Coherent Coupling and Emergent Geometry: A Variational Closure of Modal Dynamics

In previous formulations of structural modal dynamics, the evolution of mode states Ri relied on a fixed coupling matrix Kij, leaving open the fundamental origin of geometry, gauge structure, and physical observables. In this work, we resolve this incompleteness by introducing a variational principle that dynamically determines Kij from the current configuration of Ri. The resulting coupled system defines a self-consistent evolution in which metric structure, gauge potentials, and curvature naturally emerge from coherence-based relations. Numerical simulations demonstrate quantum-like phenomena such as interference, tunneling, Zeno suppression, and decoherence. The theory supports stable coherent clusters, reproduces Dirac-like spectra via spinor kernels, and admits a continuous field limit. This variationally closed framework thus elevates the structural modal approach into a predictive, internally coherent physical theory.