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.