The Mathematical Soul of the Equivalence Principle: Physical Manifestation of the Local Linearization Principle — Re-founding the Cornerstone of General Relativity via Time Field Theory (TFT)

Einstein's equivalence principle — the local indistinguishability between gravity and acceleration — is the cornerstone of general relativity, yet Einstein never explained why it holds. Based on the axiomatic system of measurement theory in Time Field Theory (TFT), this paper presents a rigorous mathematical proof of the equivalence principle: gravity is not a force, but a manifestation of the time field gradient; the time field is continuously differentiable in space; according to the local linearization property of differentiable functions, any continuously differentiable function approximates to linearity within an infinitesimal neighborhood. Therefore, within a locally infinitesimal region, the time field gradient approximates to a constant, and both gravitational acceleration and inertial acceleration reduce to the same constant gradient — the equivalence principle naturally holds. This paper further reveals that the root cause of the equivalence principle being valid locally but broken globally is precisely the trivial fact of differential calculus that linear approximation is only valid locally. This argument not only provides a deeper logical foundation for general relativity, but also offers a novel answer to the philosophical proposition of "why mathematics can effectively describe the physical world". This paper further explores the quantum-classical boundary of the equivalence principle: the Planck mass, as the critical scale of the time field transitioning from the fluctuation-dominated phase to the condensation-dominated phase, provides a mapping interpretation based on metric autonomy for "why macroscopic objects exhibit no quantum superposition".