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.