Local Spectral Closure of the Capacity–Metric Bridge

We close the local bridge between the capacity radius and the temporal co-metric component in the Lorentz-capacity formulation of the Cosmochrony programme. The local theory introduces an occupancy field ε(x) through the residual Born–Infeld capacity radius R(x)²=1-ε(x), while the metric formulation introduces the same radius through R(x)²=-2gττ(x). Their equivalence was previously stated as an open analytic bridge in. We prove that the identity \[ ε(x)=1+2gττ(x) \] follows directly from local admissibility data. The key input is that ε(x) is a quadratic occupancy in the rank-three sector Sym²(Vρ), normalised by the maximal Born–Infeld capacity σₘₐₓ=cBI². A local version of the admissibility-weight argument of then gives \[ Aτ(x)=2(1-ε(x)). \] Using the principal-symbol identification of the effective operator, together with the flat-space normalisation Aτ∞=2, gττ,∞=-1/2, yields \[ gττ(x)=-{Aτ(x)}{4}. \] Therefore R(x)²=1-ε(x)=-2gττ(x). No use is made of the Einstein equation. The bridge is closed spectrally from the local admissibility form alone.