# 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.