Explicit confinement bounds and vacuum-response localization in Hamiltonian SU(3) lattice gauge theory

We study the untruncated Hamiltonian 𝐻Λ(𝑢) = ∑𝑒 𝐶𝑒 − 𝑢 ∑𝑝 Re Tr 𝑈𝑝 on finite open rectangular subgraphs of the three-dimensional cubic lattice, with 𝐶2(3) = 4/3. For 0 ≤ 𝑢≤1/37,weobtainalowerbound𝐸𝑞𝑞,̄Λ(𝑅;𝑢)−𝐸0,Λ(𝑢)>0.656𝑅inthefullfundamental– antifundamental static-source sector, uniformly in the box and its transverse area. A rooted Fourier expansion of the true vacuum gives a separate kinematic spectral-gap bound 51/740, a volume-independent rate for logarithmic imaginary-time convergence, and summable maxi- mal correlations for a hierarchy of overlapping strips. At the larger coefficient 𝑗⋆ = 6145/3072, a positive confinement coefficient is certified on a fixed 2×6 tube. For growing cross-sections, the first exponential trial vacuum has an ordered three-state Potts restriction, which obstruct- s a fixed-size center heat-bath certificate. A Feynman–Kac polymer representation cancels the common bulk contribution before estimating the boundary response and gives an explicit short-time spatial localization bound. It also preserves the certified negative initial response on large finite cross-sections. Exact rational arithmetic and outward transcendental bounds accompany the analytic proofs. The target-coupling, area-uniform long-time estimate and the continuum limit remain open.