Spectral Vacuum Mechanism – Part XVII Spectral Confinement under Truncated SU(3) Gauge Embedding: Toward a Constructive QCD-Motivated Lattice Framework

ABSTRACT

This work develops the next stage of the Spectral Vacuum Mechanism (SVM), extending the constructive spectral-confinement framework from a truncated SU(2) gauge embedding (Part XVI) to the non-Abelian SU(3) gauge group, which is fundamental to Quantum Chromodynamics (QCD). We construct a finite-dimensional, truncated SU(3) gauge–matter Hamiltonian on a d-dimensional cubic lattice, incorporating matter fields, truncated link variables, and plaquette interactions. All local Hilbert spaces are kept finite-dimensional, and all interaction operators are uniformly bounded, enabling a fully constructive spectral analysis.

The central result of this study is the derivation of an explicit, dimension-dependent lower bound on the spectral gap

Delta: Delta >= m0 - 2 z(d) ( g Vmax + kappa Vmax ) - C_box(d) lambda Wmax

We introduce a group-dependent scaling factor g_G, revealing that the SU(3) interaction density introduces a "penalty" twice as large as the SU(2) case (g_SU(3) = 2 vs g_SU(2) = 1). This results in a more aggressive "magnetic stiffness" of the vacuum as the lattice dimensionality d increases.

Our analysis provides two major physical predictions:

1. Dimensional Stability: The vacuum remains stable and gapped for d = 3 and d = 4, but the gap bound formally vanishes for d >= 5, suggesting that four-dimensional spacetime represents the upper stability limit for this confinement mechanism.

2. Scaling Laws: We predict a specific correlation-length ratio between dimensions, R43 = xi_4 / xi_3 approx 1.41 +/- 0.15, and a corresponding increase in the string-tension scaling factor sigma_4 / sigma_3 approx 2.0.

These results demonstrate that the SVM framework remains robust under the complexity of SU(3) color symmetry, providing a rigorous path toward a constructive proof of the mass gap in QCD-motivated models.

Keywords

Spectral Vacuum Mechanism • SU(3) gauge theory • spectral confinement • mass gap • truncated gauge embedding • plaquette dynamics • constructive quantum field theory • lattice Hamiltonians • Block‑EGC • spectral gap bounds • confinement mechanism • correlation length • string tension • non‑Abelian gauge structure • QCD‑like models