The Identity of Motion and Geometry: A Rigorous Mathematical Proof that Spacetime is the Kinetic Field

We present a rigorous mathematical proof that the spacetime metric g_μν of General Relativity and the fundamental kinetic field ν^μ of the Fundamental Speed Theory (FST) are not two distinct entities, but rather two mathematically equivalent descriptions of the same underlying physical reality. Starting from the foundational FST paper—validated on 171 SPARC galaxies with χ²_ν = 0.170 across five hierarchical validation levels—we establish six central theorems: 1. The Metric-Kinetic Map: The spacetime metric is uniquely determined by the kinetic field and its gradients via an explicit, closed-form expression. 2. The Einstein Correspondence: The Einstein tensor computed from this metric satisfies the Einstein field equations exactly when the kinetic field obeys the FST field equation. 3. The Invertibility Theorem: The map from the kinetic field to the spacetime metric is bijective—no information is lost or gained. 4. The Cosmological Extension: For a homogeneous, time-dependent kinetic field, the same map generates the FLRW metric with modified Friedmann equations. 5. The Gravitational Wave Correspondence: Gravitational waves are identified as propagating perturbations of the kinetic field. 6. The Gravitational Lensing Theorem: Weak lensing masses equal FST dynamical masses without dark matter. For 15 reliable SPARC galaxies, the median lensing-to-dynamical mass ratio at 300 kpc is 1.02. We provide complete dimensional analysis in SI units for every equation, trace all derivations back to the five foundational equations of FST, and supply exhaustive appendices with step-by-step calculations. The results establish that spacetime is not a fundamental entity but the perceptual manifestation of the kinetic field.