# ratio-set-saving

- Decides exactly, at no level cap, which coprime directions lie in the ratio set of the gasket `G = {(u,v) : u, v binary base 3, supports disjoint}`, and measures the counting function the ratio-set power saving asks to bound.
- The decision is automaton reachability, not enumeration, so `A(X)` here is the limit over every level: one quantifier stronger than the level-coupled census in `lab/py/occupancy-decay`, and the honest form of the lemma.
- Conventions, printed by the sweep: height is `max(z_1, z_2)`, counts are of ordered directions, weight is `w = z_1 + z_2`, `lev(z)` is the digit length of the smallest witness, `c = lev / log_3 height` is per direction, and the share column instead thresholds `lev` against `1.8073 log_3 x` at the window cap `x`, not per direction.
- `Z(w)` counts the occupied directions of weight exactly `w`; `D_n(q)` counts the binary base-3 multiples of `q` below `3^n`.
- `Zinf(w)` relaxes occupancy from an integer witness to a 3-adic one, `N_P(delta)` is the cover of the slope set `P = {u/(u+v) : (u,v) in G}` at scale `delta`, and reach is the forward reachable set of the band automaton from `z_1/3`.
- `R_k = {u v^(-1) mod 3^k : (u,v) in G_k, u > 0, 3 does not divide v}` is indexed by the modulus `3^k`, so `R_1` is empty under `u > 0` and the sequence `|R_k| = 1, 3, 9, 23, 63, 168, 457, 1245, 3423` starts at `k = 2`; `lab/py/occupancy-decay` carries the same definition and the same offset.

## POSITIONING

- The gasket is Kenyon's one-dimensional Sierpinski gasket, and the direction question is the radial analogue of his projection theorem: an orthogonal projection in a reduced direction `p/q` has dimension below 1 unless `p + q = 0 mod 3`, and since every occupied ray here has weight coprime to 3, every direction this lemma counts sits in the dimension-below-one half of his dichotomy.
- The counting statement is the missing-digit rational-counting problem of Chow, Varju and Yu, whose `T^(2 kappa - rho)` power saving is the only theorem of the right shape and whose hypotheses cover every `base >= 5` with `base - 1` digits and base 4, excluding base 3 with two digits: the case here is precisely the excluded one, and the obstruction is the same resonance at `3^h` that makes the uniform `D_n(q)` bound false.
- No fractal-geometric input can help: the middle-thirds quotient set `C/C` is a union of intervals by Athreya, Reznick and Tyson, so any saving is arithmetic and comes from the reduction to lowest terms, not from thinness.
- Sum-product and multiplicative energy point the wrong way by construction, since `|A/A| >= |A|^4 / E` is a lower bound and an upper bound of the form `|A/A| << |A|^(2-eps)` would force multiplicative structure a missing-digit set does not have.
- The counting is not metric in either place it could be. Archimedean: the slope set `P = {u/(u+v)}` is a compact set whose cover at scale `3^-n` is measured at `3^n / n`, so a covering bound gives `w / log w` and nothing more. Non-archimedean: occupancy forces `z_1 z_2^(-1)` into the 3-adic ratio set `R_inf = intersect_k preimage(R_k)` in `Z_3`, whose 3-adic dimension is 1 on the same `sigma_k ~ C/k` reading; the honest transplant of a missing-digit rational-counting theorem is therefore to `R_inf`, not to the slope variable, and it needs an input beyond dimension because the dimension is 1.

## THE BAND

- For `3 | z_1` the pair carry state `(c_1, c_2)` of `m z_1, m z_2` is determined by the single integer `j = c_1 z_2 - c_2 z_1`, and `3^k j = E_2 z_1 - E_1 z_2` where `E_1, E_2` are the emitted low digits; disjoint supports make `E_1 + E_2` binary base 3, hence at most `(3^k - 1)/2`, so `-z_2/2 < j < z_1/2` and at most `(z_1-1)/2 + (z_2-1)/2 + 1` states are reachable, about `w/2`.
- The transition is `j -> (j + a)/3` with `a = 0` or `a = z_1` when `3 | j`, `a = -z_2` when `j = z_2 mod 3`, and none when `j = -z_2 mod 3`; the origin is the only state of the fibre `j = 0`, and a direction is occupied exactly when `0` is reachable from `z_1/3`.
- One coordinate must carry the 3: if `3` divides neither `z_1` nor `z_2` then, taking `3` out of `m` first, both `m z_1` and `m z_2` have low digit in `{1, 2}`, binary forces both to 1, and the supports collide - so no such direction is occupied and the automaton is only ever built with `3 | z_1`.
- Over the integers of the band the out-degrees `2, 1, 0` fall one to each residue class mod 3, so the mean out-degree is `1 + O(1/w)`: the survivor process is critical, and the whole difficulty of the lemma is that a critical walk must hit one target state in a band of width `w`.
- The band is forward-invariant for every integer it contains, not only for the reachable ones, so the automaton is the induced subgraph on `(-z_2/2, z_1/2)` and its core - the states admitting an infinite forward path - is computed by removing states of out-degree zero to fixation in `O(w)`.
- Read backward the moves are `k -> 3k`, `3k - z_1`, `3k + z_2`; the first two are both `0 mod 3` because `3 | z_1`, the third is `z_2 mod 3` and the class `-z_2 mod 3` has no preimage at all, so in-degree is at most 3 but the classes are not one apiece as they are forward. A backward path of length `L` lands on `z_2 C - z_1 A` for a gasket pair `(A, C)` of level `L`, so the backward cone of `0` is `{z_2 C - z_1 A} intersect band` and occupancy is `z_1/3` lying in it.

## RUN

- `uv run python research/lab/py/ratio-set-saving/ratio.py check` matches occupancy and `lev` against direct gasket enumeration at `n = 6, 9, 12`, matches the band automaton against the two-carry automaton to height 120 with no band violation, confirms the `[1/3, 2/3]` gap on both the automaton and the raw rays, confirms that every occupied direction reaches the core, rebuilds `|R_k| = 1, 3, 9, 23, 63` at `k = 2..6` and finds no failure of the residue recovery, no residue outside `R_k` and no weight below 729 breaking `Z(w) <= 2 |R_k|`, pins the binary-weight floor at `w = (3^k-1)/2` for `k = 2..8`, and asserts the Fourier identity `N_T = (1/m_T) Sum_u Prod_p (1 + e(u 3^p / m_T))` against the residue DP and the meet-in-the-middle count over every `T` at `k = 7, 8` with the antipodal family's exact count at `k <= 8`, asserts the block ladder at `k = 2..5` with its four-block branch at `k = 3`, and matches the depth-3 lift census against `Z(R_k)` at `k = 7, 8, 9`; 3.6 s.
- `... levels 243` and `... levels 2187 --lo 9 --hi 9` are the cross-checks against the earlier census; `... sweep 2048`, `... band 3000`, `... multiples 20`, `... box 12 16` and `... core 121 244 355 730 757 1093 2188 3271 3280 6562 9841` are the kept rows, together about a minute.
- `... layers 8192` is the exact weight-layer scan, 31 s, and is the generator for every per-weight exponent and constant here; `... weights W` gives one weight at `O(phi(w) sqrt(w))` and reaches `w = 1594324` in 191 s.
- `... repunit` is the repunit family `w = (3^k - 1)/2`, `k = 2..13` in 20 s: the floor three ways, `Z(R_k)` with minimal witnesses, the lift union, and the non-submask directions at `k = 7, 8, 9`; `--kmax 15` adds two rows in about 12 minutes.
- `... lifts --kmax 19 --zmax 15` is the lift census `k = 2..19`: the union `U_k = |Union_T Occ_T|`, the sum `Sum_T |Occ_T|`, the model `L_k = Sum_T 1/m_T`, the aggregate `agg_k` against it after the coprime cut, the per-`T` peak with its multiplier, the cyclotomic count against its closed form, and `Z(R_k)` with the deep tail where the band automaton reaches; meeting the two halves of a submask in the middle decides `m_T | A` in `O(2^(k/2))` per `T`, so the union alone runs `k = 2..19` in 21 s and the whole 10 min 43 s is the `Z` column, 557 s of it at `k = 15`.
- `... tail` is the depth census of the repunit layer, 2 min 5 s at the defaults: the depth of an occupied direction is the number of `k`-blocks its minimal witness lift `m R_k` fills, so `U_k` is depth at most 2 and the verb adds the depth-3 census `V_k`, `Z(R_k)`, the deep tail, the share of the tail captured at depth 3 and the tail's depth histogram, with the `2 * 3^k` depth-3 lifts enumerated as an independent generator and asserted against the automaton; `--kmax 15` adds `k = 14, 15` for ten minutes more, `--famax 13` is the int64 ceiling and costs about a quarter hour, and a one-position lift meets its submasks in the middle at `2^(k/2)` against `2^k` for a two-position one.
- `... agg` is the Fourier side of the lift family, 21 s: for every `T` at `k = 5..11` the residue distribution of the submasks of `K_T` mod `m_T` by `k` rolls and its FFT, checked against the direct product `Prod_p (1 + e(u 3^p / m_T))` at `k <= 9`, giving the raw count `N_T` with `A = 0` and `A = K_T` kept, the aggregate `u != 0` share, the absolute sum `Abs_k` and the largest `u != 0` term; the mass at the cyclotomic `T` for `t = 2..5`; the antipodal family `m_T = (3^(pt) + 1)/(3^t + 1)` at every odd `p <= 19` and every shift `s <= t` with `k <= 19`; and the cut-free aggregate `M_k = Sum_T N_T` to `k = 19` by meet in the middle with its bands. `--umax` past 12 is FFT time growing sixfold per step; never needs the `Z(R_k)` column.
- `... sweep 16384` is the deepest box sweep, 373 s against 15.0 s at 4096 and 0.6 s at 1024, so cost grows near `X^2.3`; `... box 12 18` is the deepest cover, 115 s and 0.4 GB, and `cover` refuses anything past `1.5 * 3^(2n + extra) >= 2^63` because the cell arithmetic is int64.

## CLAIMS

- The halved band is proved and essentially attained: to height 3000 no pair violates the cap `(z_1-1)/2 + (z_2-1)/2 + 1`, the largest reachable set is 2921 states at `(2997, 2926)` against a cap of 2961, a fraction `0.9865`, and the cap is met outright at `(3, 1)`; that fraction is the maximum and not the rule, the mean reach being `sqrt(w)`-sized below; the two automata agree on every pair to height 120 (`ratio.py band`, `ratio.py check`). **Proved.**
- No direction with `3` dividing neither coordinate is occupied, by the low-digit collision above; the gasket rays to height 60 are symmetric under the coordinate swap and every one of them has exactly one coordinate divisible by 3 (`ratio.py check`). **Proved.**
- The decision procedure reproduces the earlier census exactly: `A(n, 3^5) = 384, 408, 426, 436, 446, 460, 468, 472, 474` at `n = 10..18` against the `occupancy-decay` reading `384 .. 474`, and `A(9, 3^7) = 2818` reproduces the pinned regression from a generator that never enumerates the gasket. **Verified.**
- The uncapped counts are `A(X) = 32, 80, 206, 572, 1404, 4124, 9832, 26638, 72014, 184266` at `X = 32, 64, ..., 16384`, with `log A / log X` inside `[1.2057, 1.2494]` and the local exponent inside `[1.3554, 1.4380]` over `X = 2048..16384`, against the box exponent 2 and the `1.8073` that Conjecture O asks; saturation is genuine, `A(243) = 482` and `A(2187) = 10854` (`ratio.py sweep`, `levels`). **Verified.**
- The inequality `A(3^n) >= R(n)` holds with `R(n)` the number of occupied non-fibre rays of level `n`, since every such ray is primitive with both coordinates below `3^n`, hence of height below `3^n`, and no two distinct rays are the same rational. **Proved.**
- With the ray totals `R(13) = 1044840` (`lab/rs/dimension-one-ladder`) and `R(16) = 28545340` (`lab/py/occupancy-decay`) that inequality reads `A(3^n) >= 0.655 * 3^n` at `n = 13` and `>= 0.663 * 3^n` at `n = 16`, so the exponent is at least 1 and no route can win more than `eps = 1`. **Verified.**
- The weight layer is the sharp reduction: `Sum_{w <= X} Z(w) <= A(X) <= Sum_{w <= 2X} Z(w)`, so a pointwise `Z(w) <= C w^beta` gives `A(X) <= C' X^(1+beta)`, hence Conjecture O at every `alpha < 1/(1+beta)`; `beta < 1` gives `eps > 0` and `beta < 0.8073` gives O whole. **Proved.**
- The layer has a floor on the binary weights: if `w` is itself binary base 3 then every submask splits it, so `Z(w) >= #{a submask of w : 0 < a < w, gcd(a,w) = 1}`, and the floor is attained exactly at `w = 4, 13, 40, 121, 364` and missed by 6 at `1093` and `3280` (`ratio.py check`). **Proved.**
- The floor does not yet prove an exponent, and the corridor's lower end is open: 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`, so the floor beats the missing-digit exponent at odd `k` and loses at even `k`, and no family is proved to supply infinitely many good weights. What is proved is only the target `beta < 0.8073`; that `beta` cannot fall below `log 2 / log 3 = 0.6309297` is **Conjecture**, and the corridor above it is measured out below.
- The floor is where the layer actually sits, and the corridor holds across the sweep: `(W, Z_max, argmax)` reads `(32, 6, 13), (64, 8, 40), (128, 30, 121), (256, 30, 121), (512, 32, 355), (1024, 66, 757), (2048, 132, 1093), (4096, 136, 3271), (8192, 266, 7381), (16384, 500, 9841)`, every argmax a binary base-3 integer and `121` extremal at two consecutive caps, with `log Z_max / log W` inside `[0.5000, 0.7010]` printed by the sweep itself, a margin of `0.106` below the `0.8073` the reduction needs. **Verified.**
- **The top digit fixes the shape of every occupied direction.** The highest base-3 digit `3^t` of `m(z_1 + z_2)` belongs to exactly one of the disjoint binary numbers `m z_1`, `m z_2`, and the other is a sum of distinct powers below `3^t`, hence at most `(3^t - 1)/2`; so `max(z_1, z_2) > 2 min(z_1, z_2)` and the slope `z_1/w` never lies in `[1/3, 2/3]`. Nothing violates it: no occupied direction of weight at most 8192, no occupied pair to height 120, and no gasket ray at `n = 12` (`ratio.py layers`, `ratio.py check`). **Proved.**
- **The layer is one congruence class per direction, so the congruence route and the weight-layer route are the same bound.** From `r = z_1 z_2^(-1) mod 3^k` and `z_2 = w - z_1` comes `z_1 (1 + r) = r w`, and `r = -1 mod 3` would force `3 | w`, so `1 + r` is a unit and `z_1 = r w (1 + r)^(-1) mod 3^k` is determined; with `0 < z_1 < w < 3^k` the map `z_1 -> r` is injective and `Z(w) <= 2 |R_k|` at the least `k` with `3^k > w`. Hence `beta < 1` from this side needs `sigma_k` to fall geometrically, which is exactly what criticality forbids. The bound is not weak at the start: `sigma_k = |R_k|/3^k` equals `1/9` exactly at `k = 2, 3, 4` and first falls below at `k = 5`, where `23/243 = 0.0947`, and `2 |R_k|` already beats the trivial `(2/3) phi(w)` at `w = 13` (6 against 8.0) and at `w = 121` (46 against 73.3). It is weak at the end: on the `occupancy-decay` reading `|R_13| = 73440` it allows `146880` at `w = 797161` against the true `Z = 10388`, an overshoot past a factor fourteen (`ratio.py weights`, `ratio.py check`). **Proved.**
- **The metric route is exactly a covering number, and the covering number measures too large.** In the slope coordinate the ratio set is `P = {N/D : D binary base 3, N a submask of D}`, two of its points of denominator `w` differ by at least `1/w`, so an interval of length `1/w` holds at most two and `Z(w) <= 2 N_P(1/w)`; that sandwich is the proved half. The cover itself is only measured: counting both swap halves and three digits past the scale gives `N_P(3^-n) = 106874, 297974, 834624, 2347108, 6627474, 18775754, 53345672` at `n = 12..18`, each a lower estimate of the saturated count, since at `n = 12` the count climbs `103248, 105878, 106626, 106874, 106958, 106988, 106994` over zero to six extra digits with the increments falling by a factor near three. `n N_P / 3^n` rises `2.4132 -> 2.4785`, `log_3 N_P / n` rises `0.8783 -> 0.8997` and the step exponent rises `0.9333 -> 0.9504`, every reading monotone and every one already above the `0.8073` the reduction needs. So on the data the cover is `3^n / n` up to a constant, the route yields `O(w / log w)` and cannot reach `w^0.8073` - the same `1/log` ceiling that stops the congruence seed, and the reason a missing-digit import belongs at the 3-adic ratio set and not at the slope. Seven points with a constant still rising is a measurement and never an asymptotic (`ratio.py box`). **Proved / Verified.**
- **Returning to `0` is barely harder than surviving.** Let `Zinf(w)` count the directions whose start reaches the core, that is those with a 3-adic witness rather than an integer one; then `Z(w) <= Zinf(w)`, and the excess is small: `Zinf/Z` reads `1.0, 1.0, 1.3125, 1.0417, 1.2425, 1.2122, 1.1516, 1.5295, 1.356, 1.3489, 1.524` at `w = 121, 244, 355, 730, 757, 1093, 2188, 3271, 3280, 6562, 9841`, worst `1.5295` at `w = 3271`, with no occupied direction to height 120 missing the core. So the target may be relaxed from hitting one state to surviving at all without losing the exponent (`ratio.py core`, `ratio.py check`). **Proved / Verified.**
- **The layer band, per weight and deeper.** The exact layer scan to 8192 prints per octave the tuple `(octave, argmax Z, Zmax, argmax of log Z / log w, that exponent, argmax of Z / w^(log 2 / log 3), that constant)`: `(32, 40, 8, 40, 0.5638, 40, 0.7804)`, `(64, 121, 30, 121, 0.7093, 121, 1.4556)`, `(128, 244, 30, 244, 0.6188, 244, 0.9351)`, `(256, 355, 32, 283, 0.6025, 283, 0.8516)`, `(512, 757, 66, 757, 0.632, 757, 1.0071)`, `(1024, 1093, 132, 1093, 0.6979, 1093, 1.5975)`, `(2048, 3271, 136, 2188, 0.6349, 2188, 1.031)`, `(4096, 7381, 266, 6562, 0.6319, 6562, 1.0078)`. All twenty-four argmaxes are binary base-3 integers, a flag the scan prints for itself; the largest per-weight exponent is `0.7093` at `w = 121` and the largest constant `1.5975` at `w = 1093`, a margin of `0.098` below the `0.8073` the reduction needs, read weight by weight rather than off a running maximum (`ratio.py layers`). **Verified.**
- **The two extremal families hold that exponent seven octaves further out.** On the repunits `w = (3^k - 1)/2` at `k = 9, 11, 13` the layer reads `Z = 500, 2360, 10388` with `log Z / log w = 0.676, 0.6818, 0.6806` and `Z / w^(log 2 / log 3) = 1.5124, 1.7845, 1.9637`; on the shift weights `w = 1 + 3^h` at `h = 7, 9, 11, 13` it reads `Z = 132, 470, 2500, 11056` with exponents `0.6349, 0.6223, 0.6475, 0.652` and constants `1.031, 0.918, 1.2207, 1.3497`. Off those families the layer collapses: `Z = 26, 64, 68, 332` at `w = 88571, 88574, 797159, 797162` with exponents `0.2861, 0.3651, 0.3106, 0.4272`, one to two orders below their structured neighbours. So the whole weight of the conjecture sits on the binary weights, and up to `w = 1594324` their exponent never exceeds `0.6818` (`ratio.py weights`). **Verified.**
- **The square-root law, and where it breaks.** The forward reachable set of the critical band walk has mean size `0.2249` to `0.2947` times `sqrt(w)` at the four unstructured weights above, flat across a factor nine in `w`, which is the total-progeny law of a critical branching process confined to a band of width `w/2` and predicts `Z(w) ~ sqrt(w)` - the measured floor of the corridor. The structured weights are exactly where it fails: the same ratio reads `0.4137, 0.5046, 0.6422` on the repunits at `k = 9, 11, 13` and `0.9665, 1.4225, 1.9994, 2.9183` on the shifts at `h = 7, 9, 11, 13`, growing like `w^0.16` there (`ratio.py weights`). **Conjecture.**
- **The corridor closes to one value.** Below `8192` nothing beats `Z(w) <= 1.5975 w^(log 2 / log 3)`, the maximum being taken at `w = 1093`, the repunits beat it from `k = 11` on (`1.7845`, `1.9637` at `k = 11, 13`), and the exponent on the extremal families is flat at `0.68` and falling out to `w = 1594324`; so `beta = log 2 / log 3 = 0.6309297` is conjecturally both ends of the corridor, which is `0.1763` clear of the `0.8073` that gives Conjecture O whole and yields `alpha < 0.6131`. **Conjecture.**
- The criticality fixes the congruence seed at a polynomial law and predicts its constant: mean out-degree 1 gives survival `sigma_k ~ C/k`, hence `c_k = log_3((k+1)/k)` and `k c_k -> 1 / log 3 = 0.9102392`, against the `occupancy-decay` reading `[0.8418, 0.8628]` rising over `k = 13..18`; on this mechanism no congruence route can ever buy an exponent, and the `0.2618596` Cauchy-Schwarz cap is never approached. **Conjecture.**
- **The repunit layer, exactly as far as it goes.** On `w = R_k = (3^k - 1)/2` the floor `Phi_k = #{S : {} != S != [0,k-1], gcd(a_S, R_k) = 1}` is `Sum_{q | rad R_k} mu(q) N_k(q)` with `N_k(q) = q^(-1) Sum_{t mod q} P_{q,t}^(k / ord_q(3))`, `P_{q,t} = Prod_{r < ord_q(3)} (1 + e(t 3^r / q))`, so `Phi_k = 2^k - 2` at prime `R_k` and `N_k(p) = (2^k + p - 1)/p` whenever `2` is a power of `3` mod `p`; the verb prints the direct count, the residue DP and the Fourier form and they agree at `k = 2..15`: `2, 6, 8, 30, 24, 126, 112, 450, 460, 1958, 1344, 8190, 8064, 27360`. Every binary multiple of `R_k` below `3^(2k)` is `K_T = a_(T^c) + 3^k a_T` with multiplier `1 + 2 a_T` or `R_(2k)`, the lift sets `Occ_T` are occupied, and their union equals `Z(R_k)` at `k <= 10` and falls short by `18, 16, 108, 162, 624` at `k = 11..15`; `Z(R_k) = 2, 6, 8, 30, 24, 132, 118, 500, 530, 2360, 1634, 10388, 10440, 36190`, the excess ratio `X_k = (Z - Phi) / Phi` rises `0.0476 -> 0.3227` over `k = 7..15`, and the minimal witness reaches `436` digits with a column used `27` times at `k = 13`, so no witness family of bounded height is exact; the drift factors as `2^(log 2 / log 3) (1 - 3^(-k))^(-log 2 / log 3) (Phi_k / 2^k) (1 + X_k)` (`ratio.py repunit`). **Proved / Verified.**
- **The equidistribution model for the lifts is summable, and cannot be enforced one lift at a time.** `m_T = 1 + 2 a_T > 2 * 3^(max T)` and exactly `2^(t-1)` sets `T` inside `[1, k-1]` have `max T = t`, so `L_k = Sum_T 1/m_T < 1 + (1/4) Sum_{t >= 1} (2/3)^t = 3/2` at every `k`, reading `1.14285` at `k = 2` and `1.41723` at `k = 19`. At `k = 2t + 1` and `T = [t, 2t-1]` the multiplier is `m_T = 3^(2t) - 3^t + 1 = Phi_6(3^t)` with `(3^t + 1) m_T = 3^(3t) + 1`, so the `2^(t-1)` sets `S` inside `[1, t-1]` give submasks `A = (3^(3t) + 1) a_S` of `K_T` divisible by `m_T` and their `2^(t-1)` complements `K_T - A` are distinct, at least `2^t` in all against a model below `1`: no uniform `C 2^k / m_T^c` survives past `c = log 2 / (2 log 3) = 0.3154649`, while convergence of `Sum_T m_T^(-c)` needs `c > log 2 / log 3 = 0.6309297`, so every exponent that would close the lift half is refuted for that shape. `Occ_T` there is exactly `{(3^t + 1) a_S}` with its complements, of size `2(2^(t-1) - 1)` at prime `R_k` and `2, 6, 12, 30, 62, 100, 254, 510` at `t = 2..9`, an unconditional version needing `#{S inside [1, t-1] : gcd(a_S, R_k) = 1} >= 2^t / poly(t)` which is nowhere proved, and `max_T |Occ_T| m_T / 2^k` reads `4.562, 32.953, 151.898, 861.43, 4016.626, 14589.791, 83406.073, 376843.283` at odd `k = 5..19`, at that `T` every time (`ratio.py lifts`, `ratio.py check`). **Proved / Verified.**
- **The lift union to `k = 19`, and the aggregate that decides it.** `U_k` reads `2342, 1618, 10280, 10278, 35566, 31910, 175314, 128698, 715322` at `k = 11..19` against the floor `1958, 1344, 8190, 8064, 27360, 24384, 131002, 95040, 523982`, and exactly `U_k <= Sum_T |Occ_T| = agg_k L_k Phi_k`; over `k = 11..19` `agg_k` sits inside `[1.01748, 1.11457]` with no trend, `L_k` inside `[1.41043, 1.41724]`, `U_k / Phi_k` rises monotonically across `[1.19611, 1.36517]` and `U_k / Sum_T |Occ_T|` sits inside `[0.76088, 0.93128]`. So `U_k = O(2^k)` is the boundedness of `agg_k` and nothing else, an on-average equidistribution over the lift family; the same table reproduces `Z(R_k)` and the deep tail `18, 16, 108, 162, 624` at `k = 11..15` (`ratio.py lifts --kmax 19 --zmax 15`). **Verified.**
- Refuted as an input: no bound `D_n(q) <= C 2^n / q` holds uniformly over `q` coprime to 3. Every binary `m < 3^h` gives a binary `m(1 + 3^h) < 3^(2h)`, so `D_2h(1 + 3^h) >= 2^h`, read as equality for `h = 1..8`, and `D_2h(q) q / 4^h = (3/2)^h (1 + 3^(-h))` reads `2.0, 2.5, 3.5, 5.125, 7.625, 11.406, 17.094, 25.633` at `h = 1..8`; at `n = 20` the worst modulus below 500 is `q = 244 = 1 + 3^5` at ratio `1.8094`. The moduli that break equidistribution are exactly the shift-ray weights, so the divisor route to the power saving is closed (`ratio.py multiples`). **Refuted.**
- Refuted as a route: the witness is not short. Mean `lev` runs `3.875 -> 27.287` and max `lev` runs `6 -> 204` over `X = 32..16384`, mean `c` rises `1.553 -> 3.305`, and the share of occupied directions whose `lev` is below `1.8073 log_3 x` at the window cap `x` falls `0.875 -> 0.3102`, so bounding `A(X)` by `3^(max lev)` or by any level cap below `2 log_3 X` fails on a majority of the count. **Refuted.**

## WITNESSES

- coprime.md THE WINDOW AT DIMENSION ONE: the uncapped `A(X) = 32 .. 184266` at `X = 32 .. 16384`, `log A / log X` inside `[1.2057, 1.2494]`, and `A(3^n) >= 0.655 * 3^n` at `n = 13`.
- coprime.md THE WINDOW AT DIMENSION ONE: the band, the weight-layer sandwich and the reduction `Z(w) <= C w^beta` giving O at every `alpha < 1/(1+beta)`.
- coprime.md THE WINDOW AT DIMENSION ONE: the band cap unviolated to height 3000, largest reach `0.9865` of it, and `log Z_max / log W` inside `[0.5000, 0.7010]`.
- coprime.md THE WINDOW AT DIMENSION ONE: the top-digit law `max(z_1, z_2) > 2 min(z_1, z_2)` and the empty slope band `[1/3, 2/3]`.
- coprime.md THE WINDOW AT DIMENSION ONE: the congruence-layer identity `Z(w) <= 2 |R_k|`, sharp at `k = 2, 3, 4` and slack at `w = 797161`.
- coprime.md THE WINDOW AT DIMENSION ONE: the metric route, `n N_P(3^-n) / 3^n` rising `2.4132 -> 2.4785` over `n = 12..18`.
- coprime.md THE WINDOW AT DIMENSION ONE: the per-weight corridor, `0.7093` at `w = 121`, `1.5975` at `w = 1093`, the repunit and shift families to `w = 1594324`, and `beta = log 2 / log 3`.
- coprime.md THE WINDOW AT DIMENSION ONE: the divisor route refuted by `D_2h(1 + 3^h) >= 2^h` and worst modulus `q = 244` at `1.8094`, the short-witness route by mean `lev` running `3.875 -> 27.287`.
- coprime.md THE WINDOW AT DIMENSION ONE: the criticality reading `k c_k -> 1 / log 3 = 0.9102392` against `[0.8418, 0.8628]`.
- coprime.md THE WINDOW AT DIMENSION ONE: the repunit floor formula and its values to `k = 15`, the lift family and the deep tail `18, 16, 108, 162, 624`, the excess ratio `X_k` and the drift factorisation.
- coprime.md THE WINDOW AT DIMENSION ONE: the cyclotomic lift `m_T = Phi_6(3^t)` dividing `3^(3t) + 1`, the peak `376843.283` at `k = 19`, and the refuted exponent range against the one convergence needs.
- coprime.md THE WINDOW AT DIMENSION ONE: the lift union to `k = 19`, `L_k < 3/2` with its band, `agg_k` inside `[1.01748, 1.11457]` and `U_k / Phi_k` across `[1.19611, 1.36517]`.
- coprime.md THE WINDOW AT DIMENSION ONE: the Fourier form of the lift count, the antipodal family `(3^(pt) + 1)/(3^t + 1)` with `N_T = 2^(((p-1)/2)(t-s) + s)`, the cut-free aggregate `agg'_k` inside `[1.03919, 1.3403]`, and the absolute sum growing by `1.26` or more per step (`ratio.py agg`).
- coprime.md THE WINDOW AT DIMENSION ONE: the block ladder, the `2 * 3^k + 1` binary multiples of `R_k` below `3^(3k)`, the depth census `V_k` and the share of the deep tail it captures falling `0.4444 -> 0.1025` over `k = 11..15` (`ratio.py tail`).
- research/claims/ the top-digit, congruence-layer, occupancy-relaxation and `R_k`-offset rows.