Time-Field Theory: Unified Derivation of Gravitational Redshift and Cosmological Redshift from Wavelength Invariance

Based on the core framework of Time-Field Theory (TFT) with flat spacetime and spacetime flux conservation, this paper establishes a rigorous and self-contained derivation of the wavelength invariance theorem for electromagnetic waves, and from it provides a unified physical explanation for both gravitational redshift and cosmological redshift. We prove that the wavelength of light is an absolute invariant during propagation—independent of the spatial distribution and temporal evolution of the time field γ(r,t). All observed redshift phenomena, whether gravitational or cosmological, are pure measurement effects arising from differences in local time benchmarks between the emitter and the observer. The physical mechanism is identical in both cases: the intrinsic wavelength of the photon never changes; only observers at different positions or different cosmic epochs, using clocks that run at different rates, measure different frequencies for the same invariant wavelength. This unified interpretation requires no spacetime curvature, no spatial expansion, and no additional entities beyond the scalar time field γ.