Failure of the Scalar Free-Fraction Closure of the Local Capacity–Metric Bridge

The Lorentz–capacity formulation carries an open bridge, LC-O2-O1: the equivalence between the capacity radius $R^{2}=1-\epsilon$ and the metric radius defined by the temporal lapse. A proposed closure reduces the local temporal coefficient to the free capacity fraction, $A_{\tau}=A_{\tau}^{\infty}(1-\epsilon)$. This note shows that the proposal fails. Assuming a diagonal temporal block, this law together with the principal-symbol identification $g^{\tau\tau}=-A_{\tau}$ makes the metric radius increase with occupancy, $R_{\mathrm{metric}}^{2}=1/(1-\epsilon)$, so the two definitions agree only at $\epsilon=0$; no constant renormalisation repairs this. What survives is the co-metric statement $g^{\tau\tau}/g^{\tau\tau}_{\infty}=1-\epsilon$, the metric inverse of the lapse, which does not govern proper-time dilation. Two further defects are independent of the inversion: the partial-trace form of the coefficient is a fresh postulate, and the occupancy is defined by the norm of a vector while the trace step treats it as a subspace. The result is narrow — a reciprocal scalar law would close the bridge algebraically — so what fails is the free-fraction law, not scalar closure in general. LC-O2-O1 is reopened. Interpretively, and as no part of the formal result, capacity appears to constrain the co-metric more directly than the lapse.