From Metric Self‑Consistency to the Constancy of Light Speed: A Unified Derivation of Inertial and Non‑Inertial Frames within Time Field Theory (TFT)

Within the axiomatic framework of Time Field Theory (TFT), this paper rigorously proves that the constancy of the speed of light is not an independent fundamental postulate, but an inevitable corollary of metric self‑consistency under the limit of a homogeneous time field. The derivation starts from the operational definition of time: the second is a count of atomic transition periods. This self‑referential structure enforces the self‑consistency of metric benchmarks. Furthermore, when an observer measures a moving object, the motion of the object naturally superimposes the fundamental velocity along the time dimension and the translational velocity along the spatial dimension. The requirement of consistent physical laws compels the two velocities to follow an orthogonal superposition rule, which leads to time dilation. This paper strictly proves the uniqueness of the orthogonal superposition rule: the Pythagorean theorem is the only mathematical form satisfying the self‑referential structure, isotropy and a universal upper speed limit. Mathematically, consistency is equivalent to the vanishing first‑order variation. In inertial frames with a homogeneous time field, this condition reduces to the light taking the extremal path, and the locally measured speed of light is always c. In gravitational fields with an inhomogeneous time field, the condition naturally extends to the flux conservation equation, which uniformly describes gravitational lensing and optical refraction. The whole derivation removes the distinction between special relativity and general relativity, discards the postulate of constant light speed as well as the equivalence principle. Metric self‑consistency holds universally from the beginning. The only difference between inertial and non‑inertial frames lies in whether the time‑field distribution is homogeneous, which is a quantitative rather than qualitative distinction.