# 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.