Particle Coherence and Relativistic Metric Closure in Time–Scalar Field Theory

Time–Scalar Field Theory (TSFT) models time as a scalar field governing propagation, coherence, and the persistence of physical structures. Previous TSFT work derived relativistic weak-field geometry and recovered special relativistic structure from scalar-time dynamics. However, the origin of the specific relativistic metric closure remained unresolved. In particular, the admissible scalar-induced weak-field geometry permits independent temporal and spatial response coefficients, leaving open the question of why the Einstein-compatible branch is physically realized. In this work, we derive relativistic metric closure directly from particle coherence. Starting from scalar-time wave propagation and the TSFT particle coherence framework, we show that massless coherent excitations propagate along null phase structures whose first-order deformation under scalar-induced geometry must remain compatible with single-scalar coherence. For the general admissible weak-field metric family, we demonstrate that unequal temporal and spatial response coefficients generate irreducible phase shear, which cannot arise from a single scalar deformation and therefore violates particle coherence. Eliminating phase shear uniquely selects equal temporal and spatial response coefficients, yielding the Einstein-compatible weak-field branch and PPN parameter γ = 1. This result unifies the TSFT particle coherence spine with previously derived relativistic geometry and provides a non-circular closure mechanism connecting particle persistence, scalar-time propagation, and relativistic gravitational structure within a single theoretical framework.