# 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= 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.