# 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`.