Ganainy Temporal Primacy Theory (GTPT) Coercive Temporal Dynamics and Emergent Lorentzian Causality

This work establishes a mathematically rigorous and closed framework for the Ganainy Temporal Primacy Theory (GTPT), demonstrating that relativistic causal geometry can emerge directly from coercive temporal dynamics without assuming any prior spacetime metric structure. Starting from a second-order temporal operator with positive energy density and a uniformly positive-definite spatial rigidity tensor, we uniquely derive: (i) the principal symbol, (ii) the induced Lorentzian contravariant metric, (iii) the real non-degenerate characteristic cone, and (iv) a finite propagation (domain-of-dependence) theorem obtained through energy methods. We further prove that lower-order contributions (damping, memory effects, and potentials) leave the causal geometry invariant. The theoretical framework is supported by high-fidelity numerical simulations including heterogeneous and anisotropic media in two and three spatial dimensions, convergence verification, energy diagnostics, interface scattering, anisotropic lensing, and inverse parameter reconstruction. The theory produces concrete experimental predictions, including direction-dependent time-of-flight anomalies, causal refraction governed by a Snell-type law, and a program for what we term Temporal Material Physics — the classification of physical media through operator-induced causal invariants. This work provides the mathematical core for subsequent GTPT developments, including extended temporal geometries and multi-field operator structures.