Coprimality at dimension one

• 2026-08-28 [Conjecture] No fixed modulus decides mixed-radix coprimality: the smallest moduli labelling coprimality exactly on the n = 12 sets are 27994 and 20736, at which all 4096 values occupy distinct residues, an encoding of the finite set rather than a transfer matrix; a prime not dividing M is invisible modulo M, so no fixed finite state space decides coprimality on an unbounded family, and a finite matrix tracks a fixed finite prime set exactly and nothing beyond, which is why the truncated Euler product through 13 misses by 0.008977 and -0.031676 on the two alternating schedules.
• 2026-08-28 [Refuted] The universal pair-prefix transfer matrix is a route to Conjecture W - its Perron root is k^2 = 4 for every dim under coupled digit vectors, or 4^dim under the scalar tensor reading, never 3; the 3 in W belongs to the shift multiplier-pair automata, where lambda(1, 3^r) = 3 exactly and every other coprime pair has lambda <= 2, with 2 attained at (1,4); the octave census that was fitted is the unweighted count, not W's weighted (3/2)^K sum, its exponent on the stabilised octaves j = 0..3 is 9.36, and the all-octave fit alpha = 2.956, CI [2.682, 3.257], leans on right-truncated high octaves with its constant drifting C = 1.042, 1.136, 1.244, 1.356 at n = 13..16, residual Durbin-Watson 0.261. Witness: gasket-ray-machine.
• 2026-08-28 [Refuted] Weil's theorem reaches the coprimality window - over F_3[t] the restricted coprime count grows like 4^n, a positive density among ordered pairs, so gamma = log_3(4) = 1.261860 and the window analogue (gamma/2, 1/2] is empty, gamma/2 = 0.630930 already above 1/2; the quantity is positive-density counting with no zeta error term, so the framework is sound over a field where the Riemann hypothesis is a theorem while carrying no zeta content; the like-for-like test is the F_q[t] analogue at a prime power q of the ray-multiplicity second moment, still undone. Witness: lab/py/function-field-density.
• 2026-09-06 [Proved] At digit length k = 2t+1 the lift T = [t, 2t-1] has multiplier m_T = 3^(2t) - 3^t + 1 = Phi_6(3^t) dividing the binary 3^(3t) + 1, so K_T carries at least 2^t submasks divisible by m_T against an equidistribution model below 1; no uniform bound of the shape C 2^k / m_T^c survives c > log 2 / (2 log 3) = 0.3154649 while Sum_T m_T^(-c) converges only for c > log 2 / log 3 = 0.6309297, so every exponent that would close the lift-union half is refuted for that shape. Witness: lab/py/ratio-set-saving, ratio.py check 2.5 s and ratio.py lifts --kmax 19 21 s.
• 2026-09-06 [Proved] Antipodal lift family: for odd p, 0 <= s <= t, k = (p-1) t + s, the set T = Union_(i odd) [ti, ti + t - 1] has m_T = (3^(pt) + 1)/(3^t + 1) and exactly 2^(((p-1)/2)(t - s) + s) submasks of K_T divisible by m_T, by antipodal pairs, blocks and balanced-ternary uniqueness. Witness: lab/py/ratio-set-saving, ratio.py agg asserts all 74 triples to k = 19, check at k <= 8.
• 2026-09-06 [Proved] A binary K with support inside [0, bk - 1], b blocks of k digits, is a multiple of R_k exactly when its column counts satisfy Sum_r c_r 3^r = j R_k, and for b <= 3 that forces the column vector constant, so the binary multiples of R_k below 3^(3k) are exactly 2 * 3^k + 1 lifts, 3^k with one position per column and multiplier 1 + 2 a_(E_1) + 2 (3^k + 1) a_(E_2), 3^k with two, and R_(3k), which yields only submask directions; at b = 4 the column vector branches, 24 non-constant vectors at k = 3. Witness: lab/py/ratio-set-saving, ratio.py tail and ratio.py check.
• 2026-09-06 [Verified] Occ_T at the cyclotomic T is exactly the set {(3^t + 1) a_S} with its complements, of size 2(2^(t-1) - 1) whenever R_k is prime and 2, 6, 12, 30, 62, 100, 254, 510 at t = 2..9, and max_T |Occ_T| m_T / 2^k reads 4.562 to 376843.283 at odd k = 5..19 at that T every time; an unconditional statement needs #{S in [1, t-1] : gcd(a_S, R_k) = 1} >= 2^t / poly(t), nowhere proved. Witness: lab/py/ratio-set-saving, ratio.py lifts --kmax 19 21 s.
• 2026-09-06 [Verified] The lift union to k = 19 satisfies U_k <= Sum_T |Occ_T| = agg_k L_k Phi_k with L_k = Sum_T 1/m_T < 3/2 Proved; over k = 11..19 agg_k sits inside [1.01748, 1.11457] with no trend, L_k inside [1.41043, 1.41724] and U_k / Phi_k rises monotonically across [1.19611, 1.36517], so U_k = O(2^k) is the boundedness of agg_k alone. Witness: lab/py/ratio-set-saving, ratio.py lifts --kmax 19 --zmax 15 10 min 43 s.
• 2026-09-06 [Verified] The absolute-value route on the u != 0 Fourier terms of the lift count is dead: Sum_T (1/m_T) Sum_(u != 0) |F_T(u)| / 2^k reads 1.3839 to 7.9155 at k = 5..11, step ratios all above 1.26. Witness: lab/py/ratio-set-saving, ratio.py agg, the Abs column.
• 2026-09-06 [Verified] The cut-free aggregate agg'_k = Sum_T (N_T - 2)/(2^k L_k) sits inside [1.03919, 1.3403] and M_k / 2^k inside [2.47278, 2.89356] over k = 11..19, no upward trend. Witness: lab/py/ratio-set-saving, ratio.py agg, 20 s.
• 2026-09-06 [Verified] Writing b(z) for the number of k-blocks the minimal witness lift m(z) R_k fills, U_k = #{b(z) <= 2} (Proved) and the depth-3 census V_k = #{b(z) <= 3} gives (U_k, V_k, Z(R_k)) = (2342, 2350, 2360), (1618, 1624, 1634), (10280, 10310, 10388), (10278, 10310, 10440), (35566, 35630, 36190) at k = 11..15, so depth 3 captures 8, 6, 30, 32, 64 of the deep tail 18, 16, 108, 162, 624, a share falling 0.4444, 0.375, 0.2777, 0.1975, 0.1025, and the one-position lifts add nothing at any k <= 13. Witness: lab/py/ratio-set-saving, ratio.py tail.