The ratio-set power saving

• 2026-08-31 [Proved] Every occupied direction of the gasket ratio set obeys max(z_1, z_2) > 2 min(z_1, z_2): the top base-3 digit of m(z_1 + z_2) lies in exactly one of the disjoint binaries m z_1, m z_2 and the other is a sum of distinct lower powers, hence at most (3^t - 1)/2, so the slope z_1/w never lies in [1/3, 2/3]; checked against every occupied weight to 8192, every pair to height 120 and every ray at level 12, and shown sharp and strict by the adversarial pass at minimum ratio 2.0000004 over 14.3 million pairs at level 15, extremal at (3^14, (3^14 - 1)/2). Witness: lab/py/ratio-set-saving.
• 2026-08-31 [Proved] The digit-congruence bound and the weight-layer reduction are one bound: z_1 (1 + r) = r w for r = z_1 z_2^(-1) mod 3^k, and r = -1 mod 3 would force 3 | w, so z_1 = r w (1 + r)^(-1) is determined and z_1 -> r is injective for 3^k > w, giving Z(w) <= 2 |R_k| with no failing weight to 8192; beta < 1 from this side would need sigma_k to fall geometrically, exactly what criticality forbids. Witness: lab/py/ratio-set-saving, lab/py/occupancy-decay.
• 2026-08-31 [Proved] Two relaxations of occupancy, both tight enough to keep the exponent: Z(w) <= 2 N_P(1/w) because slopes of denominator w are 1/w apart, and Z(w) <= Zinf(w) because the band is forward-invariant on every integer it contains, so a 3-adic witness suffices; the backward cone of 0 is {z_2 C - z_1 A : (A, C) a gasket pair} intersected with the band, rebuilt by an independent carry-pair dynamic programme with zero mismatches, and Zinf/Z is at most 1.5295 over the eleven weights tested. Witness: lab/py/ratio-set-saving.
• 2026-08-31 [Proved] R_k is indexed by the modulus 3^k and R_1 is empty under the hypothesis u > 0, so |R_k| = 1, 3, 9, 23, 63, 168, 457, 1245, 3423, ... starts at k = 2; both studies carry the identical definition and the same offset, which pins the offset the submission candidate needs. Witness: lab/py/occupancy-decay, lab/py/ratio-set-saving.
• 2026-08-31 [Verified] The weight layer read per weight rather than off a running maximum: log Z(w) / log w peaks at 0.7093 at w = 121 and Z(w) / w^(log 2 / log 3) at 1.5975 at w = 1093 over every w <= 8192, with all twenty-four octave argmaxes binary base 3 as the scan prints for itself; on the repunits (3^k - 1)/2 at k = 9, 11, 13 and the shifts 1 + 3^h at h = 7, 9, 11, 13 the exponent holds inside [0.6223, 0.6818] out to w = 1594324 while unstructured neighbours collapse to [0.2861, 0.4272], and the mean forward reach is 0.2249 to 0.2947 times sqrt(w) off the structured families. Witness: lab/py/ratio-set-saving.
• 2026-08-31 [Refuted] A uniform D_n(q) <= C 2^n / q on the binary base-3 multiples of q coprime to 3, the divisor input to the power saving - every binary m < 3^h makes m(1 + 3^h) binary below 3^(2h), so D_2h(1 + 3^h) >= 2^h while 4^h / q is only (2/3)^h of it, ratio (3/2)^h (1 + 3^(-h)) reading 2.0, 2.5, 3.5, 5.125, 7.625, 11.406, 17.094, 25.633 at h = 1..8, and at n = 20 the worst modulus below 500 is q = 244 = 1 + 3^5 at 1.8094; the breaking moduli are exactly the shift-ray weights. Witness: lab/py/ratio-set-saving.
• 2026-08-31 [Refuted] The short-witness route, bounding the count by 3^(level cap) - mean minimal witness length runs 3.875 to 27.287 and max 6 to 204 over height 32..16384, mean lev / log_3 height rises 1.553 to 3.305, and the share of occupied directions with lev <= 1.8073 log_3 x falls 0.875 to 0.3102, so every cap below 2 log_3 x loses a majority of the count. Witness: lab/py/ratio-set-saving.
• 2026-08-31 [Refuted] Two write-up claims of the first pass, caught by the adversarial read and corrected in place - the log Z_max / log W band was printed as [0.5000, 0.6404] when the script's own W = 128 row reads 0.70099, true band [0.5000, 0.7010] and margin 0.106 not 0.167; and ten Z_max values were listed against nine arguments, the duplicate Z_max = 30 at both W = 128 and W = 256 having been dropped, shifting every later argument onto the wrong weight. Both are transcription, not mathematics. Witness: lab/py/ratio-set-saving.
• 2026-08-31 [Refuted] beta >= log 2 / log 3 as a proved lower end of the weight-layer corridor - the binary-weight floor Z(w) >= #{coprime submasks} is proved, but at w = (3^k-1)/2 the coprime cut leaves 2, 6, 8, 30, 24, 126, 112 against w^(log 2 / log 3) = 2.4, 5.0, 10.3, 20.6, 41.3, 82.6, 165.3 for k = 2..8, beating the exponent at odd k and losing at even k, and no family supplies infinitely many good weights; the lower end is Conjecture. Witness: lab/py/ratio-set-saving.
• 2026-08-31 [Refuted] The metric route to the weight-layer saving: the sandwich Z(w) <= 2 N_P(1/w) is proved, but the cover of the slope set measures too large, n N_P(3^-n) / 3^n rising 2.4132 -> 2.4785, log_3 N_P / n rising 0.8783 -> 0.8997 and the step exponent rising 0.9333 -> 0.9504 over n = 12..18, every reading monotone and every one above the 0.8073 needed, so the covering route caps at O(w / log w) exactly like the congruence seed; a missing-digit rational-counting import belongs at the 3-adic ratio set R_inf and not at the slope variable. Witness: lab/py/ratio-set-saving.
• 2026-08-31 [Refuted] Three printed statements of the second pass, caught by the adversarial read and corrected in place - sigma_k < 1/9 was dated to k = 13 when it is 1/9 exactly at k = 2, 3, 4 and first below at k = 5, the bound already beating the trivial count at w = 13 (6 against 8.0) and w = 121 (46 against 73.3); the slope cover was computed on one swap half only, 51624 against the saturated 106994 at n = 12; and the backward moves were called one per residue class when 3k and 3k - z_1 are both 0 mod 3 and the class -z_2 mod 3 has no preimage. A fourth broke on the fix: the cover at n = 19 with three extra digits overflows int64 and printed a false 1.9533, so the generator now refuses past 1.5 * 3^(2n + extra) >= 2^63. Witness: lab/py/ratio-set-saving.
• 2026-09-06 [Refuted] The digit-congruence containment as printed: z_1 z_2^(-1) mod 3^k in R_k fails whenever 3^k | z_1, witness the occupied ray (9,1) at k = 2, where R_2 = {3} and the residue is 0; the true image is R_k union {0}, the counting bound's tail terms doubling to pay for the adjoined class, and the mod-3 dichotomy is the case k = 2 and not k = 1, R_1 being empty under u > 0. Witness: coprime.md, lab/py/occupancy-decay, lab/py/ratio-set-saving.
• 2026-09-06 [Refuted] The adversarial pass on the weight-layer run: an independent carry-pair dynamic programme reproduced every table to the last digit, the top-digit gap was checked over 14.3 million pairs at level 15 and found sharp and strict at minimum ratio 2.0000004 with extremal witness (3^14, (3^14 - 1)/2), Z(w) <= 2 |R_k| was checked at every weight to 8192 with no failure, the backward cone was rebuilt with zero mismatches, and three printed statements plus the |R_k| offset were broken and fixed in place. Witness: lab/py/ratio-set-saving.
• 2026-09-06 [Refuted] The adversarial pass on the run itself: the band was rederived on paper and sharpened to the halved interval -z_2/2 < j < z_1/2, an independent all-coprime-pairs automaton reproduced every A, Zsum and Z_max row to height 1024 and the full sweep, witnesses were reconstructed digit by digit for all 716 occupied directions to height 300 with zero failures, and Chow-Varju-Yu Theorem 1.2 and Kenyon were both verified accurate at source. Witness: lab/py/ratio-set-saving.
• 2026-09-07 [Proved] The block rate of the critical band automaton is bracketed by the parity of the depth b. Every column sum of every block B(b, j) is Sum_(c = a mod 3) binom(b, c) = (2^b + 2 cos(pi (b - 2a)/3))/3, so its deviation from 2^b/3 takes only two values, -1/3 and 2/3 at even b and -2/3 and 1/3 at odd b; hence lam_b lies in 2^b/3 + [-1/3, 2/3] at even b and in 2^b/3 + [-2/3, 1/3] at odd b, and the two-sided abs(3 lam_b/2^b - 1) <= 2^(1-b) holds at every b. The parity refines which edge is which and not the rate, and the computed excess 3 lam_b - 2^b is positive at every depth reached, so the upper edge is the live one. Witness: lab/py/band-return-times ladder.
• 2026-09-07 [Proved] At the horizon n = bk, k blocks of depth b, the column transfer is one matrix per residue fixed in k and L(k, bk) = Sum_j w_j B(b, j)^(k - r0(j)) h_j, the head length r0(j) free of k but not equal to 2: it is 1 at every sector below b = 5, at most 2 at b = 5..10 and at most 3 at b = 11..14. So L(k, bk) obeys a constant-coefficient linear recurrence in k and the block rate lam_b is an algebraic integer. Witness: lab/py/band-return-times ladder, with an independent residue DP reproducing L(k,4k) and L(k,5k) to k = 12 and factoring both characteristic polynomials in exact arithmetic.
• 2026-09-07 [Verified] The block rates are exact algebraic integers: lam_4 = 6 from (x-1)(x-3)(x-5)(x-6), lam_5 = 3(5 + sqrt 5)/2 from (x-1)(x^2 - 15x + 45), lam_6 = 13 + sqrt 79, lam_8 = (99 + 9 sqrt 65)/2, every lam_b to b = 14 having an exact minimal polynomial that divides the characteristic polynomial with zero remainder, the degree-four-and-up ones at b = 9, 11, 13, 14 irreducible over the rationals by mod-p distinct-degree factorisation; and the block rate is the largest block spectral radius itself, max_j rho(B(b,j)) agreeing with the certified interval to a relative 1e-9 at every b. Witness: lab/py/band-return-times ladder.
• 2026-09-19 [Proved] The carry transfer on the slot profile s_r = ceil((n - r)/k) gives the primitive return count L(k, n) of the weight R_k exactly at every k and every n, past the rigid depth the block ladder stops at, reading L(k, 4k) = 185, 1002, 5573, 31506, 180125, 1038402 at k = 3..8. Witness: lab/py/band-return-times, verb returns.
• 2026-09-19 [Verified] The support of the return time of the weight R_k, the lengths at which some primitive return exists, is {k} union [k + 2, 8k] at every k = 2..12, one gap at k + 1 and no other inside that range, with nothing past n = 8k or k = 12 decided. Witness: lab/py/band-return-times, verb returns.
• 2026-09-19 [Conjecture] The FIRST return time is a far thinner object than the return count and takes 16, 16, 59, 80 distinct lengths at k = 11..14, a single unpinned reading of the first-return sweep that no README prints and no pinned test carries. Witness: lab/py/band-return-times, verb hist.
• 2026-09-19 [Verified] The minimal polynomial of the block rate lam_b is exact at every b <= 14, lam_7 the dominant root of x^3 - 63x^2 + 945x - 3402, lam_9 of x^4 - 255x^3 + 16065x^2 - 293787x + 1299078, lam_10 of x^3 - 392x^2 + 17469x - 96228, lam_11 of a quintic, lam_12 of x^3 - 1551x^2 + 257256x - 5629338, lam_13 of a sextic, lam_14 of x^4 - 6176x^3 + 3963141x^2 - 335533914x + 2583866142, certified by exact bisection to a width below 1e-9 at 10.854101966, 21.888194417, 42.760932540, 85.780159867, 170.715620440, 341.700429300, 682.692831036, 1365.640975936, 2730.680876219, 5461.594643683 over b = 5..14, with minimal recurrence order b at even b and (b+1)/2 at odd b, the even ladder staying in radicals through b = 14 and the odd leaving them at b = 11. Witness: lab/py/band-return-times, verb ladder.
• 2026-09-19 [Conjecture] The degree of the minimal polynomial of lam_b is ceil(b/4) at even b and (b-1)/2 at odd b >= 3, a pattern observed on the thirteen rungs b = 2..14, b = 1 printing degree 1 against the rule's 0, and licensed at no b >= 15. Witness: lab/py/band-return-times, verb ladder.
• 2026-09-19 [Verified] A block ratio reads the block rate at odd depth and at no even one: the second root of the recurrence is 0.959422 of lam_6, 0.991055 of lam_8 and 0.999909 of lam_14, so L(k+1, b(k+1)) / L(k, bk) carries at most two correct digits at k = 160 at every even b <= 14, while at odd b = 5..13 that root falls from 0.381967 to 0.333404 and the same ratio carries 66 to 76 correct digits there. Witness: lab/py/band-return-times, verb ladder.
• 2026-09-19 [Proved] The repunit sweep meets each direction (z, R_k - z) twice, at z and at R_k - z, so Phi_k, Z(R_k), U_k and V_k are counts of z values with the distinct directions half of each, every first-return count being even for that reason, and a sample size quoted without halving is doubled. Witness: lab/py/band-return-times, verbs hist and check.
• 2026-09-19 [Verified] The deep tail's survival has no law: the survival in distinct directions S(b) = (1/2) #{z : d(z) > bk} at k = 14 runs 81, 65, 58, 56, 55, 52, 48, 42, 39, 37, 35, 30, 27, 20, 18, 14 from b = 2 and reaches 1 at b = 42, the local exponent -log_2(S(2b)/S(b)) reads 0.481, 0.273, 1.415, 3.169 at b = 2, 4, 8, 16 with the sharpest resting on the two directions of S(32), and a maximum-likelihood geometric fits ratio 0.8958 with pooled chi2 = 29.0 on at most 16 degrees of freedom once the fit is carried from the doubled z counts to the directions, so on 81 directions over 41 depths the survival is neither geometric nor shown not to be. Witness: lab/py/band-return-times, verb hist.
• 2026-09-19 [Conjecture] The equidistribution model D(k, N) = Sum 2^(#supp K) / m over the primitive lifts K = m R_k of base-3 length at most N equals 2^k L_k + 4^k / (3^k + 1) at N = 2k, the second term the primitive lift R_(2k) of multiplier 3^k + 1 and cancelling in every deep part below, and puts the deep part D(k, N) - D(k, 2k) at the critical cutoff N = floor(sqrt(R_k)) inside [0.1476, 0.4429] * 2^k at k = 8 and inside [0.0373, 0.1122] * 2^k at k = 16, so on the model the deep tail is o(2^k); it is never a prediction of Z(R_k) - U_k, which it exceeds by the witness multiplicity, and 0, 0, 0, 0, 5, 32, 51, 64, 73 percent of that deep part at k = 8..16 is carried by the free 4/9 per-digit increment extrapolated past 40 blocks, both ends leaning low because the return excess rho is above 1 at every depth reached and the upper end holding only while rho < 3. Witness: lab/py/band-return-times, verb model.
• 2026-09-19 [Proved] The inequality N_K(m) <= #packings is strict from k = 5, where T = {1}, m = 7 and K = 847 carry the support {0, 2, 3, 4, 6} with four irreducibles, two decompositions of the whole and six distinct unions against seven packings, so bounding Sum_T #packings_T suffices for the lift half and is strictly the harder target. Witness: lab/py/band-return-times, verbs lift and check.
• 2026-09-19 [Proved] A column transfer for the lift count with a state set free of k is a linear representation of that count as a series over the column word, so its state count is at least the series' Hankel rank, finite Hankel rank over a free monoid being exactly a linear representation with that rank as the minimal dimension. Witness: Schutzenberger 1961 in REFS.md.
• 2026-09-19 [Verified] That Hankel rank reaches 253 at word length 7, so the floor bites at k <= 15 where 2^7.5 = 181 is below it, and the floor is neither an impossibility nor a second check read twice: the reversed reading is the transpose of the same matrix at equal side lengths, and a machine whose state set may grow with k always exists, the residue automaton on m states computing N_K(m) at cost 3^k per T. Witness: lab/py/band-return-times, verb lift.
• 2026-09-19 [Verified] The first-return sweep reaches no k past 15 and the lift-family generator stops at k = 13, where 3^(3k) passes 2^63, so the deep tail stands on five points. Witness: lab/py/band-return-times, verb hist, and lab/py/ratio-set-saving, verb tail.