Coherent Emergence: A Regularized Quantum Field Framework for Induced Gravity

We present a rigorous, standalone effective field theory (EFT) framework wherein macro-gravitational potentials emerge directly from the quantum expectation values and fluctuations of a real quantum scalar field. Initiating our framework from the fundamental quantum field and canonical quantization on a flat background, we completely eliminate the necessity for ad-hoc, external classical filtering kernels. Instead, we introduce an alternative paradigm of intrinsic quantum regularization via coherent states, demonstrating that the ultraviolet (UV) convergence of the emergent field and its gradients is naturally and dynamically governed by the spectral weight functions of macroscopically excited quantum states. By evaluating the normal-ordered bilinear gradient operators, we isolate the quantum-derived coherence correlation tensor, providing a fully gauge-invariant interface that couples the filtered microscopic energy budget directly to the macroscopic spacetime geometry. To account for backreactions beyond tree-level expectations, we extend the theory into the Feynman path integral formalism and functional loop corrections, showing that the resulting functional determinant regularizes vacuum polarization and self-interaction loops through the state's internal localization scale. From this non-perturbative foundation, we perform an analytical derivation of the effective macroscopic mass from coherent states, obtaining an exact closed-form solution expressed via modified Bessel functions of the second kind (K_ν). We prove that this mass profile exhibits an inherent quantum self-regularization at short scales while recovering the linear, additive classical particle picture over large astronomical horizons. Finally, we project this quantum-Poisson operator onto the interior structure of low surface brightness galaxies, achieving a comprehensive quantitative resolution of the cusp-core anomaly and inner velocity deficit. We show that our state-derived framework suppresses the unphysical power-law velocity surges typical of bare dark matter halos, yielding a precise numerical velocity matching the high-resolution empirical kinematics of the SPARC database within the 1σ observational error margins purely from first principles.