Anchored metric-dilation extraction of spatial and time couplings in a single-field $S^3$ ontology: operational test of $a_s=a_t$

This paper presents a numerical test of the equality $a_s = a_t$ within a single-field $S^3$ framework. Using an anchored metric-dilation insertion $\tau(x)=\tau_0\chi(x)$, the study extracts the spatial gravitational coupling ($a_s$) and compares it to the time-channel inertial response ($a_t$) derived from the fixed-field translation inertia. Main Results: The spatial coupling is calculated using a profile-free energy partition diagnostic ($a_{\text{part}}$), while the time coupling is derived from the linear inertial response of the soliton. For grid refinements $N=65$ and $N=81$, the ratio of the two couplings is found to be $|a_t|/|a_s| \approx 0.99$. The data supports the consistency of the inertial and spatial channels to within $1\%$ in the weak-field regime. Version 2 Update: This version adopts the profile-free partition coupling ($a_{\text{part}}$) as the primary spatial diagnostic. This replaces the slope-based estimator ($Q/M$) used in the previous version, resolving the normalization discrepancy and demonstrating near-unity agreement between the energy and inertia channels.