Certified bounds on the connected vacuum energy of a three-plaquette path in Hamiltonian SU(3) lattice gauge theory

We study an open path of three plaquettes in Hamiltonian SU(3) lattice gauge theory at the fixed Wilson coefficient j = 6145/3072, with C₂(3) = 4/3. Let E₁, E₂, E₃ be the neutral ground energies of one plaquette, an adjacent pair and the path. We prove −4.6177250 × 10⁻⁵ ≤ E₃ − 2E₂ + E₁ ≤ −2.555412 × 10⁻⁶. Thus the second outer plaquette lowers the ground energy more than the first. The neutral excitation gap on the same ten-link graph satisfies 1.171238937859 < Δpath < 4.726131 in lattice energy units. The estimates apply to the untruncated Hamiltonian. We retain interference terms by adding all amplitudes for a common output before taking its squared Haar norm. Rational trial vectors, inertia tests and an offline verifier reproduce the bounds. The connected energy gives one local coefficient for a plaquette expansion. Extension to a volume-uniform estimate requires control of larger clusters and their boundary interactions.