Time Field Theory (TFT): Local Effective Speed of Light, Local Mass-Energy Relation and Classical-Quantum Compatibility

Based on the axiomatic system of conservation of local intrinsic total speed and the canonical quantization framework of Time Field Theory (TFT), this paper conducts rigorous and self-consistent derivations on the local effective speed of light, the time passage rate of stationary particles, and the local mass-energy relation in gravitational fields. Starting from the first principles of TFT, this paper strictly clarifies the dual physical identity of the vacuum speed of light c, independently derives the local mass-energy relation in gravitational fields E₀ (r) =mc² (r), and verifies the compatibility of TFT with standard quantum mechanics and the relativistic mass-energy relation. The full text strictly abides by the original axioms without additional ad hoc assumptions, and clearly distinguishes the conclusions independently derived by TFT from the external input of the standard quantum field theory paradigm. The results show that TFT can reasonably explain the physical origin of c in the mass-energy equation, realize the synchronous modulation of space-time scales and particle energies by the time field (r), and maintain complete mathematical self-consistency and observational consistency in both classical gravity and quantum compatibility. This paper provides a rigorous, detailed and academically standardized special paper for the TFT system, focusing on the intrinsic connection between local speed conservation, time field dynamics and mass-energy equivalence.