Metretike of the Time Field: From Self-Referential Measurement to Gravitation, Euler's Identity, and Quantum Outlook

This paper constructs a complete metric-based optical framework rooted in Time Field Theory (TFT), achieving a self-consistent unification of conventional geometric optics, medium refraction, gravitational lensing, Fermat’s variational principle and Riemannian differential geometry under a single primitive ontology: self-referential perfect circular periodic oscillation. Conventional scalar gravitational theories and general relativity separate optical refraction and gravitational light deflection into independent physical effects, failing to explain the factor-of-2 discrepancy in gravitational bending angle from fundamental logic. TFT eliminates this theoretical split by redefining the measurement benchmark: every spacetime point’s metric scale corresponds to the local oscillation period of the time field γ. Light deflection gains its double coefficient naturally from the synchronous distortion of both light propagation trajectories and local metric benchmarks, without ad-hoc correction terms. This work further realizes the trinity unification of three core mathematical systems within TFT: Euler’s identity acts as the existential grammar of individual self-closed periodic primitives; Fourier transform serves as the superposition grammar for synthesizing arbitrary complex physical states via periodic decomposition; Riemannian geometry is re-interpreted as the spatial distribution grammar of discrete periodic benchmarks, thoroughly liberated from the substantival "curved spacetime" framework. All core Riemannian geometric quantities (metric tensor, covariant derivative, connection, curvature tensor, geodesic) acquire clear physical images rooted in periodic self-consistency constraints, with every mathematical structure proven logically inevitable rather than arbitrary auxiliary tools. The entire theoretical deduction relies solely on the core axiom of metric self-consistency and the flux conservation field equation of TFT, introducing no extra hypothetical entities such as dark matter, dark energy or absolute spacetime. This paper fills a long-standing ontological vacancy in traditional optics and gravitational theory, and consolidates the foundational mathematical architecture of Time Field Theory as a unified monistic physical framework.