Infinite-Volume Vacua and Gap Transfer for Weakly Coupled SU(3) Cube Lattices

We study a three-dimensional SU(3) Hamiltonian lattice partitioned into disjoint cube blocks and single bridge links. Taking the untruncated, volume-uniform finite-box estimates of G4A as input, we construct the infinite-volume vacuum for inter-block plaquette strength 0 ≤ s ≤ 1/6144 and prove a gap at least 2/9 in its GNS representation. The full sequence of open-box vacua converges locally in trace norm to a pure, locally normal state. The gap also holds on the nontrivial cyclic space generated by bounded local gauge-invariant observables. The argument separates local energy tightness, the Nachtergaele–Sims–Young spectral flow for infinite-dimensional sites with unbounded on-site terms, and a centered spectral-measure proof of gap transfer. We state these requirements as an abstract transfer theorem. Additional quantitative consequences are an exponential spatial correlation bound with rate 32/599 in a specified coarse-site metric, convergent finite-history imaginary-time correlations with time-reflection positivity, and an isometric Spectral Chaos representation with an exact depth-norm identity. A local inverse-operator estimate gives excitation probability O(s2); combined with inherited one-cube rational data, it yields variance at least 1/300 for every internal real Wilson plaquette. Exact arithmetic and finite-geometry checks are supplied separately from the analytic proofs. All results concern a fixed lattice spacing and the stated deformed Hamiltonian. They do not establish boundary-condition-independent uniqueness, the isotropic point s = 1, or a continuum Yang–Mills theory.