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.