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 Superseded in the cell frame: the decay coefficient is a closed form at every band width, see the width family rows under Diagonal slice stack in SETTLED.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.