Iterated Coarse-Graining of the Vacuum and Confinement Bounds in Hamiltonian SU(3) Lattice Gauge Theory

We study spatial blocking of the true ground-state measure for the untruncated three- dimensional lattice Hamiltonian 𝐻 = 𝐾 − 𝑢 ∑𝑝 Re Tr 𝑈𝑝 with continuous time and 𝐶2 (3) = 4/3. For 0 ≤ 𝑢 ≤ 1/67, its first dyadic marginal has logarithmic vacuum norm 𝑋1/67(𝑆2) < 0.014848. Every subsequent blocking reduces this norm by a factor at most 20/21, uniformly in the open box and its admissible blocking depth. On any fixed number of coarse links the marginal converges in total variation to product Haar measure along boxes accommodating increasing depths. The entrance estimate uses a hidden link on each plaquette and a factor of 1/3 from the vertex gauge projections in a partially evaluated Fourier norm. A separate comparison in the static fundamental-pair sector gives 1.11259𝑅 < 𝑉Λ(𝑅; 𝑢) ≤ 4𝑅/3 at every graph distance 𝑅 ≥ 1, and the neutral gauge-invariant gap exceeds 4.45036. The estimates use a rooted fine-vacuum bound from G4E. Exact Schur elimination controls the generated energy forms in a common fine-lattice spectral window. All energy and distance bounds are in fixed lattice units.