Construction of Four-Dimensional $SU(3)$ Yang-Mills Theory and Non-Perturbative Spectral Mass Gap in the Fundamental Pocket Model Discrete-Iteration Continuum Theory (FPM-DICT)

We present a non-perturbative construction of a four-dimensional $SU(3)$ quantum Yang-Mills theory formulated within the Fundamental Pocket Model Discrete-Iteration Continuum Theory (FPM-DICT). The framework establishes a discrete, integer-indexed scale hierarchy $N \in \mathbb{N}_{0}$ governed by geometric scale recursion $R_{N+1} = q R_{N}$ ($q \approx 0.951229$) while preserving continuous 4D Euclidean spacetime $\mathbb{R}^{4}$ at every discrete tier. We construct the microscopic state space $\mathbb{F}_{N}$ and derive a gauge-covariant state mapping operator $\Pi_{N}: \Psi_{N} \to H_{N}$ that generates local $SU(3)$ gauge connections from the 21-Element Anchor tensor $A_{21}$. Osterwalder-Schrader reflection positivity is proved for the regulated lattice path integral measure, enabling physical Hilbert space reconstruction and self-adjoint Hamiltonian transfer matrix construction $\hat{H}_{N}$. By evaluating the connected Euclidean two-point correlation function of the scalar glueball operator $C_{N}(x) = \text{Tr}[F_{\mu\nu}^{(N)} F^{\mu\nu,(N)}]$, we demonstrate exponential correlation decay in the continuum limit $a \to 0$, establishing a strictly positive gauge-invariant spectral mass gap $m_{gap}(N) > 0$ across all iteration depths.