Emergent Tensorial Propagation Geometry from Coupled Scalar-Time Closure Dynamics

Previous work in Time–Scalar Field Theory (TSFT) established that relativistic weak-field structure and the linearized Einstein equations emerge from scalar-time dynamics under controlled closure conditions. In particular, scalar-time gradients were shown to induce an effective weak-field metric structure whose trace-reversed perturbation satisfies the linearized Einstein system in the appropriate asymptotic regime. However, the propagation-level origin of the corresponding tensorial structureremained incompletely analyzed. The present work addresses this missing step directly. The objective is not to rederive Einstein gravity from the beginning, nor to postulate a fundamental tensor field independently of scalar-time structure. Rather, the goal is narrower and mathematically precise: to determine whether coupled scalar-time closure sectors generate an emergent effective rank-2 propagation geometry at the level of principalsymbol dynamics. Starting from the scalar-time field Θ(x^μ) and its associated second-order fluctuation operator, we construct the most general local propagation tensor generated by coupled scalar-time closure gradients. We show that single-sector closure backgrounds generate only rank-1 anisotropic propagation deformations, whereas coupled closure sectors naturally produce irreducible symmetric rank-2 propagation structure. Using geometric-optics analysis of the generalized closure operator, we derive the associated characteristic propagation cone and show that the admissible propagation geometry is governed by an emergent tensor G^μν = aη^μν + sigma(A,B) b_AB u^(A)μ u^(B)ν, where u^(A)μ = ∂^μ Θ^(A). We further demonstrate that when the coupled closure gradients span the local tangent space, the resulting propagation tensor acquires full rank and generates an effective hyperbolic characteristic geometry possessing independent anisotropic propagation sectors. The derivation establishes that effective tensorial propagation geometry can emerge rigorously from coupled scalar-time closure dynamics without independently postulating a fundamental tensor field. The result supplies the missing propagationlevel structural mechanism underlying the previously derived weak-field Einstein closure results of TSFT. The analysis presented here does not establish full nonlinear Einstein dynamics, graviton quantization, or complete transverse-traceless polarization closure. Rather, the central claim is narrower: coupled scalar-time closure sectors naturally generate the effective tensorial propagation geometry required for weak-field relativistic closure within the broader TSFT framework.