We construct a self-consistent mathematical-physical framework starting from five fundamental axioms of time, based on category theory, functional analysis and analytic number theory. Within this framework we rigorously prove the existence and uniqueness of the global time field, derive the energy-time duality relation of quantum mechanics, obtain natural boundary conditions of Berry-Keating Hamiltonian from first principles, and give a conditional proof of the Riemann hypothesis under the Hilbert-Pólya conjecture. Furthermore, Maxwell’s equations are fully derived as transverse vibration equations of the time medium, while gravitational, strong and weak interactions are interpreted as different geometric excitations of the same fundamental time field. Three feasible experimental verifications are proposed to falsify or validate core predictions of the model. This work provides a novel unification picture treating time as the unique fundamental physical medium of the Universe.