Width-uniform transfer-matrix gap certificates for the half-spin factor of a lattice gauge toy row: audited computer-assisted bounds at strong coupling and beyond the strong-coupling wall, with structural notes on Weil positivity

We study an explicit 2^w × 2^w transfer matrix Γ_w(β), the half-spin factor (all link spins ≤ 1/2) of the complete SU(2) transfer operator of an auxiliary open row of w plaquettes, and ask for a spectral gap that is uniform in the width w. (i) At β = 28/25 a Kotecký–Preiss polymer-current certificate gives |λ_k| ≤ e−2.90 λ_0 for all k ≥ 1 and all w ≥ 1 (computer-assisted, blind-audited). (ii) The bare current method stops at β ≈ 1.5 although the numerical gap stays between 3.4 and 4.5. (iii) After an exact “dressed” decomposition of Γ_w into connected interval operators, the connected parts obey an all-length geometric bound ‖K̂_L‖ ≤ (2009/50000)(381/2000)L−1 at β = 5/2 (exact rational certificate, not audited), with a closed-form decay rate. (iv) Beyond this wall, a time-ordered transfer certificate for the dressed gas with open time boundaries gives |λ_k| ≤ e−3.37 λ_0 for every w at β = 5/2, that is 98.8% of the numerical gap 3.4095, and |λ_k| ≤ e−5.38 λ_0 at β = 28/25 (computer-assisted, blind-audited except for its inputs from the tail certificate in (iii); one proof gap was found and repaired at no numerical cost). A second route (m ≥ 3.25) is still under audit. All certificates concern the half-spin factor only: with all spins the gap at β = 5/2 is 3.358, 3.312, 3.291 for w = 2, 3, 4, below 3.37 (5.433, 5.428, 5.426 at 28/25; certified by Astra at w = 2 and by our own, not yet audited, engine at w = 3, 4). These ratios follow a nearest-neighbour band law; a single triplet channel carries most of the higher-spin gain and, at β = 5/2, defeats a route-C certificate for the complete row. (v) We record exact structural facts: the both-open stretch of the row is a singlet/triplet mixer with a neutral singlet channel, and positivity of Γ_w forces λ_s ≥ 4p_2² − 3p_3, with physical margin Var(χ1/2) ∼ 6/β². A final part collects observations on Weil positivity for the Riemann zeta function: in the theta-dressed jump formulation the dressed hop rate equals the critical constant 1/2 up to an explicit correction localised where the kernel Φ is not negligible (an exact consequence of W ∗ Φ = 0), so null-function dressings are critical and admit no dilute regime. This indicates why a current-type method does not transfer directly. The Sonin compression of Connes–Consani, the natural candidate ground state with a margin, stops bounding the Weil form on our test banks once the prime 2 enters. We also report finite-size comparator and lane data, experiments on the arithmetic side, and a dictionary between the two problems with explicit open target inequalities. Nothing here is a statement about continuum Yang–Mills theory or a proof of the Riemann hypothesis. Author: Andrei Ursachi (A. Ursachi; Ursachi, Andrei), independent researcher, ORCID 0009-0002-6114-5011. Version 2 (31 October 2026). Supersedes version 1; corrects an error in Definition 1 and extends the results.