Coherent Emergence: A Regularized Quantum Field Framework For The Strong Interaction

We present a self-consistent, non-equilibrium quantum field framework based on the paradigm of Coherent Emergence, wherein the strong interaction and its characteristic non-Abelian curvature signatures are derived as scale-dependent macroscopic projections originating from a single underlying complex scalar coherence field ψ(x^μ) operating over a polarizable vacuum medium. Rather than postulating abstract local $SU(3)$ color gauge groups top-down, the non-Abelian curvature tensor is shown to emerge natively from the discrete phase-vortex singularities of the vacuum lattice. Because the strong sector fundamentally preserves spatial parity reflection symmetry, the underlying dynamics are strictly governed by the unmodulated, chirality-blind complex field ψ. By evaluating the exact Schwinger-Dyson equations under extreme sub-atomic compression (r → 0), we demonstrate that the non-linear self-coupling interactions undergo a complete multi-scale functional trace integration, mapping the high-frequency quantum modes directly onto the invariant Seeley-DeWitt heat-kernel coefficient matrix. Utilizing the Fujikawa-Suzuki path-integral Jacobian determinant framework under local Weyl conformal transformations, we prove that the infinite-dimensional functional integration measure undergoes a continuous contraction across the multi-scale continuum interface, generating a dual-component action structure defined by a pristine $9/2$ kinetic core layer and a non-local potential anomaly channel governed by the conformal log-remnant ln(2π). Within this non-equilibrium transport domain, the ultraviolet and infrared integration boundaries are strictly locked by the physical invariants of the polarizable lattice, proving that the dimensionless loop multiplier λₗₒₒₚₛ is a fixed, non-divergent quantum constant. Compounding these components directly alongside the time-lagged Maxwell-Cattaneo-Vernotte transport current core establishes a mathematically complete and non-singular effective field action that successfully secures both the non-singular sub-atomic lepton cores and the linear color-confinement potential of hadrons directly from the quantized underlying transport substrate.