Metric Optics: A First-Principles Unification of Geometrical Optics, Gravitational Lensing, and Calculus of Variations within the Time Field Theory Framework

Based on the metric axiom system of the Time Field Theory (TFT), this paper constructs a unified theoretical framework covering geometrical optics, gravitational lensing, and the physical legitimacy of mathematical tools. Starting from the meta-axiom that "time is the only self-consistent metric benchmark", this paper rigorously proves the mathematical equivalence between flux conservation and Fermat’s principle, derives Snell’s law of refraction and the law of reflection from first principles, establishes a unified reflection-refraction equation, and reveals the identical physical mechanism underlying gravitational lensing and optical refraction. This paper clarifies the ontological limitation of conventional geometrical optics, establishes the physical equivalence between gradient-index media and weak gravitational fields, and explains the physical origin of the double light deflection angle in TFT. It further demonstrates that the essence of photons is the extremal geodesic of the metric field; the zero intrinsic time of photons is the physical premise for the validity of the calculus of variations; and the constancy of the speed of light constitutes the physical foundation of calculus.