Sharper vacuum-contraction constants and extended ranges for iterated coarse-graining in Hamiltonian SU(3) lattice gauge theory -A supplement to G3A and G4E

We revisit the explicit constants in the rooted-norm contraction for the logarithm of the vacuum of the Hamiltonian H = K − u Pp Re Tr Up on open cubic boxes (G4E) and in the entrance estimate for the iterated spatial coarse-graining of the vacuum (G3A). Three representation-theoretic observations drive the improvements: (i) a channel-share identity: when a fundamental plaquette trace is fused into an arbitrary Fourier label at a link, Schur’s lemma distributes the coefficient norm over the fusion channels in proportion to their dimen- sions; (ii) an endpoint Casimir bound for gauge-invariant labels, cτ ≥ c(τe)+max(4, 38 (p+q)) when the link e carries (p, q); (iii) exact integration of the marked edge in two-insertion stars and in single-face attachments of the G3A entrance expansion. As consequences, the mixed constant of G4E improves from 189/8 to 543/80, the quadratic constant from4331/10000) to 3/8, and the linear relative constant of G3A Lemma 6.1 from 81/8 to 15/4; the invariant-ball radius of the scalar majorant grows from 1/36.40 to 1/18.67. The second-order constant A = 753/8 is shown to be exact. Consequently Theorem 2.2 of G3A (static-source confinement and the neutral gap) holds for 0 ≤ u ≤ 1/19, with VΛ(R) ≥ 0.7406R and a neutral gap at least 2.9626 at u = 1/19, and Theorem 2.1 of G3A (convergence of the iterated dyadic marginals to product Haar measure) holds at v = 1/35 after optimizing the weights of its norm. All numerical claims are certified in exact rational arithmetic with outward round- ing; the verification code is included. In physical terms, static fundamental sources in the pure gauge theory are linearly confined, uniformly in the volume, on an interval of couplings about three and a half times wider than before, and on the original intervals the confinement coefficients and the neutral gap are sharpened. The results concern a fixed lattice spacing in the strong-coupling region; no statement about the continuum limit is made.