Spectral Gap Construction and Vacuum Uniqueness in Non-Commutative $\hat{\Omega}_G$ Operator Mechanics

This paper constructs a self-contained Hilbert space representation for quantum field dynamics governed by the non-commutative operator $\hat{\Omega}_G$ over flat 4D Minkowski spacetime ($\mathcal{H} = L^2(\mathbb{R}^4) \otimes \mathbb{C}^{16}$). By evaluating the internal 16-channel coupling algebra and enforcing geometric cutoff invariance through dual-toroidal core boundaries ($V_{\text{core}} = 2\pi^2 R a^2$), we prove that the Hamiltonian spectrum possesses a unique zero-energy vacuum state $\vert{}0\rangle$ and a strictly positive spectral gap $\Delta > 0$. The construction utilizes a complete 64-element projector basis $\hat{\Pi}_{64} to eliminate negative-norm states and satisfies all Wightman axioms—including Poincaré covariance, the spectral condition, and microcausality—while preventing ultraviolet spectral collapse in the continuum limit.