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 $g_{\mu\nu}$ of General Relativity and the kinetic field $\nu^\mu$ of the Fundamental Speed Theory (FST) are mathematically equivalent descriptions of the same physical reality. FST, validated on 171 SPARC galaxies ($\chi^2_\nu = 0.170$), establishes six central theorems: (1) Metric-Kinetic Map: $g_{\mu\nu}$ is uniquely determined by $\tilde{\nu}$ and its gradients. (2) Einstein Correspondence: $G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}^{(V)}$ holds when $\tilde{\nu}$ obeys the FST field equation with $\beta_{\text{eff}} = |\lambda|\nu_0^2/(6c_1)$. (3) Invertibility: The map from $\tilde{\nu}$ to $g_{\mu\nu}$ is bijective in the weak-field regime. (4) Cosmological Extension: A homogeneous kinetic field generates the FLRW metric with modified Friedmann equations and effective dark energy. (5) Gravitational Waves: Field perturbations satisfy $\Box h_{\mu\nu}^{\text{TT}} = 0$, identifying gravitational waves as kinetic field waves. (6) Gravitational Lensing: Weak lensing masses equal FST dynamical masses without dark matter (median lensing-to-dynamical mass ratio $\mathbf{0.88}$ for 15 reliable SPARC galaxies). We further prove a screened near-Schwarzschild limit showing that kinetic-field stress is exponentially suppressed in the high-density phase, yielding an exterior metric indistinguishable from Schwarzschild in all Solar-System tests. The results establish that spacetime is not fundamental but the perceptual manifestation of the kinetic field.