone in the layer count, falling from 0.405084502 at N = 13 to 0.376271381 at N = 15 and 0.353981924 at N = 19, because 587 of the 1024 subsets of the odd scales 3..21 carry joint ink measure zero, the smallest being {3, 5, 7}, whose floors are never all odd at one point; the exact fill sits below the independent approximation at every N from 7 to 21 with the deviation ratio growing to 29.337331; every odd layer is invariant under the quarter turn about the centre (chi_n(1 - u) = chi_n(u) at odd n), so the fixed-increment parity fill obeys fill(d) = fill(90 - d) (Proved), 45 of 46 mirror pairs bit-equal on the raster; the square raster climbs 0.320427, 0.375175, 0.424397, 0.444069 at L = 4, 8, 14, 28. Witness: lab/py/parity-fill mass_table, section_independent, section_sweep, fill_raster.
2026-09-09 Conjecture The parity fill of the odd carpet stack tends to 1/2: the raster climbs toward it while the exact deviation at L = 11 is still 9.47e-02 and non-monotone, no rate derived; the moire law controls pairs while the expansion needs every subset mass, and which subsets of odd scales have m_S = 0 is a covering question about the intervals [k/n, (k+1)/n) with k odd, open. Witness: lab/py/parity-fill fill_raster, mass_table.
2026-09-09 Refuted The rational-increment eyes stand out in the parity fill: at N = 55 and R = 512 the 89 nonzero whole-degree increments span 0.490165 to 0.511161 and the eyes at 18, 30 and 45 degrees read 0.500185, 0.494657 and 0.500282, inside the band at neither end, the one outlier being the unspun stack at 0.475647; coincident-class layers carry different scales and share a sublattice, not a picture, so nothing cancels; the independent approximation's O(2^-L) decay does not predict the exact deviation either, whose ratios run 0.680 to 1.304 and exceed 1 twice. Witness: lab/py/parity-fill section_sweep, section_independent.

The parity-carpet stack spectrum

2026-08-28 Proved The flat odd-scale parity-carpet stack's spectrum is the divisor field of the frequency gcd and nothing more: the sine coefficient of the L-layer stack G_L at odd (a,b) is (1/(pi^2 ab))[1 - sigma_1^S(a)/L - sigma_1^S(b)/L + sigma_2^S(gcd(a,b))/L] and vanishes at any even index, the interaction part carrying exactly sigma_2(gcd(a,b))/(ab); Parseval splits the variance blockwise into the two terms of the carpet law, re-proving the moire variance formula; every spectral statistic is an Estermann-Ramanujan zeta quotient, sum sigma_2(gcd)(ab)^(-w) = lambda(w)^2 lambda(2w-2) and sum sigma_2(gcd)^2 (ab)^(-w) = lambda(w)^2 lambda(2w-2)^2 lambda(2w-4)/lambda(4w-4) with lambda the odd zeta; a stack weighted n^(-s) renders sigma_(2-s) as its spectrum; divisor information only, no new L-function; coefficients checked cell-exactly at L = 14 to 47 digits and Parseval against the exact rational variance; the object is the flat odd-scale stack, not the all-scales Farey stack. Witness: moire-correlation-laws.

The radix dial

2026-09-13 Proved The fill law survives the radix dial: a radix design accepts every word over F, so it has (card F)^level words of length level at every ring, every base, every digit set and every twist, the twists appearing nowhere in the accept slot. The row restates the accept slot rather than proving anything beyond it. (witness: mrlynum::radix::Radix::fill against words in lab/rs/radix-designs verb named)
2026-09-13 Proved Every place map phi_d(x) = (u_d x + d) / base of a base of norm q >= 2 is a similarity of ratio q^(-1/2) because a unit has modulus one, so all card F maps contract equally and the similarity dimension is s = 2 log card F / log q whatever the twists; this is Hutchinson 1981 5.1(2) and 5.1(3) at a ring base. (witness: mrlynum::radix::Radix::dimension in lab/rs/radix-designs verb named)
2026-09-13 Proved Today's plane designs are the untwisted real-base row of the dial: at R = Z[i], base = m, digits the box {a + c i : 0 <= a, c < m} and every u_d = 1, the place map is x -> (x + d)/m on each coordinate, so the word d_1 ... d_level lands on the level-level cell of the plane design of the same code at base m, the code read in box row-major order bit r q + c and not in the canonical residue order. (witness: mrlynum::radix::tile against mrlymath::bang::factory::create in lab/rs/radix-designs verb today)
2026-09-13 Proved A twist keeps every count and moves only the place: the accept slot mentions no u_d so the word count stays (card F)^level, every ratio stays q^(-1/2) since a twist has modulus one so the dimension is untouched, and the twists enter the definition only through where an image sits. (witness: mrlynum::radix::Radix::words in lab/rs/radix-designs verb named)
2026-09-13 Proved An untwisted design on pairwise incongruent digits has no glue: reducing sum_i d_i base^(level-i) mod base recovers d_level because the digits are distinct residues, and induction on level gives the rest, so the distinct-point count equals (card F)^level at every level. The hypothesis is a hypothesis of the statement and not of the generator: Radix::new accepts any digit list, and from_code and tile are the two constructors that enforce it. (witness: mrlynum::radix::Radix::distinct in lab/rs/radix-designs verb named)
2026-09-13 Proved The canonical least-norm residue system of a real base m on Z[i] is the box {a + c i : 0 <= a, c < m} at m = 2 and at no larger m: the box holds m-1 of norm (m-1)^2 >= 4 while its own class holds -1 of norm 1. (witness: mrlynum::radix::Base::residues in lab/rs/radix-designs verb today)
2026-09-13 Proved The four maps z/3, e^(i pi/3) z/3 + 1/3, e^(-i pi/3) z/3 + 1/2 + i sqrt(3)/6, z/3 + 2/3 are phi_d coefficient for coefficient at base 3 on Z[omega] with digits 0, 1, 2+w, 2 and twists 1, 1+w, -w, 1: e^(i pi/3) = 1 + w, e^(-i pi/3) = -w and (2 + w)/3 = 1/2 + i sqrt(3)/6, so (u_d x + d)/3 at (d, u) = (0, 1), (1, 1+w), (2+w, -w), (2, 1) is that list in order. The 1.241e-16 printed at level 5 is a float evaluation of the same four maps and is a self-check of the crate's ring arithmetic, not an independent identification. (witness: lab/rs/radix-designs verbs koch and compare)
2026-09-13 Verified The Sierpinski gasket is the untwisted code 7 at base 2 on Z[omega]: against the three similarities of ratio 1/2 fixing the vertices of an equilateral triangle, written independently in f64 and placed at (1, 1), (3, 1), (2, 1 + sqrt 3), the 3^9 = 19683 words agree to 4.441e-16 after the translation and positive scaling that the statement leaves free, pinned by the two corresponding words 0^9 and 2^9 and then measured at every word, with turn residual 0. (witness: lab/rs/radix-designs verb compare)
2026-09-13 Verified That level-system reading is the code 7 at base 2+w on Z[omega] with twists 1, w, 1: its 3^8 words are the segment starts word for word at level 8 to 7.511e-16. The reading is the terdragon's own level-system and is carried by no source read here, so the name stays [Conjecture] until one is. (witness: lab/rs/radix-designs verb compare)
2026-09-13 Verified The five codes as printed: code 7 at base 2 on Z[omega] of dimension 1.584963, code 3 at 1+i on Z[i], code 7 at 2+w and code 127 at 3+w on Z[omega] each of dimension 2, all four untwisted with card F = 3, 2, 3, 7, and code 147 at base 3 on Z[omega] twisted, card F = 4, dimension 1.261860. (witness: lab/rs/radix-designs verb named)
2026-09-13 Verified Every plane code at q = 2 and q = 3 is the untwisted real-base radix design of the same code at level 2: 528 codes checked, 0 mismatches, through the pixel map bit r q + c of the code is the cell at row r and column c, the column the real part and the row the imaginary part, which is box row-major order and not canonical residue order. (witness: lab/rs/radix-designs verb today)
2026-09-13 Verified The distinct-point count equals the fill for code 7 at base 2 to level 11, code 3 at 1+i to 17, code 7 at 2+w to 11, code 127 at 3+w to 6 and the twisted code 147 at 3 to 8, so the Koch twist glues nothing inside that reach. (witness: lab/rs/radix-designs verb named)
2026-09-13 Verified The classes of digit CODES under the residue action are 12, 4, 12, 8, 6, 84, 28 over 16, 4, 32, 16, 8, 512, 128 codes at Z[i] bases 2, 1+i, 2+i and Z[omega] bases 2, 2+w, 3, 3+w; a Burnside count over the group and a direct orbit walk over all 2^q codes agree at every base. These are classes of codes and never designs up to similarity. (witness: lab/rs/radix-designs verb census)
2026-09-13 Verified The group of a base is the units acting on residues by multiplication, joined by conjugation exactly when conj(base) is an associate of base, the mirror failing at 2+i and at 3+w: the abstract group R^* semidirect <conj> has order 8, 8, 4 on Z[i] at 2, 1+i, 2+i and 12, 12, 12, 6 on Z[omega] at 2, 2+w, 3, 3+w, and it acts on the residues through an image of order 2, 1, 4, 6, 2, 12, 6. The action is not faithful: at 1+i every element is the identity permutation. Burnside over the abstract list stays correct, because the list is the image of one abstract group with each element once. (witness: mrlynum::radix::Base::group in lab/rs/radix-designs verb census)
2026-09-13 Verified The canonical residue systems printed: 0, 1, i, 1+i at Z[i] base 2; 0, 1 at 1+i; 0, 1, i, -1, -i at 2+i; 0, 1, 1+w, w at Z[omega] base 2; 0, 1, 1+w at 2+w; 0, 1, 1+w, w, -1, -1-w, -w at 3+w; and 0, 1, 1+w, w, -1, -1-w, -w, 2+w, 1+2w at 3. (witness: mrlynum::radix::Base::residues in lab/rs/radix-designs verb census)
2026-09-13 Verified The twist vectors at base 3 on Z[omega] number 7^9 = 40353607 summed over the 512 codes, not over the 84 classes: sum_k binom(9, k) 6^k = 7^9 counts one 6^(card F) for each code, and the two quotients are different quotients. No class count is claimed for twists, because no action of the group on twist vectors is defined here, and 7^9 counts only designs whose representative vector is canonical. (witness: lab/rs/radix-designs verb census)
2026-09-13 Verified The twisted glue witness holds as stated: Z[i], base 2, F = {0, 1}, twists 1, -1, level 2 has fill 4 and 3 distinct points, the words 01 and 11 both landing on 1/4, that is on 1 after scaling by base^2. (witness: mrlynum::radix::Radix::distinct in lab/rs/radix-designs verb named)
2026-09-13 Proved The distinct-point count of that twisted design is 2^(level-1) + 1 at every level: the scaled point of the word whose 1s sit at positions j_1 < ... < j_t is sum_(k=1..t) (-1)^(k-1) 2^(level - j_k), an alternating sum of strictly decreasing powers of two with top exponent at most level-1; such a sum is 0 or lies in [1, 2^(level-1)], since the alternating tail is smaller than the leading term; and every integer n of [1, 2^(level-1)] is reached by exactly one choice, the greedy one, taking 2^a for the least a with 2^a >= n and recursing on n - 2^a, whose modulus is below 2^(a-1). Printed and checked at every level to 16. (witness: lab/rs/radix-designs verb named)
2026-09-13 Verified The code census is not the design census: at base 3 on Z[omega] the 84 three-digit codes fall in 13 orbits of the residue action and in 9 similarity classes of the untwisted canonical digit sets, computed in exact arithmetic over Q(w), and the two partitions cross. Codes 131, digits 0, 1, 2+w, and 137, digits 0, w, 2+w, share an orbit and are not similar, though they are affinely conjugate; codes 7, digits 0, 1, 1+w, and 42, digits 1, w, -1-w, are similar and sit in different orbits. Among the 36 two-digit codes the census gives 7 orbits where similarity gives 1 class. (witness: lab/rs/radix-designs verb affine)
2026-09-13 Proved The Hausdorff dimension of a radix design is at most its similarity dimension 2 log(card F) / log q, with no hypothesis at all, by Hutchinson 1981 5.1(4)(i), which gives H^s(K) < infinity and dim K <= s for arbitrary contractions. (witness: mrlynum::radix::Radix::dimension in lab/rs/radix-designs verb named)
2026-09-13 Verified the Koch quintuple places 256 words on 256 distinct points at level 4, similarity dimension 1.261860, no digit canonical - mrlydemo::radix::radix_read, site/check.ts row radix koch design.
2026-09-13 Verified the twindragon quintuple places 1024 words on 1024 distinct points at level 10, similarity dimension 2.000000 - mrlydemo::radix::radix_read, site/check.ts row radix twindragon tiles.
2026-09-13 Verified the carpet at base 3 on Z[i] with the BOX digits fills 64 words at level 2 and has similarity dimension 1.892789 - mrlydemo::radix::radix_read, site/check.ts row radix carpet fill.
2026-09-13 Verified that same carpet codes 479 over the canonical classes where it codes 495 in box row-major order, so the two readings of one design differ - mrlydemo::radix::radix_read, site/check.ts row radix carpet fill.
2026-09-13 Verified the digits 0, 1 at base 2 on Z[i] twisted by 1, -1 glue 4 words onto 3 points at level 2, and untwisted they glue nothing - mrlydemo::radix::radix_read, site/check.ts row radix twisted glue, crates/mrlydemo/tests/radix.rs.
2026-09-13 Verified four digits reach level 8 and eight digits level 5 at the budget of 2^16 points - mrlydemo::radix::radix_cap, site/check.ts row radix koch dimension.
2026-09-13 Proved An untwisted radix design obeys the fill law: with every unit u_d = 1, the word d_1 ... d_level lands on base^(-level) sum_i d_i base^(level-i), and reducing that integer modulo base recovers d_level because the digits are distinct residues, so induction gives distinct points for distinct words and fill(level) = card F^level at every ring, base and digit set. (witness: beneath.md, The fill law, and where it stops)
2026-09-13 Proved No plane radix design carries a rotation of order 5 or 8: a unit of Z[i] solves a^2 + c^2 = 1 with four solutions and a unit of Z[omega] solves a^2 - ac + c^2 = 1 with six, so every available twist has order 1, 2, 3, 4 or 6; and a rotation preserving a rank-2 lattice is an integer matrix in a lattice basis with trace 2 cos theta in {-2,-1,0,1,2}, the classical crystallographic restriction. (witness: beneath.md, What a plane lattice will not carry)
2026-09-13 Proved A twist keeps every count the accept slot computes and every contraction ratio: the accept slot is the full shift on F and mentions no u_d, and every place map phi_d(x) = (u_d x + d)/base has ratio q^(-1/2) because a unit has modulus one, so the similarity dimension 2 log(card F) / log q is untouched and the twists move only where an image sits and which words collide. (witness: beneath.md, The twist law)
2026-09-13 Conjecture That design is the Koch curve: Hutchinson 1981 3.3(2) gives the Koch curve as the attractor of four similitudes each carrying a_1 a_5 to a_i a_(i+1) with positive determinant, and the four maps above are exactly those for the polyline 0, 1/3, 1/2 + i sqrt(3)/6, 2/3, 1, but that polyline is read from its Figure 3.2 and not from its text, so the name rests on a figure and not on a sentence. (witness: lab/rs/radix-designs verb compare)
2026-09-13 Conjecture The twindragon is the untwisted code 3 at base 1+i and the flowsnake is the untwisted code 127 at base 3+w: each is compared only against the maps (z + d)/base over the residues of its own base, which is its definition as a radix set, so the comparison is a self-check at 0 and 2.259e-16 and no independent witness for either name exists here. (witness: lab/rs/radix-designs verb compare)
2026-09-13 Conjecture That bound is an equality: equality needs the open set condition, Hutchinson 1981 5.3(1), which is a hypothesis per base and per twist and is checked at no base here. (witness: mrlynum::radix::Radix::dimension in lab/rs/radix-designs verb named)
2026-09-13 Refuted A code over residue classes does not name a radix design: the place moves with the chosen representative, d + base m shifting the image of phi_d by m, and the Koch digits 0, 1, 2+w, 2 are not the canonical representatives of their classes, since 2 and -1 share a class mod 3 on Z[omega] and the canonical system holds -1. A design is a quintuple, ring, base, code, representative vector, twist vector. (witness: mrlynum::radix::Radix::canonical in lab/rs/radix-designs verb koch)
2026-09-13 Refuted The terdragon is the untwisted code 7 at base 2+w: read the terdragon's own level-system F -> F + F - F at 120 degrees as a turtle, three segments to a level, normalise by the endpoint, and the untwisted design misses the 3^8 = 6561 segment starts by 1.060 at level 8. (witness: lab/rs/radix-designs verb compare)
2026-09-13 Refuted The fill law is not inherited by a twisted radix design: at R = Z[i], base = 2, canonical residues 0, 1, i, 1+i, digit set F = {0,1}, twists u_0 = 1 and u_1 = -1, the words 01 and 11 both land on 1/4, so two words of length two name one point and the cell count is 3 where card F^level is 4. A twisted design owes its fill law a proof of its own. (witness: beneath.md, The fill law, and where it stops)
2026-09-14 Proved The conjugacy group of the untwisted canonical base family is the centraliser of 1/base extended by translations. Conjugating phi_d(x) = (x + d)/base by an invertible real affine h(x) = H x + s gives (y + H d + s(base - 1))/base, again an untwisted place map exactly when H commutes with multiplication by 1/base, and s(base - 1) sweeps the plane since N(base) >= 2 forces base != 1. At a non-real base that centraliser is C, so the group is the similarity group; at a real base 1/base is the scalar (1/base) I and the group is all of GL_2(R) semidirect R^2. This bites at 2 on Z[i] and at 2 and 3 on Z[omega]. (witness: lab/rs/radix-designs verb affine)
2026-09-14 Proved The mirror x -> v conj(x) + t preserves the untwisted base family exactly when conj(base) = base, and being an associate is not enough: a direct conjugacy keeps the derivative 1/base and a mirror one sends it to 1/conj(base), so the mirrored object is a place map at base conj(base). The code census admits conjugation at 1+i and 2+w, where the conjugacy group does not. At a real base the mirror is one element of the full affine group and not the only new one, so it is load-bearing for the similarity quotient alone. (witness: lab/rs/radix-designs verb affine)
2026-09-14 Verified The similarity census of the untwisted canonical digit sets runs at every base of the code census and every digit count card F from 0 to q, in exact arithmetic over Q(i) and Q(w): the classes total 5, 3, 8, 6, 4, 117, 22 at Z[i] bases 2, 1+i, 2+i and Z[omega] bases 2, 2+w, 3, 3+w, against the code classes 12, 4, 12, 8, 6, 84, 28, and by card F = 0 to 9 at base 3 on Z[omega] they are 1, 1, 1, 9, 23, 30, 29, 16, 6, 1 against code classes 1, 3, 7, 13, 18, 18, 13, 7, 3, 1. (witness: lab/rs/radix-designs verb affine)
2026-09-14 Verified The conjugacy census of the same sets, the similarity group at the four non-real bases and GL_2(Q) semidirect Q^2 at the three real ones, totals 5, 3, 8, 5, 4, 88, 22 over the seven bases, 135 over the 41 cells against 165 similarity classes and 154 code classes. At base 3 on Z[omega] the affine classes by card F = 0 to 9 are 1, 1, 1, 2, 11, 23, 26, 16, 6, 1, so 512 codes give 84 code classes and 88 affine classes and the design count still exceeds the code count under either name. (witness: lab/rs/radix-designs verb affine)
2026-09-14 Verified No two of the three quotients are comparable. Over the 41 cells the similarity count is below the code count in 17, equal in 19 and above it in 5, the two partitions crossing in 6; the affine count is below in 19, equal in 18 and above in 4, crossing in 5. The lost crossing is Z[omega] base 2 at card F = 3, where two orbits meet two similarity classes with codes 7 and 11 split and codes 11 and 14 merged, while all four triples are non-degenerate and fall in one affine class. (witness: lab/rs/radix-designs verb affine)
2026-09-14 Verified At base 3 on Z[omega] and card F = 3 the 84 codes fall in 13 orbits, 9 similarity classes and 2 affine classes, the collinear triples against the rest. Codes 131, digits 0, 1, 2+w, and 137, digits 0, w, 2+w, share an orbit and are not similar, squared side lengths 1, 1, 3 against 1, 3, 4, yet are affinely conjugate by H = [[0, 2], [1, -1]] of determinant -2; codes 7, digits 0, 1, 1+w, and 131 are affinely conjugate by the unimodular H = [[1, 1], [0, 1]]; codes 7 and 42 are similar in different orbits. (witness: lab/rs/radix-designs verb affine)
2026-09-14 Verified The census is controlled by two explicit conjugacies asserted in the verb, H = [[0, 2], [1, -1]] carrying code 131 onto code 137 and H = [[1, 1], [0, 1]] carrying code 7 onto code 131, and by the assertion that the similarity classes refine the affine classes pair by pair in every cell. The per-size orbit counts summing to 12, 4, 12, 8, 6, 84, 28 constrains the code column alone, and the f64 rerun of the similarity normal form checks the exact arithmetic and not the group. (witness: lab/rs/radix-designs verb affine)

The ratio-set power saving

2026-08-31 Proved Every occupied direction of the gasket ratio set obeys max(z_1, z_2) > 2 min(z_1, z_2): the top base-3 digit of m(z_1 + z_2) lies in exactly one of the disjoint binaries m z_1, m z_2 and the other is a sum of distinct lower powers, hence at most (3^t - 1)/2, so the slope z_1/w never lies in [1/3, 2/3]; checked against every occupied weight to 8192, every pair to height 120 and every ray at level 12, and shown sharp and strict by the adversarial pass at minimum ratio 2.0000004 over 14.3 million pairs at level 15, extremal at (3^14, (3^14 - 1)/2). Witness: lab/py/ratio-set-saving.
2026-08-31 Proved The digit-congruence bound and the weight-layer reduction are one bound: z_1 (1 + r) = r w for r = z_1 z_2^(-1) mod 3^k, and r = -1 mod 3 would force 3 | w, so z_1 = r w (1 + r)^(-1) is determined and z_1 -> r is injective for 3^k > w, giving Z(w) <= 2 |R_k| with no failing weight to 8192; beta < 1 from this side would need sigma_k to fall geometrically, exactly what criticality forbids. Witness: lab/py/ratio-set-saving, lab/py/occupancy-decay.
2026-08-31 Proved Two relaxations of occupancy, both tight enough to keep the exponent: Z(w) <= 2 N_P(1/w) because slopes of denominator w are 1/w apart, and Z(w) <= Zinf(w) because the band is forward-invariant on every integer it contains, so a 3-adic witness suffices; the backward cone of 0 is {z_2 C - z_1 A : (A, C) a gasket pair} intersected with the band, rebuilt by an independent carry-pair dynamic programme with zero mismatches, and Zinf/Z is at most 1.5295 over the eleven weights tested. Witness: lab/py/ratio-set-saving.
2026-08-31 Proved R_k is indexed by the modulus 3^k and R_1 is empty under the hypothesis u > 0, so |R_k| = 1, 3, 9, 23, 63, 168, 457, 1245, 3423, ... starts at k = 2; both studies carry the identical definition and the same offset, which pins the offset the submission candidate needs. Witness: lab/py/occupancy-decay, lab/py/ratio-set-saving.
2026-08-31 Verified The weight layer read per weight rather than off a running maximum: log Z(w) / log w peaks at 0.7093 at w = 121 and Z(w) / w^(log 2 / log 3) at 1.5975 at w = 1093 over every w <= 8192, with all twenty-four octave argmaxes binary base 3 as the scan prints for itself; on the repunits (3^k - 1)/2 at k = 9, 11, 13 and the shifts 1 + 3^h at h = 7, 9, 11, 13 the exponent holds inside [0.6223, 0.6818] out to w = 1594324 while unstructured neighbours collapse to [0.2861, 0.4272], and the mean forward reach is 0.2249 to 0.2947 times sqrt(w) off the structured families. Witness: lab/py/ratio-set-saving.
2026-08-31 Refuted A uniform D_n(q) <= C 2^n / q on the binary base-3 multiples of q coprime to 3, the divisor input to the power saving - every binary m < 3^h makes m(1 + 3^h) binary below 3^(2h), so D_2h(1 + 3^h) >= 2^h while 4^h / q is only (2/3)^h of it, ratio (3/2)^h (1 + 3^(-h)) reading 2.0, 2.5, 3.5, 5.125, 7.625, 11.406, 17.094, 25.633 at h = 1..8, and at n = 20 the worst modulus below 500 is q = 244 = 1 + 3^5 at 1.8094; the breaking moduli are exactly the shift-ray weights. Witness: lab/py/ratio-set-saving.
2026-08-31 Refuted The short-witness route, bounding the count by 3^(level cap) - mean minimal witness length runs 3.875 to 27.287 and max 6 to 204 over height 32..16384, mean lev / log_3 height rises 1.553 to 3.305, and the share of occupied directions with lev <= 1.8073 log_3 x falls 0.875 to 0.3102, so every cap below 2 log_3 x loses a majority of the count. Witness: lab/py/ratio-set-saving.
2026-08-31 Refuted Two write-up claims of the first pass, caught by the adversarial read and corrected in place - the log Z_max / log W band was printed as [0.5000, 0.6404] when the script's own W = 128 row reads 0.70099, true band [0.5000, 0.7010] and margin 0.106 not 0.167; and ten Z_max values were listed against nine arguments, the duplicate Z_max = 30 at both W = 128 and W = 256 having been dropped, shifting every later argument onto the wrong weight. Both are transcription, not mathematics. Witness: lab/py/ratio-set-saving.
2026-08-31 Refuted beta >= log 2 / log 3 as a proved lower end of the weight-layer corridor - the binary-weight floor Z(w) >= #{coprime submasks} is proved, but at w = (3^k-1)/2 the coprime cut leaves 2, 6, 8, 30, 24, 126, 112 against w^(log 2 / log 3) = 2.4, 5.0, 10.3, 20.6, 41.3, 82.6, 165.3 for k = 2..8, beating the exponent at odd k and losing at even k, and no family supplies infinitely many good weights; the lower end is Conjecture. Witness: lab/py/ratio-set-saving.
2026-08-31 Refuted The metric route to the weight-layer saving: the sandwich Z(w) <= 2 N_P(1/w) is proved, but the cover of the slope set measures too large, n N_P(3^-n) / 3^n rising 2.4132 -> 2.4785, log_3 N_P / n rising 0.8783 -> 0.8997 and the step exponent rising 0.9333 -> 0.9504 over n = 12..18, every reading monotone and every one above the 0.8073 needed, so the covering route caps at O(w / log w) exactly like the congruence seed; a missing-digit rational-counting import belongs at the 3-adic ratio set R_inf and not at the slope variable. Witness: lab/py/ratio-set-saving.
2026-08-31 Refuted Three printed statements of the second pass, caught by the adversarial read and corrected in place - sigma_k < 1/9 was dated to k = 13 when it is 1/9 exactly at k = 2, 3, 4 and first below at k = 5, the bound already beating the trivial count at w = 13 (6 against 8.0) and w = 121 (46 against 73.3); the slope cover was computed on one swap half only, 51624 against the saturated 106994 at n = 12; and the backward moves were called one per residue class when 3k and 3k - z_1 are both 0 mod 3 and the class -z_2 mod 3 has no preimage. A fourth broke on the fix: the cover at n = 19 with three extra digits overflows int64 and printed a false 1.9533, so the generator now refuses past 1.5 * 3^(2n + extra) >= 2^63. Witness: lab/py/ratio-set-saving.
2026-09-06 Refuted The digit-congruence containment as printed: z_1 z_2^(-1) mod 3^k in R_k fails whenever 3^k | z_1, witness the occupied ray (9,1) at k = 2, where R_2 = {3} and the residue is 0; the true image is R_k union {0}, the counting bound's tail terms doubling to pay for the adjoined class, and the mod-3 dichotomy is the case k = 2 and not k = 1, R_1 being empty under u > 0. Witness: coprime.md, lab/py/occupancy-decay, lab/py/ratio-set-saving.
2026-09-06 Refuted The adversarial pass on the weight-layer run: an independent carry-pair dynamic programme reproduced every table to the last digit, the top-digit gap was checked over 14.3 million pairs at level 15 and found sharp and strict at minimum ratio 2.0000004 with extremal witness (3^14, (3^14 - 1)/2), Z(w) <= 2 |R_k| was checked at every weight to 8192 with no failure, the backward cone was rebuilt with zero mismatches, and three printed statements plus the |R_k| offset were broken and fixed in place. Witness: lab/py/ratio-set-saving.
2026-09-06 Refuted The adversarial pass on the run itself: the band was rederived on paper and sharpened to the halved interval -z_2/2 < j < z_1/2, an independent all-coprime-pairs automaton reproduced every A, Zsum and Z_max row to height 1024 and the full sweep, witnesses were reconstructed digit by digit for all 716 occupied directions to height 300 with zero failures, and Chow-Varju-Yu Theorem 1.2 and Kenyon were both verified accurate at source. Witness: lab/py/ratio-set-saving.
2026-09-07 Proved The block rate of the critical band automaton is bracketed by the parity of the depth b. Every column sum of every block B(b, j) is Sum_(c = a mod 3) binom(b, c) = (2^b + 2 cos(pi (b - 2a)/3))/3, so its deviation from 2^b/3 takes only two values, -1/3 and 2/3 at even b and -2/3 and 1/3 at odd b; hence lam_b lies in 2^b/3 + [-1/3, 2/3] at even b and in 2^b/3 + [-2/3, 1/3] at odd b, and the two-sided abs(3 lam_b/2^b - 1) <= 2^(1-b) holds at every b. The parity refines which edge is which and not the rate, and the computed excess 3 lam_b - 2^b is positive at every depth reached, so the upper edge is the live one. Witness: lab/py/band-return-times ladder.
2026-09-07 Proved At the horizon n = bk, k blocks of depth b, the column transfer is one matrix per residue fixed in k and L(k, bk) = Sum_j w_j B(b, j)^(k - r0(j)) h_j, the head length r0(j) free of k but not equal to 2: it is 1 at every sector below b = 5, at most 2 at b = 5..10 and at most 3 at b = 11..14. So L(k, bk) obeys a constant-coefficient linear recurrence in k and the block rate lam_b is an algebraic integer. Witness: lab/py/band-return-times ladder, with an independent residue DP reproducing L(k,4k) and L(k,5k) to k = 12 and factoring both characteristic polynomials in exact arithmetic.
2026-09-07 Verified The block rates are exact algebraic integers: lam_4 = 6 from (x-1)(x-3)(x-5)(x-6), lam_5 = 3(5 + sqrt 5)/2 from (x-1)(x^2 - 15x + 45), lam_6 = 13 + sqrt 79, lam_8 = (99 + 9 sqrt 65)/2, every lam_b to b = 14 having an exact minimal polynomial that divides the characteristic polynomial with zero remainder, the degree-four-and-up ones at b = 9, 11, 13, 14 irreducible over the rationals by mod-p distinct-degree factorisation; and the block rate is the largest block spectral radius itself, max_j rho(B(b,j)) agreeing with the certified interval to a relative 1e-9 at every b. Witness: lab/py/band-return-times ladder.
2026-09-19 Proved The carry transfer on the slot profile s_r = ceil((n - r)/k) gives the primitive return count L(k, n) of the weight R_k exactly at every k and every n, past the rigid depth the block ladder stops at, reading L(k, 4k) = 185, 1002, 5573, 31506, 180125, 1038402 at k = 3..8. Witness: lab/py/band-return-times, verb returns.
2026-09-19 Verified The support of the return time of the weight R_k, the lengths at which some primitive return exists, is {k} union [k + 2, 8k] at every k = 2..12, one gap at k + 1 and no other inside that range, with nothing past n = 8k or k = 12 decided. Witness: lab/py/band-return-times, verb returns.
2026-09-19 Conjecture The FIRST return time is a far thinner object than the return count and takes 16, 16, 59, 80 distinct lengths at k = 11..14, a single unpinned reading of the first-return sweep that no README prints and no pinned test carries. Witness: lab/py/band-return-times, verb hist.
2026-09-19 Verified The minimal polynomial of the block rate lam_b is exact at every b <= 14, lam_7 the dominant root of x^3 - 63x^2 + 945x - 3402, lam_9 of x^4 - 255x^3 + 16065x^2 - 293787x + 1299078, lam_10 of x^3 - 392x^2 + 17469x - 96228, lam_11 of a quintic, lam_12 of x^3 - 1551x^2 + 257256x - 5629338, lam_13 of a sextic, lam_14 of x^4 - 6176x^3 + 3963141x^2 - 335533914x + 2583866142, certified by exact bisection to a width below 1e-9 at 10.854101966, 21.888194417, 42.760932540, 85.780159867, 170.715620440, 341.700429300, 682.692831036, 1365.640975936, 2730.680876219, 5461.594643683 over b = 5..14, with minimal recurrence order b at even b and (b+1)/2 at odd b, the even ladder staying in radicals through b = 14 and the odd leaving them at b = 11. Witness: lab/py/band-return-times, verb ladder.
2026-09-19 Conjecture The degree of the minimal polynomial of lam_b is ceil(b/4) at even b and (b-1)/2 at odd b >= 3, a pattern observed on the thirteen rungs b = 2..14, b = 1 printing degree 1 against the rule's 0, and licensed at no b >= 15. Witness: lab/py/band-return-times, verb ladder.
2026-09-19 Verified A block ratio reads the block rate at odd depth and at no even one: the second root of the recurrence is 0.959422 of lam_6, 0.991055 of lam_8 and 0.999909 of lam_14, so L(k+1, b(k+1)) / L(k, bk) carries at most two correct digits at k = 160 at every even b <= 14, while at odd b = 5..13 that root falls from 0.381967 to 0.333404 and the same ratio carries 66 to 76 correct digits there. Witness: lab/py/band-return-times, verb ladder.
2026-09-19 Proved The repunit sweep meets each direction (z, R_k - z) twice, at z and at R_k - z, so Phi_k, Z(R_k), U_k and V_k are counts of z values with the distinct directions half of each, every first-return count being even for that reason, and a sample size quoted without halving is doubled. Witness: lab/py/band-return-times, verbs hist and check.
2026-09-19 Verified The deep tail's survival has no law: the survival in distinct directions S(b) = (1/2) #{z : d(z) > bk} at k = 14 runs 81, 65, 58, 56, 55, 52, 48, 42, 39, 37, 35, 30, 27, 20, 18, 14 from b = 2 and reaches 1 at b = 42, the local exponent -log_2(S(2b)/S(b)) reads 0.481, 0.273, 1.415, 3.169 at b = 2, 4, 8, 16 with the sharpest resting on the two directions of S(32), and a maximum-likelihood geometric fits ratio 0.8958 with pooled chi2 = 29.0 on at most 16 degrees of freedom once the fit is carried from the doubled z counts to the directions, so on 81 directions over 41 depths the survival is neither geometric nor shown not to be. Witness: lab/py/band-return-times, verb hist.
2026-09-19 Conjecture The equidistribution model D(k, N) = Sum 2^(#supp K) / m over the primitive lifts K = m R_k of base-3 length at most N equals 2^k L_k + 4^k / (3^k + 1) at N = 2k, the second term the primitive lift R_(2k) of multiplier 3^k + 1 and cancelling in every deep part below, and puts the deep part D(k, N) - D(k, 2k) at the critical cutoff N = floor(sqrt(R_k)) inside [0.1476, 0.4429] * 2^k at k = 8 and inside [0.0373, 0.1122] * 2^k at k = 16, so on the model the deep tail is o(2^k); it is never a prediction of Z(R_k) - U_k, which it exceeds by the witness multiplicity, and 0, 0, 0, 0, 5, 32, 51, 64, 73 percent of that deep part at k = 8..16 is carried by the free 4/9 per-digit increment extrapolated past 40 blocks, both ends leaning low because the return excess rho is above 1 at every depth reached and the upper end holding only while rho < 3. Witness: lab/py/band-return-times, verb model.
2026-09-19 Proved The inequality N_K(m) <= #packings is strict from k = 5, where T = {1}, m = 7 and K = 847 carry the support {0, 2, 3, 4, 6} with four irreducibles, two decompositions of the whole and six distinct unions against seven packings, so bounding Sum_T #packings_T suffices for the lift half and is strictly the harder target. Witness: lab/py/band-return-times, verbs lift and check.
2026-09-19 Proved A column transfer for the lift count with a state set free of k is a linear representation of that count as a series over the column word, so its state count is at least the series' Hankel rank, finite Hankel rank over a free monoid being exactly a linear representation with that rank as the minimal dimension. Witness: Schutzenberger 1961 in REFS.md.
2026-09-19 Verified That Hankel rank reaches 253 at word length 7, so the floor bites at k <= 15 where 2^7.5 = 181 is below it, and the floor is neither an impossibility nor a second check read twice: the reversed reading is the transpose of the same matrix at equal side lengths, and a machine whose state set may grow with k always exists, the residue automaton on m states computing N_K(m) at cost 3^k per T. Witness: lab/py/band-return-times, verb lift.
2026-09-19 Verified The first-return sweep reaches no k past 15 and the lift-family generator stops at k = 13, where 3^(3k) passes 2^63, so the deep tail stands on five points. Witness: lab/py/band-return-times, verb hist, and lab/py/ratio-set-saving, verb tail.

The registry's integers

2026-08-31 Proved The registry's written set is finite at every ceiling: a row renders at most 48 terms, so the 18066 rows write at most 48 * 18066 = 867168 integers however far the ceiling is pushed, and the miss density tends to 1. The adversarial read kills the way the bound was first used - 11133 was read as 0.01284 of that cap, a comparison that is vacuous inside the census window because 867168 exceeds the ceiling 100000; the honest saturation is 11133/100000 = 0.11133, and the finiteness is the only claim of the lane untouched by a change of cap. Witness: lab/rs/integer-census.
2026-08-31 Verified The census of 1..=100000 over the whole registry: 18066 rows, 7692 closed, 5044 convolved, 2665 side grid and 2665 level grid, each tier matched against an independent derivation from SPACES, ledger::designs and Measure::applies, none unread; stops 5529 ceiling, 6802 cap, 5735 budget, 390 silent; never/once/multiple 41/31/928 at 1000, 3589/765/5646 at 10000, 88867/2897/8236 at 100000, shares written 0.9590, 0.6411, 0.1113; miss density by decade 0, 0, 0.045556, 0.394222, 0.947533; 347308 (row, integer) incidences against 360703 (row, index, integer), so 13395 double counts are refused, and 29144 terms at or below zero are excluded and reported. Two independent refolds of rows.csv by readers sharing no code with the sweep reproduce every one of these numbers with zero mismatches, and add four checks the study did not run - no duplicate key, max |written| = 48 never exceeded, every in-range head term present in written, every ceiling row's head strictly increasing. Witness: lab/rs/integer-census.
2026-08-31 Verified The miss set's arithmetic: 269, a prime, is the first missed integer and 1..268 the longest written run; the longest missed run is 447 wide on 95265..95711 with both neighbours written; 100000 is written by 103 rows; the written share on 10000..100000 by greatest prime factor falls 0.5798, 0.1406, 0.0506, 0.0313, 0.0117 over the bands 1..10, 10..100, 100..1000, 1000..10000, 10000..100000; the tail written count by residue mod 12 runs 1175, 440, 145, 194, 715, 176, 420, 224, 531, 358, 229, 116 for a ratio 10.13, and mod 6 1595, 664, 676, 552, 944, 292 for 5.46; primes 750/9592 with only 158 of the 8363 above 10000; cubes 46/46, fourth powers 17/17, fifth 10/10, sixth 6/6, squares 176/316 with the largest written 97969 = 313^2, carried by the single row sequence_dim=4_code=28662_measure=voids_axis=side on 4k^4 - 8k^3 + 8k^2 - 4k + 1. Witness: lab/rs/integer-census.
2026-08-31 Verified The first missed square is a cap artifact and not arithmetic: row multiplicity at 96^2, 97^2, 98^2, 99^2, 100^2 is 321, 19, 480, 0, 123, and deepening the rendered window to 96 terms writes at least 228 of the 316 squares and moves the first missed square from 9801 to 38809 = 197^2. The adversarial read kills the mechanism first offered for it - that 9801 is odd and so outside the family (2k+2)^2 - because 97969 = 313^2 is odd and written, and because that family was selected by grepping a head prefix out of the study's own rows; what survives is the frontier, printed by the generator over 96..100 rather than read off a chosen family. Witness: lab/rs/integer-census.
2026-08-31 Verified The rendered window is measured rather than assumed harmless: the census at 8, 32 and 48 rendered terms writes 5263, 8749 and 11133 integers, so 5870 of the written set arrive only past term 8 and 2384 only past term 32, and the 8-term miss set opens 269, 281, 302, 311 where the 48-term one opens 269, 362, 422, 443. Rebuilding a row's written column from its 8-term head and the pinned stop rule alone, by Newton forward extension, passes 1306 of 1306 ceiling-stopped rows and 1325 of 1333 cap-stopped rows, the 8 failures being exactly the degree-6 detections an 8-term head cannot certify; the 969 budget-stopped rows carry no rendered length in their head and are declared untestable. Extending only the 1325 rebuilt cap rows to 96 terms gives a strict lower bound on a deeper census: at least 11898 written, first miss moved from 269 to 362, longest written run at least 361. The adversarial read kills the first statement of the rebuild check, which reported a pass-set size as a population size and omitted the budget stop kind entirely; the generator now prints the population by stop kind and the failing degrees. Witness: lab/rs/integer-census.
2026-08-31 Verified The champions are small perfect powers: 16 at 2858 rows, 9 at 2811, 4 at 2559, 12 at 2303, 36 at 2270, 64 at 2176, 3 at 1951, 6 at 1883, 8 at 1790, 33 at 1777, all twenty of the top twenty below 65 and carrying 39007 of the 347308 incidences, a share 0.1123; the 366 perfect powers of the window carry 58906 incidences, a share 0.1696 against a density 0.003660, 46.34 times their weight. The adversarial read kills the normalisation the enrichment was first printed under - a straddle-coverage ratio whose value at the integer 1 is definitional, since a written span containing 1 must start at 1, and which supplies 27.6% of the squares mean - so the study prints unconditional means on 1..1000 instead: 193.42 over all integers, 995.26 over the squares, 920.58 over the perfect powers. Witness: lab/rs/integer-census.
2026-08-31 Verified A champion is a property of a measure column, not of a design: euler.side writes 1 in 695 of its 859 rows, peak.side writes 12 in 809 of its 1261, heights.side writes both 9 and 33 in 765 of its 1261, and the eight integers below 100 that heights.side writes most often are 9, 17, 25, 33, 41, 49, 57, 65, every one 1 mod 8, which is what puts 33 = 3 * 11 tenth in a census otherwise made of powers. The adversarial read kills the mechanism first offered - that heights.side rows are the progressions 1 + s(k-1) - since only 840 of the 1261 rows have arithmetic heads and only 5 start at 1, witness sequence_dim=2_code=6_measure=heights_axis=side reading 2, 4, 6, 8; the leader set and its 1 mod 8 law stand as printed. Witness: lab/rs/integer-census.
2026-08-31 Verified The tiers split the census cleanly: of the 11133 written integers the closed tier covers 7628 with 3983 exclusive, the side grid 6203 with 2603, the level grid 1826 with 541, the convolved tier 792 with 130; above 30000 there are 2174 written integers and the closed tier covers 1853 of them. Witness: lab/rs/integer-census.
2026-08-31 Verified Two OEIS collisions, both explained and neither an identification: the ascending champion set opens 2, 3, 4, 6, 7, 8, 9, 12, 14, 15, 16, 18, a window of A100290 and of A336231 and of no other record in a dump of 398817, all three parting at the thirteenth term with 21, 19 and the census's 20; and the written-per-decade run 9, 90, 859 sits inside A209631 alone, which continues 6689 where the census gives 5452. Both searches index every window of the census sequence and walk every record, so they are exhaustive on both sides rather than sampled at offsets. Witness: lab/rs/integer-census, A100290, A336231, A209631.
2026-08-31 Verified No recognizable family is systematically missed: 173 records of the dump hold at least ten distinct integers of 1..=100000 and lie wholly inside the miss set, the longest being A361796 at 41 terms, which at a miss density of 0.88867 has probability about 10^-2.1 and is ordinary across 398817 records. Witness: lab/rs/integer-census, A361796.
2026-08-31 Conjecture That a missed integer is written by no row at any depth. The miss set is a statement about the rendered window: 6802 rows are cut by the 48-term cap, the 96-term lower bound already moves at least 765 misses across, 269 among them, and the cost of a true deeper census is cubic in the cap on the dimension-2 side grid. Witness: lab/rs/integer-census.
2026-08-31 Conjecture That the miss set is new to the OEIS. It is clean only in its dense head: no record of the dump carries a 4-term window of the miss set at offsets 0..416, the first hit being offset 417 in A049537, and above that the miss set hits near-interval records - 852 hits at k = 4, 130 at k = 10, 37 at k = 15 and 15 at k = 20, the 20-term witnesses A112820 and A118471. The absence rests on a dump, which is a snapshot, so it stays a conjecture under the standing caveat. Witness: lab/rs/integer-census, A049537, A112820, A118471.
2026-08-31 Conjecture That the write-once set is absent from the OEIS. The 2897 integers written by exactly one row have no hit at any offset of any record at k = 4, 10, 15, 20, a cleaner absence than the miss set's because the once set is thin where the miss set is an interval complement; the same dump caveat applies. Witness: lab/rs/integer-census.
2026-08-31 Conjecture That the miss set has no arithmetic characterisation. No modulus to 64 separates written from missed, the written share has no common growth order, and the set is closed under nothing; the finiteness bound is the only theorem the lane offers. Witness: lab/rs/integer-census.
2026-08-31 Conjecture That no bounded union of named families reaches the written tail. Above 30000 the rows' written sets collapse to 953 distinct families of which 875 own a tail integer no other family writes, covering 2005 of the 2174 tail integers, so every cover needs at least 875 families; the families are de-duplicated by written set and not by generating rule, so 953 is itself a lower bound on the number of rules. Witness: lab/rs/integer-census.
2026-08-31 Conjecture That no reparametrisation of the multiplicity function a(n) = rows writing n is submittable. It is absent from the dump at every offset tested, but it is a reading of the registry's own shape - the tier mix, the cap and the ceiling - rather than a function of n, so its terms move with the instrument. Witness: lab/rs/integer-census.
2026-08-31 Refuted The miss set is a union of residue classes - all 2079 classes mod 2..64 hold a written integer on 10000..100000, exhaustively, so no modulus in that range separates written from missed. Witness: lab/rs/integer-census.
2026-08-31 Refuted The champions are the highly composite integers - on 1..1000 the mean row count is 193.42 over all integers, 995.26 over the squares and 920.58 over the perfect powers, but only 170.60 over the 413 integers with at least eight divisors, below the overall mean; being a small perfect power is what a champion is, and being divisor-rich reads slightly against it. Witness: lab/rs/integer-census.
2026-08-31 Refuted Multiplicity is driven by each row's first rendered term - dropping every row's first term removes 17036 of the 347308 incidences, 4.9%, and changes nothing: the written set stays 11133, the never counts stay 41, 3589, 88867 in all three windows, and the leaders stay 36 at 2212, 64 at 2112, 16 at 2000 and 9 at 1999, the same integers in a different order. Witness: lab/rs/integer-census.
2026-08-31 Refuted The multiplicity spectrum is geometric or a power law - with S(m) the count of integers written by at least m rows, S(1) = 11133 and S(2) = 8236 give a ratio 0.7398 predicting S(64) = 6.312e-5 against the observed 977, wrong by seven orders; the spectrum takes 410 distinct values to a maximum of 2858 and its local log-log slope is convex, so no single exponent fits. The adversarial read also kills the fit the power law was rejected by: the exponent first printed was a two-point secant with its amplitude pinned at S(1), not a fit, so the misfit figure it carried is not the minimax one - the rejection stands, the number behind it does not. Witness: lab/rs/integer-census.
2026-08-31 Refuted The written tail is the union of a few dominant families - above 30000 the rows' written sets collapse to 953 distinct families of which 875 own a tail integer no other family writes, so every cover of the tail needs at least 875 of them. Witness: lab/rs/integer-census.
2026-08-31 Refuted No OEIS record contains any contiguous window of the miss set - the search behind that universal sampled 40 offsets spread over 88867 terms, and an exhaustive walk of every window against every record finds hits at every length tested: 852 at k = 4, 130 at k = 10, 37 at k = 15 and 15 at k = 20, with A112820 and A118471 each carrying 20 consecutive misses and A043635 lying wholly inside the miss set. What survives is the restricted statement, absence at k >= 4 for offsets 0..416. Witness: lab/rs/integer-census, A112820, A118471, A043635.
2026-08-31 Refuted Every square below the ceiling is written - 140 of the 316 are missed, the first being 9801 = 99^2. Witness: lab/rs/integer-census.

The repunit layer

2026-09-06 Proved The binary-weight floor on the repunits w = R_k = (3^k - 1)/2 in closed form: Phi_k = sum_{q | rad R_k} mu(q) q^(-1) sum_{t mod q} P_{q,t}^(k/ord_q 3) with P_{q,t} = prod_{r < d} (1 + e(t 3^r/q)), C-finite in k prime by prime, N_k(2) = 2^(k-1), N_k(p) = (2^k + p - 1)/p whenever 2 is a power of 3 mod p (p = 5, 7, 23), and Phi_k = 2^k - 2 whenever R_k is prime; values 2, 6, 8, 30, 24, 126, 112, 450, 460, 1958, 1344, 8190, 8064, 27360 at k = 2..15 by three methods. Witness: lab/py/ratio-set-saving (ratio.py repunit), coprime.md THE REPUNIT FLOOR EXACTLY.
2026-09-06 Proved The repunit excess is a lift family and a deep tail: a binary K is a multiple of R_k exactly when its column counts satisfy sum_r c_r 3^r = 0 mod R_k; below 3^(2k) the multiples are K_T = a_(T^c) + 3^k a_T with multiplier 1 + 2 a_T and R_(2k); each lift set Occ_T is occupied and stable in k, so Z(R_k) >= |union_T Occ_T|, and at prime R_k (k >= 15, first at k = 71) Z(R_k) - Phi_k >= 2^k/7 - 4 F(k+1) - 126. Witness: lab/py/ratio-set-saving (ratio.py repunit), coprime.md THE REPUNIT EXCESS.
2026-09-06 Verified The repunit excess Z(R_k) - Phi_k reads 0, 0, 0, 0, 0, 6, 6, 50, 70, 402, 290, 2198, 2376, 8830 at k = 2..15; the lift union equals Z(R_k) at k <= 10 and falls short by 18, 16, 108, 162, 624 at k = 11..15; minimal witnesses reach 436 digits with a column used 27 times at k = 13; the drift factorises exactly as Z(R_k)/R_k^(log 2/log 3) = 2^(log 2/log 3) (1 - 3^(-k))^(-log 2/log 3) delta_k (1 + X_k) with delta_k = Phi_k/2^k and X_k rising 0.0476 to 0.3227 over k = 7..15. Witness: lab/py/ratio-set-saving (ratio.py repunit).
2026-09-06 Conjecture The repunit drift is unbounded: X_k = (Z(R_k) - Phi_k)/Phi_k rises at every step from k = 8 and beats the random-lift limit sum_T 1/m_T = 1.41 at k = 13, so no pointwise Z(w) <= C w^(log 2/log 3) holds on the repunits and every exponent above log 2/log 3 survives; the blocking lemma is whether |union_T Occ_T| plus the deep tail is O(2^k). Witness: lab/py/ratio-set-saving (ratio.py repunit), coprime.md THE REPUNIT DRIFT.
2026-09-06 Refuted That 1.5975 (the maximum of Z(w)/w^(log 2/log 3) below 8192, at w = 1093) bounds the layer: the repunits read 1.7845 and 1.963681 at k = 11, 13, so any pointwise C w^(log 2/log 3) needs C >= 1.9636. Witness: lab/py/ratio-set-saving (ratio.py repunit).
2026-09-07 Verified The deepest first return of the critical band automaton grows below the critical sqrt(w) as a sign and not as an exclusion: log d_max on log w over 27 weights gives 0.4055, 95 percent [0.3293, 0.4818], but one deletion moves the slope to 0.4261 and the interval to 0.5034, covering 1/2, so the leave-one-out range [0.3829, 0.4261] is what stands; the median has no single exponent, 0.1802 on every weight against 0.2798 at Z >= 8 with 4 of 27 weights having Z <= 2; and a two-predictor fit puts 0.3385 on log w and 0.1313 on log Z, so controlling for sample size lowers the exponent and the drift below 1/2 is understated. Witness: lab/py/band-return-times critical.
2026-09-07 Proved The submasks of a binary K divisible by a divisor m of it are closed under complement in supp K, under disjoint union and under nested difference, so N_K(m) is even and every solution is a disjoint union of irreducible ones; the decomposition is not unique, so N_K(m) is the number of distinct unions of pairwise disjoint irreducibles and satisfies N_K(m) <= #packings <= 2^iota for the irreducible count iota, with N_K(m) = 2^iota if and only if the irreducibles are pairwise disjoint, and then they partition supp K. Depth-free. The antipodal and run families of the repunit lift are the equality case, which is why their counts are exact powers of two, and the converse fails, k = 7 with multiplier 19 having a power-of-two count and overlapping irreducibles. Witness: lab/py/band-return-times lift and check, the three closures asserted over every one of the 2^(k-1) sets T at every k = 2..9, and both witnesses pinned.
2026-09-07 Verified The depth-2 lift census of the repunit reads M_k = 2, 6, 14, 36, 68, 172, 306, 728, 1338, 2814, 5224, 11852, 20888, 43364, 84124, 172516, 327092 at k = 1..17 with M_k/2^k inside [1, 2.89356], and its Hankel matrix is 9 by 9 of full rank on all seventeen terms, so no linear recurrence of order at most 8; the irreducible supply Sum_T iota_T / 2^k sits inside [0.738281, 0.890625] at k = 2..12, the even readings falling from k = 6, while max_T iota_T grows 2, 3, 4, 5, 6, 10, 14, 24, 31, 50, 68, and the equality case holds for 1970 of the 2048 multipliers at k = 12 against 1986 whose count is a power of two. The sweep is exhaustive over every T inside [1, k-1]; the multipliers meet the residue classes 1 and 7 mod 9 and never 4, which is forced by a_T = Sum 3^i with i >= 1 and not a reading. Witness: lab/py/band-return-times lift.
2026-09-07 Verified A column transfer for the submask count of the lift half needs at least 253 states where the return half needs b/2: a machine reading the k columns with a state set free of k is a linear representation of the count as a series over the column word, so its state count is at least that series' Hankel rank, and the rank reads 3, 7, 14, 31, 62, 126, 253 at word length 1..7 on each side against the full 3, 7, 15, 31, 63, 127, 255, deficiency 0, 0, 1, 0, 1, 1, 2. The words reach length 14, so the floor holds at k <= 15 and is already worse than the 2^(k/2) meet in the middle; whether the rank is unbounded is observed and not proved, and a rank levelling off would be a poly-time machine, so the route is blocked and not closed. The irreducible count has the same full-rank Hankel to word length 5. Witness: lab/py/band-return-times lift.

The second moment of the rays

2026-08-28 Refuted The non-shift residual of the second moment converges as R(n)/phi^(2n) -> C ~ 3 at rate phi^2 - the normalised ratio reads 2.498, 3.045, 3.238, 3.182, 3.113, 2.983, 2.895 at n = 8, 10, 12, 13, 14, 15, 16, peaking at n = 12 and falling by 0.9724 per level, and R(n)/R(n-1) sits at 2.573, 2.561, 2.509, 2.541, a rate near 2.54, strictly below phi^2 = 2.618; what survives is R(n) = O(phi^(2n)) on n <= 16, lim R/phi^(2n) undecided, R = o(3^n) with room 3/2.54 rather than 3/2.618, and Conjecture Z untouched. Witness: lemma-b-pincer.
2026-08-31 Refuted The majorant Sum_z M_n(z)(M_n(z)-1) is a route to Conjecture W - it grows 2.907 a level at n = 13 against 2.573 for R itself, because it drops the coprimality of (s,t) and so counts each collinear pair once per common divisor; the exact identity R(n) = Sum_z P_n(z) survives and the golden ceiling M_n(z) <= F(n+1) - 1 survives, but the step from P_n(z) to M_n(z)^2 - M_n(z) does not. Witness: lab/py/gasket-witness-weights.

The slice ladder: rate and dead routes

2026-08-28 Conjecture The roots-of-unity circulant correction Q_dim = fill/3 + (2/3)(-1)^(dim-1)(dim-1) cos(2 pi dim / 3) is worse than bare fill/3: mean absolute error 11.05299392007777488084818 against 0.1511524922667271359757626 over dim = 2..50, a factor 73.1248, with |rho - Q_dim| / |rho - fill/3| reaching 1067922.7 at dim = 50; its sign matches (-1)^(dim+1) only when dim = 0 mod 3, 16 of 49 cases; the order-dim root-of-unity term is cancelled by finite-boundary effects, the empirical correction factor collapsing to -9.363982332e-7 at dim = 50, so any model of the excess must derive the boundary cancellation.
2026-08-28 Conjecture The three-block DFT decomposition of the carry matrix does not exist: the span of 1, omega^c, omega^(2c) is not invariant for any dim >= 5 (relative Frobenius residual 0.318 to 0.523), carry residues mod 3 are coupled for every dim >= 3 so M does not commute with diag(omega^c), and the three-root average (P(1) + P(omega) + P(omega^2))/3 misses the Perron root by -9.191 to +11.263 while the true excess is -0.0435 at dim = 20; the mod-3 block version has off-diagonal Frobenius mass of order one, ratio 0.609 to 4.111 over dim = 3..30 with slope -0.00162 +- 0.00773 per dim, p = 0.836, and a Schur correction at fill/3 positive for every dim, 0.343 of fill/3 at dim = 30; one exact row-sum identity survives.
2026-08-28 Conjecture The saddle-point route is closed: the transfer operator is coefficient decimation, (Mv)(c) = [t^(c+dim)] P_dim(t) V(t^3) on a finite carry window, not multiplication by a scalar symbol; geometric vectors z^c are not eigenvectors, the all-ones vector is the only reflection-even one and is not an eigenvector either; on the unit circle max|P_dim(e^(i theta))| = P_dim(1) = fill, whose cube root is exponentially smaller than rho_dim, while max|P_dim|/3 is exactly fill/3 and misses the whole effect; no non-tautological f_dim(theta) with rho_dim = max|f_dim| was found.
2026-08-28 Conjecture Induction on dim is closed from both ends: the same-size correction between M_dim and M_(dim+1) at odd dim has full rank at every dim = 3..19, determinants from -54 at dim = 3 to -441065669103434214513656226772598887664331096 at dim = 19, so the matrix determinant lemma has no low-rank update to consume; the threshold determinant sequence d_dim satisfies no recurrence surviving holdout - constant-coefficient orders 1 to 6 with degrees 0 to 5 on the full sequence and each parity subsequence, all 62 identifiable holonomic pairs with r s <= 40, normalisation by fill^n and by dim^beta for beta = -4..4, Berlekamp-Massey over three primes at maximal linear complexity (20 for 39 terms, 10 per parity), the one determined fit (order 4, degree 2, odd subsequence) failing at dim = 35, 37, 39; the 2-adic valuation of d_dim fits none of the tested elementary forms.
2026-08-28 Conjecture Cauchy interlacing is closed: over all 27 pairs 2 <= dim <= 28, M_even(dim) is not the upper-left block of M_even(dim+2) and none of the (n+1)^2 row-column deletions of the larger matrix is the smaller, so no bordering u, v, alpha exist; the eigenvalue chain holds for every even start and fails for every odd one, witness lambda_1(3) = 7.372281323269 against lambda_2(5) = 16.965208741322; the threshold count it was meant to prove is nevertheless exact on dim = 2..30 - no eigenvalue above fill/3 at even dim, exactly one at odd dim.
2026-08-28 Refuted The decay rate of the slice-dimension excess is 3/4, 4/3 or 8/3 - at 320 digits over dim = 2..100 the eigenvalue-scale one-step ratio extrapolates to 0.742874554813847413, residual 0.00712544518615 from 3/4, and r_inf = 1.34612251727283689, residual 0.0127891839395 from 4/3, both far outside the 4.5643e-8 parity split and the fit-order spread; on the dimension scale 2 r_inf = 2.6922450 against 8/3 = 2.666667; the coarse dim <= 50 reading 0.373, inverse 2.68, and the sentence "the per-dimension factor approaches 3/4 from above" conflate the two scales; the constant is identified as prod_{k>=2} cos(2 pi/3^k) = 0.7428747134, within 1e-8 at dim = 61. Witness: slice-recurrence-order.
2026-08-28 Refuted The excess has the clean shape slice dimension - (solid dimension - 1) = (-1)^(dim+1) C r^(-dim) + o(r^(-dim)) with a constant C - |delta_dim| r_inf^dim climbs from 52.4976468882 at dim = 60 to 88.3872395676 at dim = 100, a log-linear fit puts the prefactor at dim^1, and the form is |delta_dim| ~ A dim r_inf^(-dim) with A ~ 0.897520192686; the linear factor is the parity factor dim - 1. Witness: slice-recurrence-order.
2026-08-28 Refuted Conjecture S stated against the threshold base^(dim-1) - at base 3, sgn(rho_dim - 3^(dim-1)) matches the hypothesised sign in 25 of 49 cases over dim = 2..50, is wrong already at dim = 3 where the difference is -1.627718676730986, and exact real-root counting finds no eigenvalue above 3^(dim-1) at any dim = 2..20; at base 5, rho_dim < 5^(dim-1) at every tested dim, rho_3 - 25 = -1.5341439003; the only threshold that carries the statement is fill/3, the eigenvalue-scale form of d - 1. Witness: slice-recurrence-order.
2026-08-28 Refuted The even carry block has exploitable matrix structure - over dim = 2..20 the banded-plus-low-rank form does not exist (Toeplitz displacement rank equal to the full dimension from dim >= 5, tridiagonal remainder of rank n-1 at odd and n at even dim), total nonnegativity fails for every dim >= 4 with an exact negative minor per row, only dim = 2 is symmetric and only dim = 3, 4 are positively diagonally symmetrizable (weights (1,6)), and M_even - (fill/3) I is Metzler rather than a Z-matrix, so the M-matrix route is circular; the matrices have n distinct real roots at every tested dim, which is spectrally useless. Witness: slice-recurrence-order.
2026-08-28 Refuted A simple positive test vector certifies the Collatz-Wielandt bound - the all-ones vector has ratios mixed around fill/3 for every dim = 3..50 (dim = 2 excepted, that matrix being one by one), one-parameter cosine, alternating and centred-quadratic corrections succeed only at dim = 2, 3, 4, and the Gaussian exp(-3 i^2 / dim) and binomial-centre profiles only at dim = 2; the Perron vector certifies at every dim <= 50 (worst discrepancy 9.15e-46), is peaked at index 0 and monotone non-increasing rather than bell-shaped, and has no closed form. Witness: slice-sign-even-half.
2026-08-28 Refuted A cheap route proves the spectral separation rho_dim/|lambda_2| -> 1 - the common-amplitude Gaussian kernel predicts a limiting ratio 9 where the truth is (dim+2)/(dim-2) -> 1, discarding an order-dim parity modulation; the zero-shift 2x2 Schur complement has median relative error 0.625 over dim = 2..40 and worst 0.9998, deteriorating with dim; the Perron profile peaks at index 0 for all 39 tested dim, a boundary-centred half-Gaussian at median R^2 = 0.99999, not near dim/6; the second eigenvector has one sign change at every even dim and at dim = 3, 5 but several at every odd dim from 7 to 39; separation holds numerically to dim = 60, and a proof must be uniform in a margin of order 4/dim. Witness: slice-recurrence-order.

The spirograph loops

2026-09-10 Proved On a circle track a/b in lowest terms, with s = -1 inside and s = 1 outside and rho = a + s b, a pencil at real seat t wheel radii draws z(phi) = rho e^(i b phi) + t b e^(i s rho phi) on [0, 2 pi). Setting sigma and delta for the half sum and half difference, z(phi) = z(psi) reads rho sin(b delta) + t b sin(s rho delta) e^(i s a sigma) = 0, so sigma is a multiple of pi / a and rho sin(b delta) = e t b sin(rho delta) with e plus or minus one. The number of unordered parameter pairs that meet is a/2 times the number of such delta in the open interval (0, pi) over both signs. Witness: lab/rs/roulette-loops.
2026-09-10 Proved Every self crossing of a trochoid on a circle track lies on one of the a mirror lines through the centre: the reflection in the line of angle b sigma fixes it. For a real seat those lines are k pi / a; a seat at angle alpha turns the curve by -s b alpha / a and turns its lines with it. Read off mrlynum::spirograph::trace at 24001 samples, the worst distance from a crossing to its line is 4.33e-7 of the frame over 30 cells at seat angles 0 and 0.3, the floor being the f32 the trace returns. Witness: lab/rs/roulette-loops, mrlynum::spirograph::trace.
2026-09-10 Proved A trochoid on a circle track has a point of multiplicity above two at exactly one reach, rho / b: a multiple point needs every pairwise delta to be a multiple of pi / a, and such a delta solves the crossing equation only there, where the curve runs through the centre at the a parameters (2j+1) pi / a. So the distinct double point count equals the parameter pair count elsewhere and falls short by C(a, 2) - 1 at that reach. Witness: lab/rs/roulette-loops, mrlynum::spirograph::trace.
2026-09-10 Proved The parameter pair count of a trochoid on a circle track changes only where the crossing equation has a double root, and those are exactly the roots delta of a sin(m delta) = m sin(a delta), equivalently sinc(m delta) = sinc(a delta), with m = a + 2 s b. Each carries the reach abs(cos(b delta) / cos(rho delta)), read as abs(rho sin(b delta)) / abs(b sin(rho delta)) where both cosines vanish. A threshold is one such angle in [0, pi), not a reach; several angles can share a reach. Witness: lab/rs/roulette-loops.
2026-09-10 Proved The ends delta = 0 and delta = pi solve the crossing equation of a trochoid on a circle track at every reach, and a root is born at each end as the reach passes one, since the derivative there is rho b (1 - e t) and rho b ((-1)^b - e t (-1)^rho). The single tangency angle at delta = 0 stands for both births, which is why the jump at reach one is a and not a/2. Read off the equation, 3/1 inside has no root in (0, pi) at reach 0.98 and two at reach 1.02, at 0.1984 and 2.9432, count three. Witness: lab/rs/roulette-loops.
2026-09-10 Proved On a tangency reach a trochoid on a circle track touches itself: the two branches meet with equal tangents, a tacnode, so the meeting count there is the transversal count just below plus a/2 for each tangency angle at that reach. The step function is read on the open intervals between tangency reaches and never on one. At reach squared 27/2 the hypotrochoid 5/1 has two tangency angles and two simple roots, so ten meetings against five below and fifteen above, the branches closing to 8.88e-16 at radius 2.041241. Witness: lab/rs/roulette-loops.
2026-09-10 Proved Swapping the wheel frequency b and the rim frequency rho of a trochoid on a circle track fixes every tangency angle, because the unordered pair {a, abs(m)} is {rho + b, abs(rho - b)} either way, and inverts every tangency reach, because abs(cos(b delta) / cos(rho delta)) inverts. So the thresholds depend on the ordered pair and the falling a < 2b staircase is the reciprocal of the rising a > 2b one: 5/1 inside steps at 3.674234614175 and 5/4 inside at 0.272165527. Witness: lab/rs/roulette-loops.
2026-09-10 Proved The tangency equation of a trochoid on a circle track integrates: a sin(m delta) - m sin(a delta) is 2 a m times the integral of sin(b t) sin(rho t) from 0 to delta, up to sign. That integral's derivative vanishes on (0, pi) only at j pi / rho and k pi / b, and there the integral is exactly (-1)^(j+1) sin(b j pi / rho) rho / (rho^2 - b^2) and (-1)^k sin(rho k pi / b) b / (rho^2 - b^2), so the threshold count is a sign count over a merged Farey sequence and carries no numerics at all. Witness: lab/rs/roulette-loops.
2026-09-10 Proved Every tangency reach of a trochoid on a circle track is an algebraic number in closed form. Expanding sin(k delta) and cos(k delta) in u = sin^2(delta) by the integer recursion of Sakhnovich 2023, theorems 2.1 and 2.5, the tangency equation becomes u Q(u) = 0, times cos(delta) when a is even, for an explicit integer polynomial Q, and the reach squared is T(u) / B(u) for explicit integer polynomials T and B. The recursion is the cited source's; Q, T and B are this study's. Witness: lab/rs/roulette-loops.
2026-09-10 Proved Two pencils at complex seats p and q on one wheel of a circle track a/b, drawing distinct curves, meet at exactly (a/2) N unordered parameter pairs, where N counts the roots in [0, 2 pi) of rho^2 sin^2(b delta) = b^2 (P sin^2(rho delta) + E cos^2(rho delta) - s X sin(2 rho delta)) with P and E the squared halves of the sum and difference of the seats and X half the imaginary part of q times the conjugate of p. The equation is pi periodic in delta so N is even and the halving is exact. Witness: lab/rs/roulette-loops.
2026-09-10 Proved The pair law counts parameter pairs, and a point count needs more: no self crossing of either curve may lie on the other, and neither seat may sit at reach rho / b, where that curve alone loses C(a, 2) - 1 points into the centre. The extra condition is codimension one and is not implied by the curves being distinct.: 3/1 inside with one seat at reach sqrt 5 - 1 and one at the wheel's centre gives six parameter pairs and three points, all at radius rho, while two percent either side gives six points. Witness: lab/rs/roulette-loops, mrlynum::spirograph::trace.
2026-09-10 Proved A pencil at the wheel's centre of a circle track a/b draws the circle of radius rho, and the pair law collapses to abs(sin(b delta)) = b t / (2 rho) against a pencil at reach t, so those two curves meet 2 a b times below reach 2 rho / b and never above; 10/3 at 8/3 inside, read as 48 then 0 off the trace. Witness: lab/rs/roulette-loops, mrlynum::spirograph::trace.
2026-09-10 Verified For reach positive, not a tangency reach, not rho / b, and m nonzero, the self crossing count of a trochoid on a circle track a/b is a (b - 1) + a sign(m) t with t the number of tangency angles whose reach is strictly below, counted with multiplicity. It runs from a (b - 1) to a (rho - 1), and the number of angles in [0, pi), counting delta = 0, is abs(rho - b), the same integer as the smaller of abs(m) and a. Witness: lab/rs/roulette-loops.
2026-09-10 Verified The step function of a trochoid on a circle track holds on 179 coprime fractions a/b with a at most 24, 357 cases over the two sides with m nonzero, against the crossing equation's root count at the midpoint of every step, and on 25578 reads of mrlynum::spirograph::trace at 4001 and 12001 samples over reach 0.5 to 4 in 203 steps for b in one to six and a in b+1 to eleven coprime, both sides, with no disagreement and every jump bracket 0.0173 wide or less. Witness: lab/rs/roulette-loops, mrlynum::spirograph::trace.
2026-09-10 Verified The exact integer sign count of the integral of sin(b t) sin(rho t) over its critical values returns abs(rho - b) on all 29450 coprime frequency pairs b and rho up to 220, with no root finding anywhere in the computation. Witness: lab/rs/roulette-loops.
2026-09-10 Verified Over all 210 swaps of the wheel and rim frequencies of a hypotrochoid with b + rho at most 26, every tangency reach times its partner under the swap is 1 to within 1.47e-13. Witness: lab/rs/roulette-loops.
2026-09-10 Verified The hypotrochoid 5/1 has tangency reaches 1 and 3.674234614175 twice, the second exactly reach squared 27/2 at u = 5/6 on Q(u) = 40 - 48 u, and its count runs 0, 5, 15. The hypotrochoid 7/2 has 1 and 2.353415666603 twice, reach squared (81 + 21 sqrt 21) / 32 at the smaller root of Q(u) = 192 u^2 - 336 u + 140, and counts 7, 14, 28. Witness: lab/rs/roulette-loops.
2026-09-10 Verified The epitrochoid 5/1 has tangency reaches 1, 4.180967894379 twice and 5.789603394549 twice, the last two exactly reach squared (102 - 7 sqrt 21) / 4 and (102 + 7 sqrt 21) / 4 on Q(u) = -320 u^2 + 448 u - 140, and its count runs 0, 5, 15, 25, four values for three distinct reaches because two angles share each of the last two. Witness: lab/rs/roulette-loops.
2026-09-10 Verified When a is even, delta = pi/2 is always a tangency angle of the trochoid on a circle track a/b and its reach is the rational rho / b, which is also the one reach where the curve runs through the centre with all a branches: 3 at 4/1 inside, 5/3 at 8/3 inside, 13/5 at 8/5 outside. There a step and the centre correction fall on the same reach and the step function is not read. Witness: lab/rs/roulette-loops.
2026-09-10 Verified At reach rho / b the distinct self crossing point count of a trochoid with a odd is the step function less C(a, 2) - 1, read off mrlynum::spirograph::trace at 24001 samples as 1 for 3/1, 5/2, 5/3 and 7/4 inside, 6 for 5/1 and 5/4 inside, 8 for 7/2 inside, 7 for 3/1, 10 for 3/2 and 21 for 5/2 outside, the two counts agreeing three percent either side. Witness: lab/rs/roulette-loops, mrlynum::spirograph::trace.
2026-09-10 Verified The pair law matches mrlynum::spirograph::trace on 48 reads over 5/1, 7/2 and 8/3 inside and 5/2 outside at reaches 0.6, 1.3, 2.4 and 3.7 against three seat kinds, equal reach, shorter reach and the wheel's centre, on the parameter pair count exactly. Witness: lab/rs/roulette-loops, mrlynum::spirograph::trace.
2026-09-10 Verified The pair count 2 a b of two distinct trochoids from one wheel survives the loop threshold, which is a tangency of one curve with itself and never of two curves with each other. For two seats at one reach and half angle nu apart on 5/1 inside it first leaves 2 a b at 2.242763, 1.741061, 1.379486, 1.143270, 1.047854, 1.010207 and 1.002194 for nu of 1, 0.5, 0.2, 0.05, 0.01, 0.001 and 0.0001. The pair threshold is not monotone in nu: 8/3 inside gives 1.117596 at nu = 1 against 1.523254 at nu = 0.5. Witness: lab/rs/roulette-loops.
2026-09-10 Verified On a tangency reach the meeting count of a trochoid is the count below plus a/2 per tangency angle there, read as 21 against 14 and 28 for 7/2 inside, 55 against 44 and 66 for 11/4 inside, 6 against 3 and 9 for 3/1 outside, and 15 against 10 and 20 and then 25 against 20 and 30 for 5/2 outside, the two branches closing to 1e-14 or better in every case. Witness: lab/rs/roulette-loops.
2026-09-10 Conjecture For every coprime a/b and both sides the trochoid's tangency angle count is abs(rho - b) and every angle moves the self crossing count by exactly a sign(m), so the count runs from a (b - 1) to a (rho - 1) in abs(rho - b) equal steps. The angle count is exhaustive to frequency 220 in exact integers and the step size to a at most eleven against the trace; what is missing is a proof that the merged Farey sign sequence changes sign exactly abs(rho - b) - 1 times inside (0, pi). Witness: lab/rs/roulette-loops.
2026-09-10 Conjecture The first reach at which two trochoids from seats at one reach and half angle nu apart stop meeting 2 a b times is above one for every positive nu, with infimum one, the excess falling like nu^(2/3). The reading is seven sampled nu at one fraction, 5/1 inside, whose excesses fall by 4.69 then 4.65 per decade against 10^(2/3) = 4.64. There is no bound and no third decade. Witness: lab/rs/roulette-loops.

The spirograph nodes

2026-09-10 Proved One curve of a circle roulette crosses itself exactly a(b - 1) times when its seat obeys 0 < abs(p) < min(1, M/b), for a/b the ring over the wheel in lowest terms and M = a - b inside, M = a + b outside: the crossing equation reduces on the torus to A abs(sin(b x)) = C abs(sin(M x)) with A = r M / b and C = r abs(p), every root carrying exactly a half sums, and that is the zero set of the imaginary parts of A e^(i b x) -+ C e^(i M x), whose arguments climb strictly while A > C and A b > C M, so each takes exactly 2b zeros and, once C > 0, the two share only x = 0 and x = pi. Witness: research/lab/rs/roulette-nodes, mrlylab::roulette::nodes.
2026-09-10 Proved Two distinct curves of one wheel on a circle track cross exactly 2ab times when both seats lie in the window, 0 < abs(p) < min(1, M/b), and share a radius: the half difference equation gains only a phase, each branch keeps its 2b zeros and the two share none, so the root count is 4b and the crossing count a 4b / 2. For seats of different radii the same count follows from the sufficient bound 2 A sqrt(1 - k^2) > r(abs(p) + abs(q)) with k = r sqrt(abs(p) abs(q)) M / (A b), far from necessary: inside 7/3 at seats 0.950 and 0.672 the bound reads 1.604 against 1.622 and fails while the count is 42. Witness: research/lab/rs/roulette-nodes.
2026-09-10 Verified A whole design carries one node count, 2ab C(k, 2) + k a(b - 1) crossings with k its distinct curves, so the design enters only through k: 5553 cells and 4455175 crossings over both tracks, every a/b in lowest terms with a at most 20 and b at most 10, four designs and seven reaches inside the window 0 < abs(p) < min(1, M/b), every count three sample counts alike; 143 cells are aligned and printed, 26 need a further doubling, none goes unsettled, and two disagree by a few crossings at a near tangency, inside 20/3 at seat 0.250 and outside 10/9 at 0.400, both read as the law by the torus. Witness: research/lab/rs/roulette-nodes.
2026-09-10 Verified A roulette cuts the plane into nodes + 2 regions, the unbounded one among them, at a generic reach with every seat in the window 0 < abs(p) < min(1, M/b), nodes counting distinct transversal double points: the picture is a connected 4-regular plane graph and Euler gives the count, k = 1 with b = 1 carrying no node and 2 regions by Jordan. A flood of the rastered walls at 1600 and at 2400 pixels returns 2, 7, 16, 8 and 32 for one seat inside 3/1, 5/2, 7/3 and two seats inside 3/1, 5/2, and 1206 for the carpet inside 7/3 at the alignment reach, where 1288 crossings sit at 1148 nodes of 2352 branches and the count is branches - points + 2. Witness: research/lab/rs/roulette-nodes.
2026-09-10 Verified The self law ends at the crest of abs(sin(b x)) / abs(sin(M x)), the least of its local maxima, never below the seat threshold C/A = 1 because the ratio reaches 1 at the midpoint of two consecutive zeros of sin(b x): over 213 cells and the 31 inside ratios with a < 2b and a at most 20, every count below the crest is a(b - 1), every count above it is smaller, always a multiple of a, never rising, and every ladder ends at a(a - b). At exactly C/A = 1 the curve runs through the centre, a branches meet, and the counts 25 at 7/5 and 31 at 7/6 are neither the law nor a multiple of a. Witness: lab/rs/roulette-nodes.
2026-09-10 Proved Two pencils on a circle track a/b in lowest terms draw one curve if and only if a rotation of 2 pi/b about the wheel's centre carries one seat to the other; the converse is read off abs(z)^2 = A^2 + C^2 + 2 A C cos(a u - arg p), whose phase runs over a turns, with Niven's theorem cutting the square lattice to the quarter turns. So curves coincide by half turns when b is even and by quarter turns when 4 divides b, and k is the pencil set modulo the rotations of order gcd(b, 4); on a line track a seat's angle is a shift, so two seats of one radius draw translates of one shape and never one curve. Witness: mrlynum::spirograph::representatives, lab/rs/roulette-reaches, spirograph.md.
2026-09-10 Refuted The loop threshold is not where a design's node laws end: inside a track with a < 2b the seat leaves the centre path first and the counts fall while abs(p) is still under 1, 7/6 reading 35, 21 and 7 self crossings at seats 0.158, 0.175 and 0.9 against a(b - 1) = 35. One seat past that threshold is enough to lose the pair law where the self law still holds: inside 7/4 at seats 0.900 and 0.636 two curves cross 42 times against 2ab = 56 while both self counts hold at 21 under the crest 1.333. Past it the pair count is no function of a and b, reading 6 against 12 at 3/2 and 8 against 24 at 4/3. Witness: lab/rs/roulette-nodes.

The spirograph reaches

2026-09-10 Proved A pencil at complex seat p wheel radii on a wheel rolling on a circle track a/b in lowest terms draws z(psi) = r e^(i b psi) (A + p e^(i eps a psi)) on psi in [0, 2 pi), with A = abs(a/b + eps), eps = -1 inside and +1 outside. Witness: lab/rs/roulette-reaches, mrlynum::spirograph::point.
2026-09-10 Proved That trochoid's picture turns a fold, since z(psi + 2 pi / a) = e^(2 pi i b / a) z(psi) with no eps because e^(i eps a 2 pi / a) = 1, and turning the seat by alpha turns the whole curve by -b eps alpha / a, since shifting psi by -alpha / (eps a) absorbs the seat turn and leaves the prefactor e^(-i eps b alpha / a); the four seats of one square orbit therefore draw four rotations of one master curve and the seat modulus is the only shape parameter. Witness: research/lab/rs/roulette-reaches.
2026-09-10 Proved Writing x for the seat's phase and m(x) = 4 a (b eps x / a + arg(A + p e^(ix))) / pi, with p not zero the radius abs(A + p e^(ix)) is strictly decreasing on [0, pi], so the trochoid meets every circle strictly between the two apex radii in exactly 2 a points, at the angles c + m/8 and c - m/8 in units of a turn over a, with seat offset c = -b eps arg(p) / (2 pi). Witness: lab/rs/roulette-reaches.
2026-09-10 Proved While the seat modulus is under A the mark runs from 0 at the outer apex to 4 b eps at the inner one, and past A the point A + p e^(ix) circles the origin so the inner value is 4 b eps + 4 a instead. Witness: lab/rs/roulette-reaches.
2026-09-10 Proved The mark is one to one in the radius whenever the seat modulus is under min(1, A): while it is under A the derivative of arg(A + p e^(ix)) in x grows with cos x, running from minus the modulus over A minus the modulus at x = pi to the modulus over A plus the modulus at x = 0, and each of those stays under b/a exactly when the modulus is under one, while past A the derivative at x = pi exceeds one and the mark turns back. Witness: lab/rs/roulette-reaches.
2026-09-10 Proved That window is sharp but for one endpoint: at modulus one the binding derivative meets b/a at the single phase x = 0 inside and at x = pi outside, so the mark is still one to one there, and every larger modulus fails. Witness: lab/rs/roulette-reaches.
2026-09-10 Proved Two trochoids of one seat modulus whose seat offsets differ by a quarter turn cross only on the circles where the mark is a whole number, and a curve crosses itself only where the mark is a multiple of four; the quarter turn is needed, since seats a fifth of a turn apart on 7/3 inside cross where the mark is plus or minus 1.2 modulo four. Witness: lab/rs/roulette-reaches.
2026-09-10 Proved Inside the window the mark's range has length 4 b, so a trochoid on a circle track a/b has b - 1 self crossing circles and a (b - 1) self crossings, and two distinct curves of one seat modulus a quarter turn apart have 2 b crossing circles and 2 a b crossings. Witness: lab/rs/roulette-reaches, mrlynum::spirograph::trace.
2026-09-10 Proved Three trochoids of one seat modulus on a circle track never run through one point unless the modulus is A: the radius squared A^2 + q^2 + 2 A q cos x for modulus q is strictly decreasing in x on [0, pi], so one radius fixes one phase and one mark for all three at once, the angles are c + m/8 and c - m/8, two of the three must share a sign, and that forces their seat offsets to differ by a multiple of a turn over a, which makes the two curves the same curve. Witness: lab/rs/roulette-reaches.
2026-09-10 Proved That argument needs only the monotone radius, so it carries past the one to one window, holds for b even, and holds at both apexes where the two signs merge. Witness: lab/rs/roulette-reaches.
2026-09-10 Proved At seat modulus A on a circle track a/b every trochoid of that modulus runs through the centre, a times each; inside this needs b < a < 2 b for a modulus under one, and outside it never happens because A = a/b + 1 exceeds one. Witness: lab/rs/roulette-reaches.
2026-09-10 Proved A meeting of trochoids from the carpet's two seat moduli on a circle track a/b happens exactly when both marks are whole numbers at one radius and one eighth class collects three or more branches; the seat offsets are the exact eighths -b eps d mod 8 for compass index d, so the test is integer arithmetic and never a tolerance, each meeting class holds a points because the picture turns a fold, and each meeting swallows five of the generic picture's double points. Witness: lab/rs/roulette-reaches.
2026-09-10 Proved A trochoid on a circle track a/b sits at whole mark congruent to j modulo four exactly where w^(a + 2 b eps) (A + p w)^a = i^j (A w + p)^a on the unit circle, which follows from abs(A + p w)^2 = (A + p w)(A w + p) / w; squaring the mark condition loses half the angle, so the law pins the mark only modulo four. Witness: lab/rs/roulette-reaches.
2026-09-10 Proved The carpet alignment reaches on a circle track a/b are contained in the real algebraic set cut out by that law for the corner modulus, the same law for the edge modulus, and the equal radius equation, three real equations in three real unknowns over the field generated by the square root of two; the containment is proper, since the law pins the mark only modulo four and the marks -10, -6 and -2 share one system on 7/3 inside, but the mod four branches are disjoint and closed inside the window because the mark is continuous there, so an isolated alignment reach is an isolated point of that set and hence an algebraic number. Witness: lab/rs/roulette-reaches.
2026-09-10 Verified The reduction agrees with mrlynum::spirograph::point to 2.138e-13 over all 98 circle tracks with b in 1..8 and a in b+1..13 coprime, both sides, on all eight carpet fill seats at reach 0.83 and 29 phases a seat, and the seat offset classes reproduce mrlynum::spirograph::distinct on all 98. Witness: lab/rs/roulette-reaches, mrlynum::spirograph.
2026-09-10 Verified The node counts a (b - 1) and 2 a b need the seat modulus under min(1, A) and not merely under one: the radius and mark census of the study gives 7/5 inside 4 self classes and 10 pair classes per pair at modulus 0.3900 under A = 0.4, and the pair count falls to 8 at 0.4100, while 4/3 inside falls from 2 self classes to 1 between 0.3267 and 0.3400, both well under the cusp threshold one. Witness: lab/rs/roulette-reaches.
2026-09-10 Verified The carpet on 7/3 inside has 24 alignment reaches over the scan, 21 transversal and 3 tangential; every transversal meeting carries four branches, twelve of them from four distinct curves and nine from three, one curve bringing two branches where its own self crossing lands on the meeting. Witness: lab/rs/roulette-reaches.
2026-09-10 Verified The 7/3 inside transversal reach near 0.79 is 0.791009415157 with marks (-6, -9), ring 1.005704332357 wheel radii and corner seat modulus 0.527339610104; the f64 bracket is 1.1e-16 and the exact law residual 4.18e-15, so the printed twelve decimals round safe. Witness: lab/rs/roulette-reaches.
2026-09-10 Verified The four curves predicted to meet at that reach do meet on the crate's own curve: the four meeting seats sit 2.167e-5, 1.441e-6, 7.697e-8 and 5.590e-9 from the point the algebra names, read in f64 from mrlynum::spirograph::point at 2000, 8000, 32000 and 128000 samples, falling like the square of the sample count, while the other four seats stay 1.206e-1 away. Witness: lab/rs/roulette-reaches, mrlynum::spirograph::point.
2026-09-10 Verified The same read off mrlynum::spirograph::trace gives 2.162e-5, 1.526e-6 and 1.109e-7 at 2000, 8000 and 32000 samples, tracking the f64 column until it reaches the f32 floor: trace returns f32, half a step at that radius is 1.192e-7, so the third column measures rounding rather than convergence and only the first two carry the square law. Witness: lab/rs/roulette-reaches, mrlynum::spirograph::trace.
2026-09-10 Verified At the control reaches 0.781009 and 0.801009 no class carries more than two branches and the four seats stand 1.767e-2 and 1.771e-2 off the point that meets at 0.791009415157. Witness: lab/rs/roulette-reaches, mrlynum::spirograph::trace.
2026-09-10 Verified The node count drops at a transversal alignment: 7/3 inside carries 184 crossing classes at a generic reach and 164 at the reach 0.791009415157, the four meetings swallowing five double points each, and since the picture turns a fold those are 1288 and 1148 nodes. Witness: lab/rs/roulette-reaches.
2026-09-10 Verified The tangential alignment family is not empty: inside the window the corner band contains the edge band, so a corner pair's crossing radius sweeps through both edge apex radii, and on 7/3 inside three such reaches sit in the scan, at 0.687455178256 with marks (-7, -12), 0.948942238176 with (-1, 0) and 1.176138007019 with (-5, -12). Witness: lab/rs/roulette-reaches.
2026-09-10 Verified The two sided falsification holds over all 98 circle tracks with b in 1..8 and a in b+1..13 coprime, inside and outside, inside the window: 2157 alignment reaches, 193 of them tangential, every one carrying a node past two branches read off mrlynum::spirograph::point in f64 and off mrlynum::spirograph::trace in f32 at 8000 samples, worst gap 2.445e-4 in f64 against a worst f32 floor of 9.537e-7, and none of the 98 control reaches carrying one. Witness: lab/rs/roulette-reaches, mrlynum::spirograph::trace.
2026-09-10 Verified Every transversal alignment carries 4 / gcd(b, 4) meeting classes, and the reach counts are stable in the scan: 7/3 inside gives 24, 12/7 inside 153, 13/7 inside 177, 9/5 inside 89, 13/8 inside 11 and 13/7 outside 61, each at both 6000 and 24000 steps. Witness: lab/rs/roulette-reaches.
2026-09-10 Conjecture Every carpet alignment reach is an isolated point of the real algebraic set that contains it, and so an algebraic number. The scan finds each reach as a simple sign change of a continuous defect, and no two of the 2157 reaches over the 98 tracks coincide, but nothing in the run certifies isolation, and a failed integer relation search to degree 48 with coefficients under 1e7 at 200 digits on the 7/3 reach 0.791009415157 is consistent with either answer. Witness: lab/rs/roulette-reaches.
2026-09-10 Conjecture The number of carpet alignment reaches on a circle track a/b inside the window follows the window's own width: inside it climbs with a while a < 2 b, where the window is A and grows with a, and falls once a > 2 b, where the window is the fixed one, b = 3 inside giving 18, 28 for a = 4, 5 and then 24, 21, 19, 18, 15 for a = 7, 8, 10, 11, 13, and b = 5 inside rising 39, 55, 71, 89 and then falling 79, 68, 64; outside it barely moves with a, b = 3 giving 9, 9, 11, 11, 11, 11, 12 and b = 7 giving 57, 57, 57, 57, 57, 61. No closed count is proved. Witness: lab/rs/roulette-reaches.

The spirograph walls

2026-09-10 Proved On a circle track R/r = a/b in lowest terms a pencil at complex seat p draws a closed curve whose signed area over the whole track, counterclockwise positive and counted with multiplicity so that it is the winding number integrated over the plane, is pi b rho (rho -+ d^2/r), minus inside and plus outside, with rho = R -+ r the centre circle's radius and d = r abs(p). Green's theorem on z(t) = rho e^(i t) + p r e^(-+ i (rho/r) t) gives it, the cross terms carrying e^(-+ i a t / b) over b centre turns and integrating to zero, and no hypothesis on the reach is needed since loops are counted with their sign. Witness: mrlynum::spirograph::signed_area, lab/rs/roulette-cover.
2026-09-10 Proved Every curve of a circle roulette lies in the closed annulus from abs(rho - d) to rho + d about the track's centre and attains both bounds, since abs(z)^2 = rho^2 + d^2 + 2 rho d cos(a t / b -+ arg p) and the phase runs over a full turns. The whole roulette therefore sits in the disc of radius rho + max d and enters no disc of radius under min abs(rho - d), the least over the seats and not the outermost seat's own, since seats on either side of rho keep their own inner radius; a trace of 200001 points per curve meets both radii on all 48 cases with worst gap 3.55e-15, the five ratios with a seat past rho included. Witness: mrlynum::spirograph::disc, lab/rs/roulette-cover.
2026-09-10 Proved The fluid poured at the centre of a circle track always fills at least the disc of radius min abs(rho - d), which no curve enters, so the hole is at least that radius over the disc's radius, squared. The raster reads no leak on all 48 cases, the slack running from 0.000111 at four quarter-turn copies inside 7/3 to 0.461499 at one pencil outside 2/1, and to 0.142005 at one pencil inside 4/1 over the inside cases alone; read instead with the outermost seat the bound is false, and one pencil inside 5/4 at reach 0.9 is the smallest counterexample, inner radius 2.6 and not rho - d. Witness: mrlynum::spirograph::disc, lab/rs/roulette-cover.
2026-09-10 Proved At b = 1 with one distinct curve below the loop threshold the roulette is a simple closed curve on both sides, so its complement has exactly two components: the shape between the walls is the wall alone and covers nothing, while the fluid poured at the centre fills the whole inside, the centre lying inside because the winding number about it is b = 1. The hole is therefore the signed area over the disc's area, rho (rho -+ d^2/r) / (rho + d)^2, minus inside and plus outside, met to 8.70e-5 at worst over the 16 ratios 2/1 to 9/1 on both sides while the cover falls like the pixel to at most 0.000124. Witness: mrlynum::spirograph::cover, lab/rs/roulette-cover.
2026-09-10 Proved For one distinct curve the shape between the walls is the union of the bounded complement components other than the one holding the centre, so it is empty for a simple curve and of positive area as soon as the curve crosses itself: one pencil at reach 0.9 inside covers -0.000030 on a bar of 0.000064, zero to its bar, at 3/1 with no self crossing, 0.003269 at 3/2 with a(b - 1) = 3 crossings, and 0.273092 at 7/3 with 14. Witness: mrlynum::spirograph::cover, lab/rs/roulette-cover.
2026-09-10 Verified The mean signed winding number of the enclosing disc's pixel centres, read by scanline against the polylines and never off a flood, meets the sum of the distinct curves' signed areas over the disc's area on all 48 cases at all four raster sides, worst gap 1.73e-3 at the carpet's corners outside 5/8 at side 256 and 4.84e-4 at side 2048, with no reading outside the perimeter bound. At side 2048: 0.282984 against 0.282996 for one pencil inside 3/1, 1.678772 against 1.678770 for four quarter-turn copies inside 7/3, 9.103089 against 9.103448 for the carpet's fills inside 7/3. Witness: mrlynum::spirograph::cover, lab/rs/roulette-cover.
2026-09-10 Verified The raster cover converges like the pixel: on all 48 cases the successive differences fall by a factor of at most 0.626 as the side doubles from 256 to 2048, so the Richardson limit of c(n) = c + A/n carries a bar of at most 0.000354. The carpet's fills inside 7/3 at reach 0.9 cover 0.800044 on a bar of 0.000343 from the ladder 0.814487, 0.807142, 0.803764, 0.801904, its corners cover 0.765594, four quarter-turn copies of one seat cover 0.629299, two cover 0.443227, one pencil covers 0.273092, and outside 5/8 the fills cover 0.758414 and the corners 0.867284. Witness: mrlynum::spirograph::cover, lab/rs/roulette-cover.
2026-09-10 Verified The cover reads the distinct curves and not the pencils: two half-turn copies of one seat inside 5/2 are one curve under the coincidence law and cover 0.213852, the single pencil's own figure, while inside 7/3 the same two seats are two curves and cover 0.443227; the closed form follows the same law, the carpet's eight fills inside 5/2 summing to 3.346939 over four curves where a sum over the eight seats would read 6.693878. Witness: mrlynum::spirograph::cover, lab/rs/roulette-cover.
2026-09-10 Conjecture No closed form for the cover shows itself on the small cases: all three candidates fail on all 48, and the walls are stitched at the parameters where the outermost and innermost arcs cross, so an exact area would be a sum over the regions those crossings cut, which no study here computes. Adding curves does not act independently either: four quarter-turn copies inside 7/3 cover 0.629299 where the independent-union guess 1 - (1 - p)^k on the single curve's 0.273092 gives 0.720798. Witness: lab/rs/roulette-cover.
2026-09-10 Refuted The full annulus between the walls' extreme radii, 4 rho d / (rho + d)^2, which is 4 d (a - b) b / (a - b + b d)^2 inside, is not the cover: the fluid poured from outside reaches into every bay between the outermost arcs, and the form overshoots on all 48 cases, closest at the carpet's fills inside 7/3 with 0.856124 against 0.800044 on a bar of 0.000343, worst at one pencil inside 2/1 with 0.997230 against 0.000017. Witness: lab/rs/roulette-cover.
2026-09-10 Refuted The sum of the distinct curves' enclosed areas over the disc's area is not the cover: it counts multiplicity, so it reads 9.103448 for the carpet's fills inside 7/3 where a cover cannot pass 1, and at b = 1 it is the hole and not the cover, 0.282996 against a cover of zero inside 3/1. Witness: lab/rs/roulette-cover.
2026-09-10 Refuted The single simple curve's form rho (rho -+ d^2/r) / (rho + d)^2 is not the cover at any ratio: at b = 1 the cover is zero and the form is the hole, and above b = 1 it misses, 0.139898 against 0.273092 on a bar of 0.000076 for one pencil inside 7/3 and 0.347206 against 0.800044 for the carpet's fills there. Witness: lab/rs/roulette-cover.

The spun stack

2026-09-09 Proved The dead-spin theorem: rotated layers (m, alpha) and (n, beta) of the line stack share a node off the origin iff cos and sin of alpha - beta are both rational, and the shared set is then R_alpha (1/gcd(m,n)) Z^2, of density gcd(m,n)^2 per unit area; at whole degrees Niven's theorem makes the condition alpha = beta mod 90, exactly 4 of 360 degrees having both cos and sin rational by reduction of zeta^d + zeta^-d mod Phi_360; the rational rotations are exactly w^2/N(w) for nonzero Gaussian w, not z/|z|, (1 + i)/sqrt 2 the counterexample, all 68 rational unit-circle points of denominator at most 60 reached from the box of side 12. Witness: lab/py/spun-stack rational_angle_degrees, dead_spin_pairs, pythagorean_hits.
2026-09-09 Proved The exact spun stack indexes layers by the nonzero associate classes of Z[i], layer z the lattice z^-1 Z[i]; its lit nodes in the unit square are the Gaussian rationals u/d in lowest terms, the node with reduced denominator d is lit by exactly the layers d divides, and its brightness is g(floor(N/N(d))) with g(t) = sum_j (floor(t/(4j+1)) - floor(t/(4j+3))), the Gauss circle count of nonzero classes of norm at most t and the Gaussian twin of floor(N/b); the lit set has sum_{[d], N(d) <= N} Phi(d) nodes, the Gaussian totient sum; 672 nodes and 0 mismatches against exact literal stacking at norm bound 50, counts 672, 10608, 168088 at norm bounds 50, 200, 800. Witness: lab/py/spun-stack literal_stack, closed_brightness, totient_sum.
2026-09-09 Proved A fixed rotation with a fixed geometric scale per layer is multiplication by one complex c: the layers c^-k Z[i] overlap off the origin iff c is in Q(i) and nest iff c is in Z[i], and then brightness is depth + 1 - address, a pure address with no moire, the base-c numeration tree (base -1 + i the twindragon); Verified at c = 1 + i depth 8 and c = 2 + i depth 4, 256 and 625 nodes, 0 mismatches, overlaps 440, 220 and 0 over the box of side 10 for 1 + i, 3/2 + i/2 and sqrt 2 e^i. Witness: lab/py/spun-stack base_depth_check, base_c_overlap.
2026-09-09 Verified No Franel-Landau theorem for the Gaussian Farey set is found in the sources read (Sayous arXiv:2407.04380 proves equidistribution on C/Z[i] with no rate and a gap law, naming neither Franel nor Landau; Estala-Arias arXiv:1908.03658 states RH for zeta_K on measures over the positive reals; Huxley Acta Arith. 18 (1971) and Kanemitsu-Yoshimoto Acta Arith. 75 (1996) unread), so an RH-equivalent for zeta_K rendered by the spun stack is unstated, neither proved nor refuted; the named obstruction is that Franel-Landau needs a rank and C/Z[i] carries no canonical linear order. Witness: lab/py/spun-stack, REFS.md. Superseded: Huxley 1971 and Kanemitsu-Yoshimoto 1996 are read at source, the rank obstruction blocks only the rank functional, and the Fourier L^2 equivalence is proved under Franel one field up.
2026-09-09 Proved The centre is a node of every spin schedule of the odd carpet stack: rotation about the centre preserves distance from it and layer n's cell containing the centre has inradius 1/(2n), so a disc of radius 1/(2N) lies inside one cell of all layers at every angle; at N = 55 the centre reads ink 14/28, the scales n = 3 mod 4, the unspun value, under every schedule and every raster. Witness: lab/py/spin-render main.
2026-09-09 Proved The unspun odd carpet stack at N = 55 attains its global ink maximum 18/28 on exactly four square cells of side 1/159, total area 4/25281 = 0.000158222: ink at (u, v) is the size of the intersection of the two scale sets, so a maximum needs them equal and maximal; the one-dimensional maxima over 636 exact breakpoints are [1/3, 18/53) and [35/53, 2/3), and the scale set is invariant under x -> 1 - x because floor(n(1 - x)) = n - 1 - floor(nx) with n - 1 even, so both intervals carry the same 18 scales and all four products are maxima. Witness: lab/py/spin-render diagonal_maximum.
2026-09-09 Verified Spinning by whole degrees destroys the unspun maximum and shrinks its cell: peak ink falls from 18/28 to 14/28 under a one-degree increment and 16/28 under the prime-degree schedule, the golden and Gaussian schedules, 17/28 under random angles, the peak cell area from 3.95523e-05 to 9.80453e-06 and 1.65596e-05, at R = 256, 512, 1024, 2048 and under a zoom at effective R = 51200; spinning leaves the fade law alone, rms sqrt(L) in the layer count L at L = 28 running 0.401417 to 0.460395 over six schedules, the unspun raster matching the exact rational covariance sum 0.309477, 0.389754, 0.426869, 0.458411 at L = 4, 8, 14, 28 to 0.4%, the lane's c = 0.522 being the limit constant and not the L = 28 value. Witness: lab/py/spin-render report, main.
2026-09-09 Proved The eyes of the fixed increment: under the schedule that turns layer k by k theta, two layers with indices j, k share an exact lattice iff (j - k) theta is a multiple of 90 degrees (the odd carpet being invariant under a quarter turn), so at theta = 90 p/q in lowest terms the layers fall into exactly q angle classes and the sharing pairs number sum_classes C(size, 2), while at an irrational theta/90 no pair shares; on the 28 odd scales to 55 the count reads 378 at theta/90 = 0, 182 at 1/2, 117 at 1/3 and 2/3, 84 at 1/4 and 3/4, 65 at 1/5 and 2/5, 52 at 1/6, 36 at 1/8, 30 at 1/9 and 0 at sqrt 2 - 1, confirmed by the pairwise exact test and by the Niven-free rational-angle test at the whole-degree increments; the moire switches on exactly past the Farey fractions of a quarter turn. Witness: lab/py/spun-stack increment_classes.
2026-09-09 Proved The node-count constant of the spun stacks: for every imaginary quadratic field K with class number h, w units and discriminant D_K, sum_{N(a) <= N} Phi(a) = (rho_K/(2 zeta_K(2))) N^2 + O(N^(3/2)) over nonzero ideals with Phi = N * mu_K and rho_K = 2 pi h/(w sqrt |D_K|), by Dirichlet convolution and Abel summation on the ideal count A(t) = rho_K t + O(sqrt t), itself derived one ideal class at a time from the lattice of covolume N(a) sqrt |D_K|/2; the Gaussian constant is pi/(8 zeta(2) G) = 0.260634696495 and the Eisenstein constant pi/(6 sqrt 3 zeta(2) L(2, chi_-3)) = 0.235217881630, the nine published node counts recounted exactly and extended to norm bound 102400 where count/(c N^2) reads 0.999746 and 1.000049, the deviation scaled by N^(3/2) never past 0.119 and the ratios oscillating about 1, with D = -20 (h = 2, ratio 1.000001117) and D = -23 (h = 3, ratio 0.999886) at 102400 as the class-number witnesses; the observed N log N size of the error stays Conjecture; the literature's complex Farey constant pi/(sqrt |D_K| zeta_K(2)) counts element denominators and is w times this one, the sets being equal. For a general number field the same argument runs from any ideal count with error O(t^theta), 0 < theta < 1. Witness: lab/py/totient-constant main, norm_totient_sum, farey_set.
2026-09-09 Conjecture The Gaussian Farey stack's node count is asymptotically pi N^2/(8 zeta(2) G) = 0.260635 N^2 with G Catalan's constant; the ratios read 0.268800, 0.265200, 0.262638 at norm bounds 50, 200, 800; the literature count of the complex Farey set is 4 times this, the order of the unit group, a convention difference unresolved. Witness: lab/py/spun-stack totient_sum. Superseded: the constant is Proved and the convention resolved, see the constant row under The spun stack in SETTLED.
2026-09-09 Refuted A whole-degree prime schedule (layer k at p_k degrees) is coincidence-free: 5 of the 435 layer pairs to N = 30 share, all at relative angle exactly 90 (7 and 97, 11 and 101, 13 and 103, 17 and 107, 19 and 109 degrees), each sharing a lattice of density gcd(m,n)^2 per unit area whose count in the open unit square with the origin excluded reads 1, 9, 49, 1, 9 and is angle-dependent (2, 3, 4 at g = 2 over degrees 1 to 89), not a formula; the other 430 pairs are dead with margin 0.003390. Witness: lab/py/spun-stack dead_spin_pairs, unit_square_shares, share_count_spread.
2026-09-09 Refuted A resolution-stable off-centre maximum in a spun render is an exact coincidence: the stack is piecewise constant on cells of positive area, so any cell wider than a pixel holds its position at every resolution, and the prime-degree schedule shows one at (0.19469, 0.15501) drifting 0.29 px from R = 1024 to 2048; raster stability measures cell area, and the discriminators are the peak value and the cell area. Witness: lab/py/spin-render drift.

The stack algebra

2026-09-09 Proved Stacking is Dirichlet convolution: the u-stack of the v-stack draws the inner scale n at scale kn with weight u(k) v(n), so the composite weight is u * v, exact at every scale m <= N under the hyperbolic cut kn <= N and only for m <= min(K, M) under a rectangular cut (10 of 16 scales differ above 24 in the 24 x 40 case at u = v = mu). The plain stack is zeta, the stack of stacks is zeta^2 with scale n drawn d(n) times, and 1 * mu = e collapses the Mobius stack of the plain stack to one layer. Checked by literal double stacking in exact rationals at N = 60 by three routes sharing no inner loop on u = v = 1, u = 1, v = mu, u = v = mu and u = 1, v = n^-1, 1102, 1102, 974 and 1102 nodes, zero mismatches. The group carries no RH content: RH sits at the inverse of 1 alone, whose b = 1 node is M(N). Witness: lab/py/stack-algebra convolution_check.
2026-09-09 Proved Every selected line stack has a closed-form node at denominator b: evens floor(N/lcm(2,b)), odds 0 at even b and ceil(floor(N/b)/2) at odd b, primes pi(N) at b = 1, 1 at prime b <= N and 0 elsewhere, squarefree the double divisor sum sum_{d^2 <= N/b, gcd(d,b) = 1} mu(d) sum_{e | b} mu(e) floor(N/(b d^2 e)), prime powers floor(log_p N) - i + 1 at b = p^i; zero mismatches against literal stacking at every b <= N for N = 30, 61, 200, 501; the primes-only stack at N = 501 lights b = 1 and the 95 primes, every prime node at brightness exactly 1. Witness: lab/py/stack-algebra selection_closed_forms.
2026-09-09 Proved The primes-only carpet stack fades at exactly the independent rate: distinct primes are coprime, so every layer pair has covariance exactly 0 by the gcd law, L Var of the L-layer mean is the mean of the per-layer variances identically and the ratio to independent layers is exactly 1 at every L; with 16 p^4 Var_p = 3p^4 - 4p^3 - 2p^2 + 4p - 1 = (p-1)^2 (3p-1)(p+1) the constant is c^2 = 3/16, c = sqrt(3)/4 = 0.4330127, approached from below at rate O(log log p_L / L), L Var reading 0.1429334753, 0.1595579958, 0.1833424270, 0.1869968711 at L = 5, 10, 100, 1000; the odd stack under the same estimator reads 0.2708541 and factor 1.202738 at L = 4000, converging to the lane's 1.2054. The criterion is pairwise coprimality, not primality, the odd primes being the densest uncorrelated selection by least-prime-factor injectivity; the squarefree-odd rival fails with Cov(C_15, C_21) = 284/99225 = 0.0028621819 and ratio 1.308596 over 1000 layers. Witness: lab/py/stack-algebra prime_carpet_variance, squarefree_carpet_variance.
2026-09-09 Proved The s-harmonic stack: weights n^-s are completely multiplicative, so the node a/b reads b^-s H_s(floor(N/b)) and tends to zeta(s)/b^s, and the total node mass sum_b phi(b) zeta(s) b^-s equals zeta(s-1) for s > 2, read at N = 16000 as 1.644872, 1.202057, 1.082323 against zeta(2), zeta(3), zeta(4). At s = 1 the renormalisation is Davenport's expansion at a = mu, sum mu(n)/n ((nx)) = -sin(2 pi x)/pi, so the renormalised Mobius stack of the sawtooth is one sine; truncated at n <= 10^5 over five rational x the max error is 5.49e-03, 1.37e-03, 2.08e-04 at cuts 10^3, 10^4, 10^5; the {nx} form differs by (1/2) sum mu(n)/n, whose vanishing is the prime number theorem, read as -0.00048723 at n <= 10^5; the identity is Verified through arXiv:2005.08279 equation 1.1, which quotes it, not at the 1937 source. Witness: lab/py/stack-algebra harmonic_stack, davenport_check.
2026-09-09 Refuted Everything inside the Dirichlet group of stacks is closed form: the group contains 1 and mu alike, so membership buys nothing and closed-form-ness is a property of the weight, not of the algebra; the group statement carries no RH content and RH sits at exactly one element, the inverse of 1. Witness: lab/py/stack-algebra.

The tent rank law

2026-08-28 Proved The palindromic module reformulation: mod 2 at odd dim (R = (dim-1)/2, n = R+1, M = 6R+2) the symbol is G = (1+t)^(4R)(1+t+t^2) with 4-block coefficients C(R, a), and nullity_2(M_even) = dim{H : no exponent == 1 mod 3, deg H <= M, G | H, t^M H(1/t) = H} - palindromy folds the R+1 kernel conditions onto one residue class mod 3; the explicit kernel vectors H_(b,i) = t^s (1+t^3)^i (1+t)^(2^b) with s = (M - 3i - 2^b)/2, i even, i + 2^b >= 4R, 3i + 2^b <= M are independent since 3R - 1 <= 2^b <= 6R - 4 forces a unique b, and number tent(dim), so nullity_2 >= tent(dim) with troughs exactly where 3R is adjacent to a power of 2; the reversal involution on the t^3-chain of the single generator gives nullity_even = ceil(nullity_full/2) exactly; the staircase submatrix (rows c' = R - j, leftmost pivots at R - 1 - 3j) yields only floor((R-1)/3) + 1 independent rows, the wrong third of the rank; without the (1+t+t^2) hypothesis the valuation lemma yields only tent + 1 (witnesses R = 2, 4, 7); matrix nullity equals tent at 21 dim through 511 and at every odd dim = 3..401, the module identities to R = 1024. Witness: slice-sign-even-half.
2026-08-28 Proved Lemma M, sharp with equality: for every d >= 2 and r in {0, 1, 2}, the maximal (1+t)-valuation over nonzero H in F_2[t] with no exponent == r mod 3 and deg H <= d is mu_r(d) = max_(2^b <= d) [floor((d - s_0 - 2^b)/3) + 2^b], s_0 = (r + 2^b) mod 3, attained by t^(s_0)(1+t^3)^i(1+t)^(2^b) - the Frobenius split H = A^2 + t B^2 gives the exact case law v(H) = 2v(A) / 2v(B) / 2 min / 2w + 1 (the equal-valuation case forced by C' = B_1^2, a unit at 1), classes move r -> (2r, 2r + 1), the recursion M_r(d) <= Phi(M_(2r)(floor(d/2)), M_(2r+1)(floor((d-1)/2))) with Phi(X, Y) = max(2X, 2Y, 2 min(X, Y) + 1) has the closed form as supersolution by lifting the child's maximising Frobenius block b -> b + 1 (six integer inequalities, X = Y forcing b_e = b_o by a numerator gap >= 2^(min+1) - 3, finite windows d = 5..12 with 24 evaluations, 8 tight, and base cases d = 2..4); the corollary Lemma M' for (1+t+t^2) | H is M'_r(d) = M_r(d - 3) + 1 for d >= 5, the cheapest purchase of valuation being one Frobenius block plus (1+t^3) padding at exchange rate 3:1, which is where sup nullity/n = 1/3 comes from; brute-forced to d <= 16000 (failure set exactly the four d < 2 pairs), the upper bound certified independently by full rank of Lucas submask matrices at d = 1023..8193, the supersolution tight at 4926 points up to 2^60 with minimum slack 0, the case law exact on all H < 2^17, and the true minimum of the module Y computed at 204 R up to 1025. Witness: slice-sign-even-half.
2026-08-28 Proved The one-class window lemma, by the parity of an index: Y = Z ∩ G F_2[t] is F_2[t^3]-free of rank 2 with generator degrees delta_1 < delta_2 in distinct classes mod 3 and delta_1 + delta_2 = 12R + 5 exactly - truncation counting gives dim_(F_2) Z/Y = (delta_1 + delta_2 - 2)/3, the projection onto the missing exponent class identifies the cokernel of Z -> F_2[t]/(G) with F_2[u]/gcd(A_0, A_1, u A_2) of dimension exactly 1 (since (1+t^3) | G but (1+t^3)^2 does not), so dim Z/Y = deg G - 1 = 4R + 1; the sum is odd, so delta_1 <= 6R + 2 = M < delta_2 in two lines, margins 0 and 2 impossible and margin 1 iff delta_1 = M; explicitly {delta_1, delta_2} = {12R - 2A + 2[a even], 2A + 3 + 2[a odd]} with a = floor(log_2(4R - 1)), A = 2^a, from A + 2 <= 4R <= 2A; hence nullity_2(M_full)(dim) = floor((M - delta_1)/3) + 1 and nullity_2(M_even)(dim) = ceil(nullity_full/2) in closed form for every odd dim; the identity generalises as delta_1 + delta_2 = 3(deg G - deg_u gcd) + 2 at 400 random G, the closed form holds to R = 200000, margins below 5 lie in {1, 3, 4} exactly as parity predicts, R = 683 = J(11) has delta_1 = M (margin 1) and R = 1365 = J(12) margin 3. Witness: slice-sign-even-half.
2026-08-28 Proved The Jacobsthal tent rank law, entire: for every odd dim = 2R + 1 >= 3, base 3, middle-digit design, nullity_2(M_even)(dim) = tent(dim) = 1 + dist(R, {J(a), J(a) + 1}) with a = floor(log_2(4R - 1)), and sharply nullity_2(M_even) <= ceil(n/3), n = (dim+1)/2, with equality exactly at dim in {3} ∪ {2^(2j) + 1}; with m = R - J(a) all four (parity of a) x (branch) cells reduce to nullity_full = 1 - 2m (m <= 0) or 2m (m >= 1), the parity of a cancelling completely, and halving gives nullity_even = 1 + d_a(R); the nearest-trough index is a itself (an a - 1 reading was rejected exhaustively), margins exactly 1 at both window endpoints propagate by 1-Lipschitzness, R = 1 is the sole reason the cap is ceil rather than floor, and the odd-a peaks miss by exactly 1; closed form equals tent equals the real transfer-matrix nullity at every odd dim = 3..1401, closed-form checks to R = 500000 with points to 2^60 (argmin strictly unique everywhere), no residue family past the cap (max excess 0); so v_2(det M_even) = tent(dim) + X(dim) with the tent capped at ceil(n/3), and base-3 strictness rides on the cascade layers X(dim) alone. Witness: slice-sign-even-half.

The tile monoid

2026-08-31 Proved Factorisation of a 0/1 tile is unique once the ordered side profile is named: if A (x) B = A' (x) B' with A, A' of side m and B, B' of side n, all non-empty, then cutting the composite into an m x m array of n x n blocks reads A off as the 0/1 indicator of the non-zero blocks and B as any one of them, since every non-zero block equals B and B is not the zero tile, so A = A' and B = B'; the non-empty tiles under the Kronecker product are therefore a monoid graded by side, cancellative and atomic, whose block test decides factorability at a named shape in O(N^2) of exact integer comparison, the 0/1 hypothesis being load-bearing since over the rationals A (x) B = (kA) (x) (k^-1 B), and the block reading being the 0/1 shadow of the Van Loan and Pitsianis rearrangement. This is a rediscovery and is cited, not claimed: it is Lemma 2.4 and section 2 of Voet and De Novellis, Identifying Kronecker product factorizations, arXiv:2510.25292, for binary matrices under equality. Witness: lab/rs/code-factorisation, magic.md, arXiv:2510.25292.
2026-08-31 Proved The ordered side profile is not recoverable from the composite, so the tile monoid has no unique factorisation, and the failure is not axis-separable: [6]{(0,0),(2,2)} is both c1 (x) c257.base3 and c17.base3 (x) c1, all four letters of prime side and hence irreducible with differing multisets, and the tile is not a rectangle; the mechanism is an infinite family rather than a side-6 accident, since I_m (x) I_n = I_mn = I_n (x) I_m and E_m (x) E_n = E_mn = E_n (x) E_m at every pair of sides by the symmetry of (nm - 1) - x = (n - 1 - i) m + (m - 1 - j) in m and n, the same digit identity the diagonal action already carries, so taking m and n distinct primes gives four irreducible letters at every side with two distinct prime factors; the existence of shape-distinct factorisations is Example 2.5 of Voet and De Novellis, and new here are the axis-separable refutation and the I/E family. Witness: lab/rs/code-factorisation, magic.md, arXiv:2510.25292.
2026-08-31 Proved Neither the length nor the side multiset of a factorisation is an invariant of the composite, first at side 12 and at no smaller side: [12]{(0,0),(3,3)} reads as three irreducible letters of sides 2, 2, 3 and as two of sides 3, 4, the side-4 letter [4]{(0,0),(3,3)} being irreducible because its one candidate cut has two unequal blocks, while every side below 12 is a prime power, where factorisation is unique, or a product of two distinct primes, where every letter has prime side; the short reading needs a letter of composite side, which no plane code is, so length and the side multiset are invariants for free inside the magic-word submonoid generated by prime-side letters and fail in the full monoid, and the alphabet of the words is not the alphabet of the monoid, 65310 of the 65535 non-empty side-4 tiles being irreducible already and the reducible share falling from 0.343328% at side 4 to 0.0000221% at side 6; a factorisation whose sizes are not all prime is Example 2.6 of Voet and De Novellis, and new here are the minimality of side 12 and the alphabet gap. Witness: lab/rs/code-factorisation, magic.md, arXiv:2510.25292.
2026-08-31 Proved Two factorisations of one tile admit a common refinement exactly when the union of their cut chains is totally ordered by divisibility, because a cut at d' dividing a cut at d factors the side-d left factor through the side-d' one; so unique factorisation holds at every prime-power side, where the divisors are a chain, and fails exactly when the cut set L(C) holds two incomparable divisors, checked with zero mismatches against direct enumeration of every irreducible factorisation over all 339795 side-12 plane-code composites, of which 7023 carry two or more factorisations and 2376 carry factorisations of unequal length. Witness: lab/rs/code-factorisation, magic.md.
2026-08-31 Proved The tile monoid is not a trace monoid, so no canonicalisation may sort or commute letters: [2]{(0,0)} (x) [3]{(1,1)} = [6]{(1,1)} = [3]{(0,0)} (x) [2]{(1,1)} uses four pairwise distinct irreducible letters, which no commutation of a letter pair can produce, and only 11 of the 171 side-6 cross-shape tiles are honest commutations against 160 rewritings. Witness: lab/rs/code-factorisation, magic.md.
2026-08-31 Proved Two letters render one tile at one side only at side 3: if a base-2 code and a base-3 code agree cell for cell at one side then row r equals row r' whenever r = r' mod 2 or mod 3, and at side >= 4 those two partitions join the whole row range, so every row and every column agrees and a non-empty constant tile is the full tile; the census is 480 pairs at side 2, 15 at side 3 and the full tile alone at sides 4, 5, 6, 7, 8, 9, 12 and 18, the carpet's side-3 partner is uniquely c495 of fill 8, and at side 9 the readings separate into fills 65, 72 and 64 on three pairwise distinct tiles, though read as level-2 fractals of the side-3 letter they do not diverge at all, since at side 3 they are one tile. Witness: lab/rs/code-factorisation, magic.md.
2026-08-31 Proved The canonical name of a composite is the code together with its ordered side profile, c<code>(side_1 x side_2 x ... x side_level), and the two non-injectivities are different objects that must be disambiguated in order: the render collision belongs to the alphabet alone and tabulates once per base and side, since fixed-shape uniqueness proves the fold never creates one, while the fold collision belongs to the profile; profiles of different length occur, so the tie-break orders profiles by length first, finest before coarsest, then lexicographically, before the diagonal-action class rep breaks what is left. Witness: lab/rs/code-factorisation, magic.md.
2026-08-31 Verified The side-6 census: the two shape images are injective at 7665 tiles each, 171 tiles lie in both, so 15159 of the 2^36 - 1 side-6 tiles are reducible once the overlap is removed and 68719461576 are irreducible; of the 171, 121 are axis-separable and 50 are not, 11 are commutations and 160 rewritings, fills run 1:36 2:64 3:32 4:16 6:14 12:8 36:1 over the 171 and 2:16 3:32 6:2 over the 50 with outer-fill signature (1,1):24 (1,2):8 (1,3):16 (2,3):2, and the 48 that are neither separable nor commuting are exactly the 48 carrying a one-cell letter in at least one reading and exactly the 48 carrying a one-cell outer factor in at least one reading, three statistics on one set checked as sets rather than as counts, since the two one-cell readings differ elsewhere (0:23 1:8 2:140 against 0:23 1:60 2:88 over the 171). Witness: lab/rs/code-factorisation, magic.md.
2026-08-31 Verified 121 = 11 x 11 is arithmetic with a checked bijection: the 121 axis-separable side-6 cross-shape tiles are exactly the products R x C of the 11 lines that factor in both radix orders, {0} {1} {0,1} {2} {0,2} {3} {4} {5} {3,5} {4,5} {0..5}, verified as set equality and not as a count. Witness: lab/rs/code-factorisation, magic.md.
2026-08-31 Verified Counting reducible tiles at prime-power side is inclusion-exclusion over the divisor chain, equivalently the series I = T/(1+T) on the grading, giving 225 at side 4, 1962675 at side 8, 261121 at side 9, 553402322215537199175 at side 16 and (2^25 - 1)^2 = 1125899839733761 at side 25, cross-validated in one dimension against exhaustive brute force at N = 4, 8, 16, 9 reading 9, 63, 1431, 49; nothing new happens at a prime-power side, where the two side-8 shape images of 983025 tiles each meet in exactly the 3375 triple products of base-2 codes, checked as set equality, so 3375 is pure associativity and never stands beside 171. Witness: lab/rs/code-factorisation, magic.md.
2026-08-31 Proved One-cell letters commute exactly when a(n - 1) = b(m - 1), giving gcd(m - 1, n - 1) + 1 singleton pairs per axis and, where no common power exists, gcd(m - 1, n - 1) + 2 commuting pairs in one dimension, checked at nine side pairs and exceeded only at (3,9) at 7 against 4 through the common-power branch; at base 2 against base 3 this gives the 11 commuting code pairs (1,1) (2,4) (3,7) (4,64) (5,73) (6,84) (8,256) (9,273) (10,292) (12,448) (15,511), nine of them a commuting row line against a commuting column line and the other two the diagonal and the antidiagonal, with base-2 codes 7, 11, 13, 14 unpartnered, so the carpet code itself does not commute. Witness: lab/rs/code-factorisation, magic.md.
2026-08-31 Conjecture The cut set L(C) is closed under gcd, with zero failures over every non-empty subset of a line at N = 1..20 and over all 339795 side-12 plane-code composites and no proof; it is the one missing structural fact, since with it L(C) is a meet-subsemilattice of the divisor lattice and the canonical name closes, and without it there is no counting theorem at non-prime-power side, where 171 and 15159 are enumeration rather than formula. Witness: lab/rs/code-factorisation, magic.md.
2026-08-31 Conjecture Two tiles commute under the Kronecker product exactly when they are powers of one common tile or the members at their two sides of one scale-free family, the one-cell case being settled by a(n - 1) = b(m - 1) and the general case tested only at (2,3) in two dimensions and at ten side pairs in one; relatedly, whether the diagonal and the antidiagonal are the only permutation tiles factoring in both radix orders at every coprime split. The next coprime test needs all 2^25 side-5 codes, so this has to be settled by proof and not by search. Witness: lab/rs/code-factorisation, magic.md.
2026-08-31 Conjecture The three-family description of cross-shape collisions at a coprime shape - axis-separable rectangles, tiles with a fill-1 outer factor, and the diagonal pair - is exhaustive at side 6 and untested anywhere else. Witness: lab/rs/code-factorisation, magic.md.
2026-08-31 Conjecture An intrinsic description of which tiles are Kronecker products, rather than the block test's algorithm and the published decomposition graph; and what the irreducible letters of composite side do, now that they are known to exist and to be generic, which is the question the plane-code word census could not see. Witness: lab/rs/code-factorisation, magic.md.
2026-08-31 Conjecture The diagonal embedding D -> {(x,x) : x in D} is an injective, divisor-closed embedding of the one-dimensional digit-set monoid into the tile monoid preserving cut sets and irreducibility, and a diagonal tile is axis-separable only at one cell; so a line sweep is a tile sweep, the whole non-uniqueness phenomenon already lives in one dimension, and it lifts to tiles no rectangle can explain. Witness: lab/rs/code-factorisation.
2026-08-31 Refuted That every shape-distinct factorisation of a tile is axis-separable, which would have closed the question with a one-line lemma - [6]{(0,0),(2,2)} factors as c1 (x) c257.q3 and as c17.q3 (x) c1 with four irreducible letters and is not a rectangle, separability covering 121 of the 171 side-6 cross-shape tiles and none of the diagonal family, and the diagonal and antidiagonal families put a non-separable witness at every side with two distinct prime factors, so the door stays open. Witness: lab/rs/code-factorisation, magic.md.
2026-08-31 Refuted That the mechanism of a shape-distinct factorisation is always a side-6 cross-shape collision sitting inside the word - the side-12 tile [12]{(0,0),(3,3)} has cut set {1,2,3,4,12} with no cut at 6, so no side-6 collision sits inside it, and its two readings differ in length; the earlier statement was read off a sample of words over plane codes and is a property of that universe, not a law. Witness: lab/rs/code-factorisation, magic.md.
2026-08-31 Refuted That the length and the side multiset of a factorisation are invariants of the composite - true inside the magic-word submonoid, where every plane code has prime side and every word over it has length exactly the number of prime factors of the side, and false in the full tile monoid, first at side 12, where [12]{(0,0),(3,3)} reads at lengths 3 and 2; the plane-code census could not have found the witness, since the short reading needs the irreducible side-4 letter [4]{(0,0),(3,3)}, which is no plane code. Witness: lab/rs/code-factorisation, magic.md.
2026-08-31 Refuted That the cut set L(C) is closed under lcm, and with it the naive reading that any two factorisations refine to a common one - the line {0,3} at N = 12 has L = {1,2,3,4,12}, holding 2 and 3 and not 6, and the first failure by mask order at that side is the line {1,2} with the same cut set; 132 failures over every line to N = 20 and 2376 over the 339795 side-12 plane-code composites, against zero failures of gcd closure in both sweeps. Witness: lab/rs/code-factorisation, magic.md.
2026-08-31 Refuted That the commuting pairs are the four corner cells plus the scale-free families of row, column, full tile, diagonal and antidiagonal - the cells [3]{(1,1)} and [5]{(2,2)} commute at side 15, both readings giving [15]{(7,7)}, and neither is a corner cell nor a member of any of those families; the four-corner picture is an artifact of gcd(1,2) = 1 at sides (2,3), the correct criterion for one-cell letters being a(n - 1) = b(m - 1). Witness: lab/rs/code-factorisation, magic.md.
2026-08-31 Refuted That the reachable literature cannot reach the tile factorisation question, an earlier positioning against graph products - the isomorphism-versus-equality gap is real for graph products, but Voet and De Novellis, arXiv:2510.25292, is a binary-matrix paper working under equality that already contains fixed-shape uniqueness, the prime vocabulary, the shape-distinct factorisation, the non-prime factor sizes and a decomposition graph enumerating every factorisation; the Proved core recorded above is a rediscovery, and the single verbatim quotation the old positioning rested on could not be recovered from its source and is withdrawn rather than carried. Witness: magic.md, arXiv:2510.25292.
2026-08-31 Refuted The annotation 1125899839733761 = 65535^2 in a draft of the prime-power counts - the integer is right and its name is wrong, 1125899839733761 = (2^25 - 1)^2 = 33554431^2 at side 25, while 65535^2 = 4294836225 is a side-16 term; transcription, not mathematics, and the generator now prints the identity beside the value. Witness: lab/rs/code-factorisation.

The zeros of the design zeta

2026-09-07 Verified This specific infinite design zeta has zeros in its own half-plane of absolute convergence, which the integers forbid, and the claim is the object and not the principle, since the positive-term Dirichlet series 1 + 2^(-s) has abscissa of absolute convergence -infinity and zeros at (2m+1) pi i/log 2. The census counts zeros of the Lyndon cofactor Z(s) = zeta_F(s)(1 - fill base^(-s)), analytic on Re s > alpha - 1, so it needs no pole-free strip and leaves no sliver against the pole line, on the single strip alpha - 0.92 < Re s < alpha + 3.02, 0.02 < Im s < 60, split at Re s = alpha exactly. Base 3 F = {0,1} at alpha = log_3 2 carries 3 zeros right of the abscissa and 20 left of it inside that strip; base 10 missing the digit 9 at alpha = log_10 9 carries 13 right and 25 left; base 3 F = {0,2} carries 3 right, in the same three boxes as {0,1}. The largest surviving phase step on any census contour is 0.9896 and the largest propagated bound met at any census evaluation is 9.99e-11, both printed beside every count. The base 2 full digit set is the control on both sides and each side names its object: zeros of zeta in alpha + 0.02 < Re s < alpha + 3.02 count 0, which is what the Euler product forbids, computed and not quoted; zeros of zeta in alpha - 0.98 < Re s < alpha - 0.02 count 13, the first thirteen below Im s = 60; and the teeth of the cofactor 1 - 2 base^(-s), which sit exactly ON Re s = alpha and are not zeros of zeta, bring the one-strip count to 19 = 13 + 6 with 6 = floor(60 log 2/2 pi). Right of the abscissa no continuation is used, since the positive series converges absolutely there and the ladder only rearranges it. The count is resolved and not certified: the largest surviving phase step is printed and nothing bounds zeta_F'/zeta_F on the contour, so a zero pair closer than the surviving spacing would stay invisible. Witness: lab/py/design-zeta.
2026-09-07 Proved Scaled digit columns share a zero set exactly: for a positive integer a with a max F <= base - 1, so that aF stays inside {0..base-1}, the carry-free bijection m -> a m gives zeta_(aF)(s) = a^(-s) zeta_F(s), an exponential factor with no zeros and no poles, so zeta_(aF) and zeta_F have the same zeros and residues in the ratio a^(-s_(m,j)), and the proof uses 0 in F nowhere. Base 3 {0,2} against {0,1} agrees to 5.6e-43 at three points, and on what was censused, the strip alpha < Re s < alpha + 3.02, 0.02 < Im s < 60, the two censuses coincide box for box: winding one in Im [22.01, 24.01], in Im [28.01, 30.01] and in Im [56.00, 58.00] for both, and zero in every other box. Left of the abscissa {0,2} is not censused and is inferred from the theorem. On the Mobius side the same bijection twists the meter by a sign, so the transfer is exact on both faces and trivial on one of them. Witness: lab/py/design-zeta, mobius.md.
2026-09-07 Proved The Euler-product bridge between the two faces of RH is absent on a design: zeta M = 1 on the full set, S_F is not multiplicatively closed for any proper F, and zeta_F M_F is not 1, so no known route runs from a zero of zeta_F to theta(F) and the zero census carries no bound on the square-root conjecture. What survives is not the zeros but the position product: G_level pairs against a = 1 and a = mu alike and the arithmetic sits entirely in the kernel. Coons 2010 Theorem 2.3 rules out the automatic-continuation route to M_F and nothing wider. Witness: mobius.md, lab/py/design-zeta, lab/py/mrly-euler, REFS.md.
2026-09-07 Proved The zeros of the design zeta are read off one analytic function and their positions near the pole lattice are forced by the residues. The Lyndon cofactor Z(s) = zeta_F(s)(1 - fill base^(-s)) is analytic on Re s > alpha - 1, since 1 - fill base^(-s) cancels exactly the m = 0 line of the digit recursion's poles and no other; the poles of Z are those s_(m,j) = alpha - m + 2 pi i j/log base with m >= 1 at which zeta_F has a nonvanishing residue, the nearest line to that half-plane being Re s = alpha - 1 with residue -s_(1,j) gamma_1 r_j/fill, and on a full digit set Z has no pole at all, being zeta(s)(1 - base^(1-s)), entire. One peel level gives Z(s) = E_1(s) + sum_(l >= 1) binom(-s,l) base^(-s-l) gamma_l zeta_F(s+l) with E_1(s) = sum_(a in F, a != 0) a^(-s) and gamma_l = sum_(a in F) a^l, checked against brute-force digit summation to 1.6e-14 with and without 0 in F, and Z(s) a_min^s -> 1 to the right. A zero of zeta_F right of alpha - 1 is always a zero of Z; conversely a zero of Z is a zero of zeta_F except at a pole s_(0,j) with r_j = 0, where Z vanishes and zeta_F is regular. At s_(0,j) one has fill base^(-s_(0,j)) = 1 exactly for every j, so with u = s - s_(0,j) the periodic factor is 1 - base^(-u) with no j dependence and zeta_F(s) = Z(s)(1/(L u) + 1/2 + L u/12 - L^3 u^3/720 + ...), L = log base, giving residue Z_0/L, regular part Z_1/L + Z_0/2 and its derivative Z_2/L + Z_1/2 + Z_0 L/12 from the Taylor coefficients of Z alone, on a disc of radius at least 1 and exactly 1 when r_j does not vanish. Witness: lab/py/zeta-locus, lab/py/design-zeta, lab/py/burnol-residue.
2026-09-07 Verified The zeros of the design zeta near the abscissa are a residue comb whose tooth position the residue and the regular part predict. A zero near the pole s_(0,j) solves u(R_j + R'_j u + ...) = -r_j, first order u_1 = -r_j/R_j and second order the near root of R'_j u^2 + R_j u + r_j = 0, both built from the Laurent data with nothing fitted. Over 20 designs to Im s = 40 (every scaling class at base = 3 and base = 4, two at base = 5, base 9 {0,1,2}, base 16 {0,1,2,3}, base 10 missing 9, base 2 full set) one assignment radius 0.45, fixed by the discs not overlapping and not by the tooth law so that every count is conditional on it, serves both the count and the tooth, discs never overlapping since the smallest period in the sweep is 2.2662: the argument principle on that circle gives 164 poles carrying one zero of Z, 40 none and 8 two, of which 21 are the residue-null pole centres of the three full-set columns and are zeros of Z that are not zeros of zeta_F, leaving 143 poles with one zero of zeta_F, 61 with none and 8 with two. All 151 poles carrying a zero have their zeros located by a polar grid inside that same disc and not by the prediction, so no tooth is selected by the law it tests and no pole carrying a zero is left without one. Comparing prediction to tooth afterwards, miss2/miss1 has median 0.1637 with miss2 < miss1 at 147 of the 151, and the accuracy is conditional on the tooth being close: the 43 teeth at abs(u) < 0.1 have largest first-order miss 0.01446 and largest second-order miss 0.00164, the 84 at abs(u) < 0.2 have 0.10815 and 0.01526, while the 31 at abs(u) >= 0.3 reach 1.64614 and the prediction says nothing. The densest column is the sharpest: base 10 missing 9 at fill/base = 0.9 locates 15 teeth to a largest first-order miss of 0.013602 and a median of 0.000841. Witness: lab/py/zeta-locus.
2026-09-07 Verified The critical line is the second family of the full digit set. On a full digit set zeta_F is zeta, whose only pole is s = 1 = alpha, so it is regular at every s_(0,j) with j != 0 and the residue there vanishes as a one-line consequence rather than a measurement; the machinery reads those residues as 1e-26 to 1e-33, which is a control of the engine, and the comb is empty. The winding of the cofactor over alpha - 0.92 < Re s < alpha + 3.02, 0.02 < Im s < 40 then splits exactly as six zeros of zeta plus floor(40 log base/2 pi) cofactor-only teeth, those teeth being the zeros of 1 - base^(1-s) on Re s = 1 by exact arithmetic: 10 = 6 + 4 at base = 2, 12 = 6 + 6 at base = 3 and 14 = 6 + 8 at base = 4. The six survivors read Re s = 0.5 at Im s = 14.1347251417, 21.0220396388, 25.0108575801, 30.4248761259, 32.9350615877, 37.5861781588 at all three bases, and the three columns share that zero set to 1e-26 because they are one arithmetic object. On a design the same split leaves a second family that is not a line at alpha/2: its real parts run -0.273079611 to 0.391038600 over the 7 zeros below Im 40 at base 3 {0,1} against alpha/2 = 0.3154648768, -0.30495894 to 0.28101268 over 6 zeros at base 4 {0,1} against 0.25, and 0.060261843 to 0.97363028 over 5 zeros at base 16 {0,1,2,3} against 0.25, the spread being the witness and no per-design mean claimed. Witness: lab/py/zeta-locus.
2026-09-07 Proved The second family of the design zeta does not depend on which comb is stripped, and the next pole line's comb is computed from the first one's residues. For m >= 1 the cofactor Z_m(s) = zeta_F(s) prod_(i <= m)(1 - fill base^(-(s+i))) has exactly the zeros of Z(s) = zeta_F(s)(1 - fill base^(-s)) inside alpha - 1 < Re s < alpha + 3.02, since each extra factor vanishes only on Re s = alpha - i for i >= 1, so the survivors of the assignment are one set under every comb. What Z_m adds is the level-i comb, and its Laurent data is forced by the level-zero data: Z(s) = E_1(s) + sum_(l >= 1) binom(-s,l) base^(-s-l) gamma_l zeta_F(s+l) is singular at s_(1,j) = alpha - 1 + 2 pi i j/log base through its l = 1 term alone, and with base^(-s_(1,j)-1) = base^(-s_(0,j)) = 1/fill and 1 - fill base^(-s_(1,j)) = 1 - base the residue of zeta_F there is r_(1,j) = s_(1,j) gamma_1 r_(0,j)/(fill(base-1)), so the level-one comb is empty wherever the level-zero comb is, and at the full digit set s_(1,0) = alpha - 1 = 0 makes it vanish, which is zeta having no pole at s = 0; the lab prints abs r_(1,0) = 0.0 with the null flag set and abs r_(1,1) = 8.89623e-29 at the base 2 full set. Witness: lab/py/zeta-family, lab/py/zeta-locus, lab/py/design-zeta.
2026-09-07 Verified The second family of the design zeta, split out and censused over twenty-two designs at a stated assignment radius, with no gap at that radius on any design. Stripping the level-zero and level-one combs at rho = 0.45, a constant fixed only by the pole discs not overlapping and not by the tooth law, which is accurate only inside abs(u) < 0.2, the twenty designs of the locus sweep plus base 5 {0,1,2,3} and base 10 missing two digits give 377 zeros wound by the argument principle, 351 located, 171 teeth of which 9 are level-one teeth, 19 cofactor-only zeros at null-residue poles and 161 second-family zeros, each design censused to its own printed height, 40 except the four base 3 designs at 42.894, base 9 {0,1,2} at 41.464 and base 10 missing two at 25.923. There is no gap at rho on a design: the distance from a second-family zero to the nearest live pole has minimum 0.45510938 at base 4 {2,3}, 0.45909168 at base 3 {0,1}, 0.48696667 at base 4 {0,1,2} and 0.50481072 at base 4 {1,3}, with base 4 {2,3} putting five of its eight inside 0.45 < abs(u) < 0.6, so every count is conditional on rho and falls as rho rises, N_2 reading 8, 7, 13, 9, 14 at rho = 0.45 against 7, 6, 12, 8, 7 at rho = 0.6 on base 3 {0,1} and the four base 4 two-digit designs. The full digit set is where the gap exists: at base 2 the nearest live pole to a second-family zero is 14.143566 away and no radius below 0.9 moves any count. Where the located count falls short of the winding, base 4 {2,3} at 12 of 18 being the worst, N_2 is a lower bound. Witness: lab/py/zeta-family verb tests.
2026-09-07 Verified Seventeen designs carry a lower bound on the Mertens exponent of their own Mobius, and the strongest bound is radius-robust. Nineteen of the twenty-two designs have a censused zero of zeta_F strictly right of alpha, twelve of them in the second family, and at the seventeen of them whose digit set contains 1, so that nu_F exists, the transport theorem gives that sum_(n <= x) nu_F(n) is not O(x^(Re rho - eps)); base 4 {2,3} and {0,2,3} omit the digit 1 and carry a zero but no nu_F. Base 10 missing two digits has a zero at 1.00151438765 + 2.77402670058 i against alpha = 0.903089987, a second base-10 column where the design's own Mobius has a Mertens exponent above 1 and so above x itself; that zero is a level-zero tooth at abs(u) = 0.1083 of the j = 1 pole, deep inside every assignment radius tested, so the bound does not depend on where the comb is cut. Base 4 {1,2} has a second-family zero at 0.940012431696 + 13.0678968771 i against alpha = 1/2, an exponent of 0.94 against a design mass exponent of 0.5, and base 3 {0,1} reads 0.720787601477 at Im 28.6056765649 against alpha = 0.630929754. Witness: lab/py/zeta-family verb tests, lab/py/mrly-pairing verb inverse.
2026-09-07 Verified What converges as a design fills is the ordinate set and not the real part. Against the derived null of a quarter of the mean gap between consecutive zeta ordinates in the range, the exact expectation for an equally spaced ordinate set of the same density and conservative for one with gap variance, the mean distance from a second-family ordinate to the nearest zeta ordinate divided by that null falls monotonically in alpha: 2.0495374 at base 5 {0,1} with alpha = 0.430676558, 1.8953371 at base 4 {0,1} with 0.5, 0.75419266 at base 3 {0,1} with 0.630929754, 0.51648744 at base 4 {0,1,2} with 0.792481250, 0.32356636 at base 5 {0,1,2,3} with 0.861353116, 0.090501352 at base 10 missing two with 0.903089987 and 1.0429899e-23 at the base 2 full set. The base and fill confounds are dead: the fall is monotone at fixed base, 2.0495374 to 0.32356636 inside base 5 and 1.8953371 to 0.51648744 inside base 4, and at fixed fill = 2 across bases, 2.0495374, 1.8953371, 0.75419266, 1.0429899e-23 at alpha = 0.430676558, 0.5, 0.630929754, 1; the nulls move only 1.0425839 to 1.3595166 across the ladder while the raw mean distance falls 2.4032315 to 0.12303809, so the denominator does not drive it. Over the same designs mean abs(Re s - 1/2) reads 0.36482392, 0.39426128, 0.3901396, 0.25540269, 0.31452367, 0.20473972 and 2.4065966e-23 and does not fall monotonically, so at alpha = 0.903 the heights are pinned to 2.3 percent of the mean gap while the real parts are still 0.20 off 1/2. alpha is a trend and not a function: the four base 4 two-digit designs at one alpha = 1/2 spread 0.79050661 to 2.8404536. The matching is nearest-ordinate and not injective, 3 distinct ordinates for 4 design zeros at base 10 missing two. Witness: lab/py/zeta-family verb limit.
2026-09-07 Proved The ordinate shadow is a first-order perturbation and its constant-free form is a Newton step from the zeta zero. The discrete position identity 1_(D_level)(n) = base^(-level) sum_(a mod base^level) G_level(a/base^level) e(-n a/base^level) on 0 <= n < base^level gives zeta_(F,level)(s) = base^(-level) sum_(a mod base^level) G_level(a/base^level) S_level(s, a/base^level) with S_level(s,x) = sum_(1 <= n < base^level) e(-nx) n^(-s), reproduced from the transform to 8.326e-40 at level = 2 on ten designs, and since G_level(0) = fill^level the a = 0 fibre carries the weight (fill/base)^level exactly against the partial sum of zeta to base^level, with no arc and no limit. That identity splits the polynomial at level level against a TRUNCATED zeta while the object is the continued zeta_F against the full zeta, and (fill/base)^level falls to 0 with level while both series tend to 1 on the right, so no level is forced and c = fill/base is the level = 1 reading and a definition. For any constant c the split zeta_F = c zeta + E_F gives E_F(rho_0) = zeta_F(rho_0) at a zero rho_0 of zeta, an identity carrying no information about c, and a first-order zero of zeta_F at rho_0 - zeta_F(rho_0)/(c zeta'(rho_0)); reading c zeta'(rho_0) as zeta_F'(rho_0) removes the constant and gives rho_0 - zeta_F(rho_0)/zeta_F'(rho_0), Taylor at a simple zero of zeta_F. The offset is one complex number, so at the zeros this law pairs the ordinate offset and the real-part offset are one quantity. The continuous form, the mass of G_level on abs(t) < 1/(2 base^level), is the exact sinc sum 1/base^level + sum_(n in D_level, n > 0) sin(pi n/base^level)/(pi n) and equals kappa_level(F) (fill/base)^level with kappa_level running 0.6015221 to 0.96774464 over the ladder at level = 1, 2, 3, so it adds no constant the fibre does not give. Witness: lab/py/zeta-shadow verb mass, lab/py/mrly-euler verb position.
2026-09-07 Verified The constant-free first-order step predicts the design zero attached to each zeta zero, and it sharpens as the offset shrinks. Over nine designs at twelve zeta zeros to Im s = 56.4462476971, six to Im s = 37.5861781588 at the two densest so the rungs do not share one height, both predictions are computed from zeta_F(rho_0), zeta_F'(rho_0), zeta'(rho_0) and the digit density alone and the zero is located afterwards by Newton from rho_0, accepted only at abs(zeta_F) < 1e-16, within 1.5 of rho_0 and 0.02 clear of the pole lattice, largest ladder bound 9.001e-23. The step's median ratio reads 1.3843088, 1.284225, 1.2481449, 1.2060106, 1.2042502, 0.89075541, 1.0195598, 1.005076, 0.99741809 at alpha = 0.430676558, 0.5, 0.630929754, 0.792481250, 0.861353116, 0.903089987, 0.954242509, 0.982877878, 0.994835739, with largest abs(ratio - 1) 0.14041 at base 20 missing one digit and 0.01734 at base 50 missing one digit, bands [0.94875, 1.14041] and [0.98266, 1.01144]; pooled over the ladder that largest deviation runs 0.01734, 0.0508884, 0.193158, 0.83912, 3.32327 over the buckets abs off < 0.05, < 0.1, < 0.2, < 0.4 and above, on 7, 4, 14, 18, 44 zeros. The level = 1 reading c = fill/base is the looser column, median ratio 1.4129353, 1.2842149, 1.0955991, 1.1806066, 1.1372453, 1.276577, 1.1347487, 1.0320127, 1.0507079 with largest abs(ratio - 1) 0.24964 and 0.0821168 at the two dense rungs, five times looser than the step at base 50, and the coupling zeta_F'(rho_0)/zeta'(rho_0) does not select it either, median abs(coupling - fill/base) reading 0.24057225 and 0.08291158 there against median abs(coupling - 1) 0.27619434 and 0.079335871, a flip between the two rungs while the candidates differ only by 0.05 and 0.02. Nine zeros at the three sparsest designs have no located zero inside the trust region, predicted offsets 0.95618855 to 3.0967393, so those rungs' medians are conditioned on Newton succeeding. The base 2 full set is the exact control, abs(zeta_F(rho_0)) between 1.85e-34 and 1.329e-25 at all twelve zeros, so E_F = 0 and both offsets are 0. Witness: lab/py/zeta-shadow verb predict.
2026-09-07 Verified The paired shadow offset carries its exponent in the missing-digit density rather than in 1 - alpha, and two new rungs sample the interval between alpha = 0.954 and 1. The median paired offset divided by m/base = 1 - fill/base reads 1.6463532, 1.2495026, 1.8345578, 1.5731321, 2.2102406, 1.6634381, 2.2424916, 1.8779239, 1.051349 across the nine rungs and divided by 1 - alpha reads 1.7350628, 1.2495026, 1.6569184, 1.8951686, 3.1883019, 3.432954, 4.9008186, 5.4839111, 4.0716338; a least squares in the logs, a fit and not a theorem, gives (m/base)^1.04544 at R2 0.957842 against (1-alpha)^0.71691 at R2 0.944011, the first column spanning 2.13297 and the second 4.38888, so m/base carries the exponent by a factor of 2.05764 inside the 4.28797 that (1-alpha)/(m/base) itself spans over this ladder, which is the whole discrimination the two normalisations admit here. The new rungs are base 20 missing its top digit at alpha = 0.9828778777 and base 50 missing its top digit at alpha = 0.9948357391, all six zeros located at each, median abs(E_F(rho_0)) 0.11830158 and 0.028066806 and median offset 0.093896196 and 0.021026979, so the PAIRED offset falls fast across that interval; this bounds no maximum over the whole second family and touches no jump clause, since the pairing selects zeros for closeness to a zeta zero and censuses nothing. Read in the form of the family row, the mean distance from a located design ordinate to the nearest zeta ordinate over a quarter of the mean gap between consecutive zeta ordinates in the range gives 0.81218635, 0.57141859, 0.50488757, 0.37447728, 0.20954319, 0.29197634, 0.14794812, 0.052888241, 0.011098646 and 0 at the full set; the pairing is zeta-zero-first where the family row's is design-zero-first, so this is a parallel ladder and not that row recomputed. Witness: lab/py/zeta-shadow verb rungs.
2026-09-07 Proved A positive Rouche margin proves exactly one zero of the design zeta in a disc about a pole, with every input bounded from the digit recursion itself. Write Z(s_0+u) = P(u) + T(u) at a pole s_0 = s_(0,j) with nonvanishing residue, where P(u) = (1 - base^(-u)) D_(P-1)(s_0+u) + E_P(s_0+u) is entire with Taylor coefficients the exact finite sums sum_n n^(-s_0)(-log n)^m/m! convolved against those of 1 - e^(-L u), and T is the l >= 1 part of the ladder numerator, bounded on abs(u) <= R_2 by B_T = sum_(l >= 1) binom(abs(s_0)+R_2+l-1, l) base^(-sigma-l) gamma_l G(sigma+l) at sigma = Re s_0 - R_2 with G the peeled majorant. That l sum is closed by a majorant ratio and not by an observed one, the term ratio itself not being monotone: since gamma_(l+1)/gamma_l <= a_max and G(sigma+l+1)/G(sigma+l) <= base^(-(P-1)) because every string in the pools is at least base^(P-1), the term ratio is at most R_l = ((abs(s_0)+R_2+l)/(l+1)) a_max base^(-P), which decreases in l once abs(s_0)+R_2 >= 1 and is below a_max base^(-P) otherwise, so stopping at the first l with R_l < 1 and adding term_l R_l/(1-R_l) is a proof. Then abs(Z_n) <= B_T/R_2^n for n >= 2 beyond the explicit part, so on abs(u) = rho one has abs(Z - (Z_0 + Z_1 u)) <= sum_(m >= 2) abs(P_m) rho^m + B_T tau^2/(1-tau) with tau = rho/R_2, while abs(Z_0 + Z_1 u) >= abs(Z_1) rho - abs(Z_0); when the first is strictly less than the second the linear model and Z have the same zero count in abs(u) < rho by Rouche, and that count is one because abs(Z_0/Z_1) < rho follows from the same inequality. Since the residue does not vanish, Z(s_0) != 0 and the zero is a zero of zeta_F. No step uses a differenced quantity: Z_0 is the ladder value with its propagated bound and Z_1 is the first Fourier mode of T on a circle of radius R < R_2 with N samples, whose aliasing is at most (B_T/R_2)(R/R_2)^N/(1-(R/R_2)^N), plus an exact p_1. The peel depth P and the radii rho and R_2 are free parameters of the proof. Witness: lab/py/zeta-locus, lab/py/design-zeta.
2026-09-07 Verified The residue comb carries exactly one zero of the design zeta at eleven certified poles, the peel depth is the lever that decides which, and the certificate fails at every pole carrying none or two. Running the Rouche margin with the peel depth raised at each pole until the certificate fires or the string pool caps, over 106 poles at base 3, base 5, base 9, base 16 and base 10 missing 9 to Im s = 40 inside a fifteen minute budget, gives 11 certified, 60 failed, 7 residue-null and excluded because there the model's zero is the pole centre, a zero of the cofactor that is not a zero of zeta_F, and 28 skipped when a design spent its budget. The certified eleven, with depth, margin and the radius the proof used: base 3 {0,1} j = 2 at P = 7, 0.13418242, rho = 0.205; j = 5 at P = 7, 0.028140545, rho = 0.16; j = 7 at P = 9, 0.00082974181, rho = 0.175; base 5 {0,1} j = 4 at P = 7, 0.15035818, rho = 0.2775; j = 5 at P = 7, 0.12269904, rho = 0.295; base 9 {0,1,2} j = 5 at P = 5, 0.038456894, rho = 0.26; and base 10 missing 9 at j = 1, 2, 3, 4, 7, all at P = 3, margins 0.047105507, 0.030062806, 0.045802462, 0.043508323 and 0.046292701 at radii 0.1275, 0.105, 0.1025, 0.09, 0.0725, each on 24 contour samples. Every certified disc agrees with the argument principle count of one and none disagrees; of the 19 poles carrying zero or two zeros in abs(u) < 0.45 that the budget evaluated none is certified, the two double poles reached, base 3 {1,2} j = 3 and j = 5, both failing, while base 5 {1,2} j = 7 and base 10 j = 15 were skipped for budget. Base 10 is not closed by any sharper majorant but by peeling: at the automatic depth P = 2 its B_T runs 1.08 at j = 1 to 38.1 at j = 15, and at P = 3 it runs 0.2096 to 1.2010 over the eight poles reached, five of which certify. Proximity of the tooth is no threshold, the certified abs(Z_0/Z_1) running 0.0282669 to 0.149708 and base 3 {0,1} j = 7 at 0.104443 failing at P = 7 and certifying at P = 9. The margins are evaluated in high precision and not in ball arithmetic, which is the one step between this row and Proved. Witness: lab/py/zeta-locus.
2026-09-07 Refuted The locus of the zeros of the design zeta is no curve Re s = f(Im s) shared by designs of equal alpha, no comb in the pole-period residue, and no law in alpha and fill/base. The witness against a shared curve is a pair of zeros of nearly equal imaginary part and very different real part on two designs of equal alpha, which a single curve cannot carry: base 4 {1,2} and base 16 {0,1,2,3}, both alpha = 1/2, hold zeros 0.015058 apart in Im s near Im s = 4.72 and 0.817047 apart in Re s; base 4 {0,1} against {2,3}, equal in alpha and in fill/base, gives 0.0136014 against 0.719693 near Im s = 17.64; base 4 {0,1} against {1,2} gives 0.0063091 against 0.280397 near Im s = 22.87 and base 4 {1,2} against {2,3} gives 0.00638631 against 0.198012 near Im s = 31.79, each pair drawn from censuses of the same box and the same height. Equality of both alpha and fill/base therefore fixes nothing. Within one design the worst real-part gap between two zeros of equal frac(Im s log base/2 pi) runs 0.077591803 at base 10 missing 9 to 0.65632474 at base 3 {0,1}, so the fractional part fixes nothing either, and the zeros per period at alpha = 1/2 reads 1.1897445 at base 16, 1.2868204 at base 9 and 1.586326, 2.0395621, 2.0395621, 2.2661801 at base 4, so no counting law in alpha alone survives. The single exception is alpha = 1, where the full digit sets at base = 2, 3, 4 are one arithmetic object and do share every zero. Witness: lab/py/zeta-locus.
2026-09-07 Refuted The second family of the design zeta is not symmetric about any vertical line Re s = c_F. Reading c_F as the midpoint of the real parts of the two second-family zeros of least Im s and testing the rest, no second-family zero in any design has a reflection partner: the reflection branch needs two second-family zeros within the 0.05 test tolerance in Im s, and the smallest ordinate gap inside a design is far above that on every design tested, so the branch cannot fire at all. Every pair the sweep records is a self-pair, a real part landing within 0.05 of c_F, and self-pairs occur below the chance rate: over the ten designs recensused the tally is 8 self-pairs and 0 reflection partners of 47 zeros tested, a rate of 0.170213 against the 0.229904 that drawing each real part uniformly from that design's own observed band predicts, and over the full sweep 22 of 117. The three full-set controls pair 13 of 13 at c_F = 1/2 to 1e-22, where the functional equation makes every zero its own partner. c_F is not a quantity either: c_F - alpha/2 runs -0.28413232 to +0.47788515 and c_F - 1/2 runs -0.78413232 to +0.28664994, so it is not alpha/2, not theta(F) and not 1/2. Witness: lab/py/zeta-family verb symmetry.
2026-09-07 Refuted There is no counting law for the second family in alpha or in fill, at either assignment radius. The four base 4 two-digit designs share alpha = 1/2 and fill/base = 1/2 exactly and give N_2(40) = 7, 13, 9, 14 at rho = 0.45 and 6, 12, 8, 7 at rho = 0.6, with N_2(80) = 20, 30, 22, 29 and 17, 26, 21, 20: a factor of two at one alpha and one fill/base at both radii, so the refutation is radius-robust even though the integers are not. The subject of the spread is comb occupancy and not the second family, base 4 {0,1} and {2,3} differing by 29 percent in total winding, 14 against 18, and by a factor of two in N_2 because 7 of 8 poles are occupied against 4 of 8. Read as N_2(T) = c_F T log T + d_F T from the two heights, c_F at alpha = 1/2 is 0.10820213, 0.072134752, 0.072134752, 0.018033688, spread 0.09016844, against the base 3 and base 4 full-set controls 0.15486803 and 0.16230319, the classical 1/(2 pi) = 0.15915494 and a control spread of 0.0074351582. Every winding is the nearest integer to a numerically integrated phase whose largest surviving step runs 0.9205 to 0.9998 against a cap of 1, so the counts are Verified and not Proved. Witness: lab/py/zeta-family verbs tests and count.
2026-09-07 Refuted The real parts of the second family do not contract to alpha/2 as a design fills, so the critical line is not the alpha -> 1 limit of MrlyMath. The refuted law is that max abs(Re s - alpha/2) -> 0 as alpha -> 1. Undivided, that statistic stays flat along the ladder carrying alpha toward 1, reading 0.2275679549 at base 5 {0,1} with alpha = 0.430676558, 0.5549589411 at base 4 {0,1} with 0.5, 0.5885444877 at base 3 {0,1} with 0.630929754, 0.4233198337 at base 4 {0,1,2} with 0.792481250, 0.5365616661 at base 5 {0,1,2,3} with 0.861353116 and 0.3151426744 at base 10 missing two with 0.903089987, then collapsing to 1.43e-22, 1.10e-21 and 1.76e-22 at the base 2, 3 and 4 full sets. At base 10 missing 9, alpha = 0.954242509, the second family reads 0.216084781875 to 0.70401657869 about alpha/2 = 0.477121255, a band of width 0.488 against 1 - alpha = 0.0458. Divided by 1 - alpha the statistic runs 0.39971647 to 5.7047812 with no monotone in alpha, falling from 3.8699872 to 3.2519104 on the last two rungs, so the refutation rests on the undivided spread and not on the ratio. Witness: lab/py/zeta-family verb limit.
2026-09-07 Refuted The ordinate shadow does not explain why a design's ordinates converge before its real parts, because at the zeros it pairs it separates neither. The first-order offset is one complex number, so for a paired zero the ordinate offset and the real-part offset are one quantity with no preferred phase: per zero abs(Im off)/abs(Re off) spans 0.137681 to 6.11895 at base 20 missing its top digit and 0.14167 to 18.7749 at base 50 missing its top digit, and rung by rung the medians median abs(Im off) against median abs(Re s - 1/2) read 0.55734029/0.43095421, 0.49670656/0.28603903, 0.48081533/0.22535435, 0.30633741/0.16748977, 0.12823995/0.36138728, 0.25093621/0.12181102, 0.11574693/0.19332005, 0.058239278/0.049215607, 0.011954894/0.010995712 at alpha = 0.430676558 to 0.994835739, the ordinate offset larger on six rungs and smaller on three with both falling along the ladder. The law binds only the zeros Newton reaches from a zeta zero inside 1.5 of it and this lab enumerates no design zero, so it neither explains nor forbids what a design-zero-first census reports; the census contrast and this row are both consistent with a mixture in which the partnered zeros approach in both coordinates while the rest of the second family does not approach at all, and that mixture has no witness until the unpartnered count is measured. Witness: lab/py/zeta-shadow verb rungs, lab/py/zeta-family verb limit.
2026-09-14 Verified The peeled continuation of a digit-design Dirichlet series runs in double precision inside the public crate. mrlynum::ladder carries Design, zeta, cofactor and residue on mrlynum::design::elements and mrlynum::zeta::Complex, and returns every value beside a bound. Against the arbitrary-precision lab at the base 2 full set the gaps are 7.4e-11 at s = 2, 4.6e-14 at s = 0.3 + 40i and 7.5e-12 at s = -1 + 2i against reported bounds 3.8e-10, 1.3e-11 and 6.8e-10; the base 3 residues meet the certified enclosures to 3.8e-15 against bounds near 8e-14. Five adversary breaks are repaired and no pinned number moved. Witness: mrlynum::ladder, lab/py/design-zeta.
2026-09-14 Proved The carried scale of a double-precision ladder majorises every intermediate magnitude of the value recursion, term by term. Beside the propagated truncation bound the module carries scale_j = (sum_(n in E_P) n^(-Re w) + cut + sum_l abs(binom(-w,l) base^(-w-l) gamma_l) scale_(j+l)) / abs(1 - fill base^(-w)). The induction is immediate: the base entries start at tail >= 0, poly_scale(E_P, Re w) >= abs(poly(E_P, w)) term by term, and every step applies the same nonnegative weights and the same divisor modulus, so scale_j >= abs(value_j) at every level. Witness: mrlynum::ladder.
2026-09-14 Verified The rounding charge of the double-precision ladder is measured and not counted, and it holds with a factor of fifteen to spare. The reported bound is truncation + ROUNDING * scale with ROUNDING = 1e-13; a count of about 75 roundings over up to 115 levels gives 9.5e-13, an order above the constant, so the charge is not a standalone bound. Measured over four designs at 28 points each plus the real axis the worst ratio of true error to returned bound is 0.0667, at base = 3, F = {0,1}, s = 0.5 + 40i, and the error never exceeded the bound anywhere probed; on a rejected rung the truncation half carries the bound and that ratio reaches 451. Witness: mrlynum::ladder.
2026-09-14 Verified The wall of the double-precision port is cancellation and not truncation, and it is visible at Re s = -1. At s = -1 + 2i on the base 2 full set the truncation bound falls to 1.1e-20 at shift 12 and to 7.8e-206 at shift 114 while the carried bound sticks at 5.6e-10, because the peeled tail G_P(-1+2i) has modulus 5.6e3 against zeta_F(-1+2i) of modulus 0.183, a loss of four and a half digits; the module raises rather than returns at any tolerance under that. The lab reaches 2.8e-32 there only by lifting the working precision with the height. Witness: mrlynum::ladder, lab/py/design-zeta.
2026-09-14 Proved The residue column below a pole of a digit-design zeta is a finite recursion in the peeled variables, Burnol's Proposition 7.1 in peeled form. With R_m the residue at s_(m,j) = alpha - m + 2 pi i j / log base, taking residues in the peel identity at w = s_(m,j) kills E_P and leaves only the terms whose shifted argument is a pole: (1 - base^m) R_m = sum_(l = 1)^m binom(-s_(m,j), l) base^(-s_(m,j)-l) gamma_l R_(m-l) with R_0 the numerator over log base, since fill base^(-s_(m,j)) = base^m. At base 3 on F = {0,1} the m = 1, j = 1 value 0.6950303416383606 + 0.37779086109705695i meets the eight-node contour average on a circle of radius 0.05. Witness: mrlynum::ladder, REFS.md Burnol 2026.
2026-09-19 Refuted The pole lattice does not force the zeros of a design zeta: at base 3 {0,1} the two polished zeros right of the abscissa are 0.665639628004 + 23.0347504431 i and 0.720787601477 + 28.6056765649 i, an ordinate gap of 5.5709261 against the pole period 2 pi/log 3 = 5.7192017, short by 0.148, because the shifted terms sum_(l >= 1) binom(-s,l) base^(-s-l) gamma_l zeta_F(s+l) of the digit recursion are not 2 pi i/log base periodic although 1 - fill base^(-s) is. Witness: lab/py/zeta-locus verb census.

Transparent fraction

2026-08-28 Conjecture The common-box transparent fraction does not grow to 24.5%: GRID(n) = (n-1)/(4n) for odd n reproduces 1/6, 2/9, 13/54, 27/110 at n = 3, 9, 27, 55, reads 0.2495463 at n = 551 and 0.2499504 at n = 5041, and its limit is 1/4; 24.5% is the n = 55 value, and at n = 0 the normalization is undefined.

Walk dimension

2026-08-28 Verified Mass does not fix music: codes 127 and 239 share fill 7 and dimension log(7)/log(3) = 1.771244 with identical density at every level, yet their walk dimensions separate by about 0.39 (2.26 against 2.64), both generators agreeing and every drift bar an order of magnitude smaller than the gap. Witness: lab/rs/walk-dimension.
2026-08-28 Conjecture Fixed mass does not fix the drum either: all nine fill-8 tiles recomputed at side = 243 and side = 729 under both boundary conventions give centre 1.2111, edge-midpoint 0.9661, corner 0.9634 (shipped convention 1.2093 / 0.9666 / 0.9633) at full held fraction, so the 0.25 spread is not a dropped-component artifact and "mass falls faster than stiffness" fails at exactly fixed mass.

Wallis sieve

2026-09-06 Proved The solid Wallis sieve, which drops the centre cube of every surviving cube cut into (2k+1)^3 at level k, keeps the limit volume prod_{n odd >= 3} (1 - n^(-3)) = pi^(3/2) / (8 |Gamma(7/4 - i sqrt(3)/4)|^2) = 0.948815486, by the Weierstrass product for 1/Gamma after m^3 - 1 = (m - 1)(m - w)(m - w^2) turns the k-th factor into k (k + (1 - w)/2)(k + (1 - w^2)/2) / (k + 1/2)^3 with the three shifts summing to 3/2; the plane sieve's limit area is Wallis's pi/4. Witness: mrlynum::sieve::solid_limit, evaluated by the log series to one ulp of 0.9488154857196796, checked against the truncated product to 1e-14 and against the ratio form cosh(pi sqrt(3)/2) / (3 pi) over the even product.
2026-09-06 Proved The Wallis sieve is a mixed-radix schedule word: letter k is the side-(2k+1) tile with its centre cell removed, the word is their Kronecker fold, and the word's fill is the product of the letters' fills exactly, so any odd schedule in any dimension is a sieve with area the product of its letters' survival ratios. Witness: mrlynum::sieve::ratio, the raster count against the product at levels 1 to 3 in both schedules.
2026-09-06 Proved A schedule of distinct odd letters buys area and a constant one buys a dimension: with strictly increasing odd sides sum s_k^(-dim) converges, the limit area is positive and the odd word's fill exponent dim + log(ratio_level) / log(side_level) walks up to dim (1.972027 at plane level 4), while the constant side-s word's stands at log(s^dim - 1) / log s forever (log 8 / log 3 = 1.892789 for the carpet); a schedule that merely varies, 3, 5, 3, 5, ..., loses its area and holds log 192 / log 15 = 1.941432. Witness: mrlynum::sieve::exponent.
2026-09-06 Verified The plane sieve's truncated product reads 0.785398262 at two million factors against pi/4 = 0.785398163, the identity being Wallis's. Witness: mrlynum::sieve::ratio, the test that pins its nine-digit rounding.

Waves

2026-09-08 Proved Riesz product: the DFT of a base-2 design mask popped at level level at frequency xi on an N torus is the product over j = 0..level-1 of the level-1 tile polynomial P(side^j xi / N) minus the tile centre value, since the offsets are digit sums and the exponential sum factorises; checked at codes 7, 6, 9 and levels 2 to 4 over all 256 x 256 frequencies, largest error 2.5e-12, the subtracted term 1 for code 9 and 0 for codes 7 and 6. Witness: lab/py/kernel-waves.
2026-09-08 Verified From the density-0.5 soup on the 256 torus, 64 steps at one seed, the Bugs, wide-survive and narrow-birth windows reach no ring still on the box r = 5, 13 masks or the design masks code 7 at levels 2 to 4 and codes 6 and 9 at level 3: 0 of 21 cells, 10 dead, 6 flat frozen soups, 5 active; a wide survive on a mask of 512 ones or more freezes the soup unchanged. Witness: lab/py/kernel-waves.
2026-09-08 Verified On box r = 5, carpet code 7 at levels 2 and 3 and diagonal code 9 at level 3 every one of 130 ring stills has its peak ring k* inside the mask first negative band of the ring-mean signed DFT, k*/k_2 from 0.77 to 1.17, k* read on the rings k <= 128 which drop a fifth of the frequencies; as a universal law it fails on 3 of 176: box r = 13 at k* = 10, code 7 level 4 at k* = 2, code 6 level 3 at k* = 8. Witness: lab/py/kernel-waves.
2026-09-08 Verified The carpet family median still wavelength over mask side is 0.729 at levels 2 and 3, equal to 256/k_2/side at both; level 4 reads 0.790 against 0.632 where the 256 torus holds 3.2 mask widths. Witness: lab/py/kernel-waves.
2026-09-08 Verified On the carpet cell graph at level = 3, 512 cells on 4-neighbour adjacency with Laplacian D - A, the strong nodal domain counts of the returned eigenvectors are 1, 2, 2, 4, 4, 4, 4, 5, 8, 8, 8, 8 for k = 1..12 with multiplicities 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, Courant nu_k <= k holds on all 512 at every zero tolerance from 1e-12 to 1e-6, the spectrum has 380 classes at gap 1e-8 of which 130 are degenerate covering 262 indices with multiplicity up to 4, and nu_k / k falls in five bins of width 0.2 as 0, 42, 217, 193, 60 at tolerance 1e-9; counts for the returned basis only. Witness: lab/py/carpet-nodal.
2026-09-08 Verified At level = 4, 4096 carpet cells, the counts are 1, 2, 2, 4, 4, 4, 4, 5, 4, 4, 8, 8 with the same multiplicity pattern, where the 4, 4 at k = 9, 10 rests on 64 cells at |v| = 1.4e-7 in each returned vector and reads 8, 8 at tolerance 1e-6 as at level = 3; Courant holds on all 4096, the spectrum has 3056 classes of which 1022 are degenerate covering 2062 indices with multiplicity up to 20, the bins are 24, 797, 2061, 971, 243, and the top eigenvector has 12 cells below the zero tolerance so its printed count is 4084 while no edge joins two nonzero cells of one sign. Witness: lab/py/carpet-nodal.
2026-09-08 Verified The nodal count on a degenerate eigenvalue is basis-dependent: at the double eigenvalue k = 6, 7 of the carpet cell graph the two returned vectors have 4 strong nodal domains each and their normalised sum and difference have 6 each, at level = 3 and level = 4 alike, and such a disagreement occurs at 108 of the 129 double eigenvalues at level = 3 and 971 of 1021 at level = 4. Witness: lab/py/carpet-nodal.
2026-09-08 Proved By the discrete nodal domain theorem every eigenvector of an eigenvalue of index k and multiplicity r on a connected graph has at most k + r - 1 strong and at most k weak nodal domains, so at the double eigenvalue k = 6, 7 of the carpet cell graph at level = 3 and level = 4, index 6 and multiplicity 2 read at gap 1e-8, no vector of the eigenspace has more than 7 strong domains, and the returned basis sum and difference reach 6. Witness: lab/py/carpet-nodal.
2026-09-08 Verified On the control grids 22 x 22 and 64 x 64 the separable cosine basis has exactly (p + 1)(q + 1) strong nodal domains on all 484 and all 4096 eigenvectors, the returned basis satisfies Courant on all of them with degenerate index fractions 0.9566 and 0.9846 and multiplicity up to 21 and 63, and the mean of nu_k / k over k >= 2 at tolerance 1e-9 is 0.605 and 0.536 on the carpet (0.602 to 0.610 at level = 3 across tolerances 1e-12 to 1e-6) against 0.508 and 0.465 on the separable grid and 0.433 and 0.330 on the returned grid basis. Witness: lab/py/carpet-nodal.
2026-09-08 Conjecture A Larger-than-Life still sits where the kernel ring-mean transform is negative because a pattern there has its count anti-correlated with its state, which a birth window below the mean and a survive window above it hold fixed; the correlation is -0.52 to -0.85 on the strongest stills of the four lobe masks; a proof, a 1024 torus for the side-81 mask and the cross mask lobe are open. Witness: lab/py/kernel-waves.
2026-09-08 Refuted The wavelength of a ring still is the mask width, k* at the first minimum k_min of the mask ring-mean transform: over 180 window rules at two seeds, 2520 runs and 176 ring stills, k* sits within one ring of k_min in 25, 21 of them on the under-resolved side-81 mask; witness box r = 5, all 25 ring stills at k* = 30..36 against k_min = 24 and k_2 = 32, wavelength 7.1 to 8.5 cells for a mask 11 wide. Witness: lab/py/kernel-waves.
2026-09-08 Refuted One k* per mask across rules: box r = 5 spreads k* = 30..36 over the grid rules and seeds, one rule alone reading 32 at one seed and 36 at the other; the spread is up to two fifths (10 to 14 on carpet level 3) and never leaves the mask negative lobe on the four masks where the lobe law holds. Witness: lab/py/kernel-waves.
2026-09-19 Verified The FFT count step on a power-of-two torus agrees cell for cell with the direct neighbour count under a wrap boundary, on the Moore mask and the level-2 carpet mask over 8 steps. Witness: mrlydemo::chladni against mrlymath::life::next_grid.
2026-09-19 Verified Under the wide-survive window rule, the Bugs birth with S[0.28, 0.60], from a density-0.5 soup on a 256 torus, the three masks of 512 cells or more in the sweep, the box r = 13 at 728 cells and code 7 at levels 3 and 4, freeze the soup unchanged. Witness: lab/py/kernel-waves.

Weighted designs

2026-09-06 Proved For a weighted design (a probability vector on |F| >= 2 cells) the Dirichlet root of sum_f w_f^s = 1 is identically 1, so it is the arithmetic class of the log w_f, never the root, that carries the mass-stopping count N(t) = #{words of mass >= t}: N is log-periodic exactly when the group generated by the log w_f is cyclic (for rational weights, when the prime-exponent matrix has rank 1) and smooth otherwise, by Lalley's renewal dichotomy on f = -log w; and every length observable keeps its log base ripple at every weight, mu(B(r)) = w_0 mu(B(qr)) at a corner fixed point for r < min_{f != 0} |f|/base. Witness: lab/py/weighted-designs.
2026-09-06 Verified The multifractal pressure of a weighted design with equal contraction 1/base under the open set condition is tau(s) = log(sum_f w_f^s)/log(base), so f(alpha) = inf_s (alpha s + tau(s)) is explicit: the box moments at level level carry it exactly as sum_i mu_i^s = (sum_f w_f^s)^level, the coarse-grained band sits under the transform at every level (f_level <= f from N_i mu_i^s <= base^(level tau(s))), exact at both endpoints and deficient by 0.176458, 0.147536, 0.127619 at the band's middle alpha = 1.077324384 at levels 6, 8, 10 on the three-cell weighted gasket (base 3, cells (0,0) (2,0) (0,2), weights 3/8, 3/8, 1/4), the deficit matching Stirling's series, and the JSR bracket validates on the hat mask at alpha = 1 and on D4 at alpha in [0.4929285, 0.5500157] against the closed form 2 - log_2(1 + sqrt 3) = 0.5500157. Witness: lab/py/weighted-designs, Cawley and Mauldin 1992.
2026-09-06 Refuted That delta, the Dirichlet root, is a weight observable of a design (it is 1 at every probability vector), and that a norm upper bound may print truncated: D4's norm upper is 0.710581107211, so the safe print is 0.7105812 and alpha's upper 0.5501 at four digits, 0.5500 sitting strictly below the closed form 0.5500156865. Witness: lab/py/weighted-designs.
2026-09-06 Refuted The Type II route through the multiplicative energy of a column: with |a|, |b| <= 1, two Cauchy-Schwarz steps give |Sigma|^2 <= M E_x(M, N) <= 2MN E_x(level)^(1/2) x^(o(1)), hence |Sigma| <= x^((1 + alpha)/2 + o(1)), missing the trivial x^alpha by (1 - alpha)/2 for every digit set with alpha < 1; the unbalanced sum has no estimate at all since a = b = 1 returns the representation count itself, and the balanced form returns the box's own trivial bound on the census (bound over trivial 1.0134 at level = 12 rising to 1.0730 at level = 14). Witness: lab/rs/rho-decoupling (the menergy module), mobius.md THE METER AND ITS YARDSTICK.