Spectral Gap Construction and Vacuum Uniqueness in Non-Commutative Ω̂G Operator Mechanics

This paper presents a constructive operator proof establishing a strictly positive spectral gap (Δ > 0) and a unique zero-energy vacuum state 0 for non-commutative Ω̂G quantum field dynamics on flat 4D Minkowski spacetime (H = L²(R⁴) ⊗ C¹⁶). Key Highlights & Updates: Constructive Proof: Fulfills Wightman positivity and vacuum uniqueness on flat space without invoking background metric curvature. Operator Dynamics: Refines the spectral cutoff framework using the prime-indexed dual dilation operator D̂ₖ across the manifold MK to prevent ultraviolet divergence and spectral collapse. Matrix Structure & Positivity: Details explicit 4 × 4 sub-block decompositions of the 16 × 16 internal coupling matrix Γab and verifies ghost-free positivity via the 64-element projector space Π̂₆₄.