Temporal-Gradient Closure in the Enchan Field: A Common Origin for Gravitational Acceleration and Finite-Tension Response

One local spatial gradient of the Enchan time-dilation field supplies both gravitational acceleration and the finite-tension kinetic response. In the static spatial weak-field effective definitions, sqrt(X) = ell0 |grad S| = |g|/a0 exactly, with ell0 = c^2/a0. This paper isolates the common-gradient identity already implicit in published Enchan ingredients and verifies the frozen quadrature kinetic vessel dF/dX = mu_quad(sqrt(X)). The observable g = c^2 grad S remains linear in the gradient; the solved source-to-gradient response becomes nonlinear through its gradient-dependent coefficient. All TG0–TG4 audits pass: exact symbolic identities, a 241-point derivative sweep over X from 10^-24 to 10^24 at 40/60/80 digits, one numerical gradient supplying two readouts on a synthetic Gaussian with second-order refinement, a matched linear mu=1 flux control, and asymptotic checks. The 80-digit maximum derivative relative error is 7.01e-67; the finest dual-readout peak-scaled residual is 1.83e-16. With the inherited spherical exterior source normalization, the quadrature branch gives g^2 = g_bar^2 + a0 g_bar. This is a mathematical clarification within the effective formulation, with an explicitly synthetic illustration and no observational data or fitted parameters. It does not derive finite tension from continuity, a microscopic kinetic origin, or an all-regime nonlinear tensor action. The full covariant invariant is distinguished from its static spatial weak-field reduction. The quadrature coefficient tends to one at high gradient and is not identified with distinct concentration-suppressing Enchan vessels. Version 1.0 includes the manuscript PDF, a compact reproducibility ZIP with standalone Python code, frozen protocol, CSVs, four figures, bibliography, provenance and manifests, plus separate LICENSE and README. The manuscript TeX is excluded from the ZIP. The PDF is the default preview. License: Enchan Research & Verification License v1.0. Copyright (c) 2026 Mitsuhiro Kobayashi. Full terms appear in LICENSE and the Rights metadata.