Local Spectral Closure of the Capacity-Metric Bridge

Overview

The paper LCII introduced the local Born-Infeld capacity radius $R(x)$ on two sides: on the capacity side, $R(x)^{2} = 1 - \epsilon(x)$ from a local spectral occupancy field; on the metric side, $R(x)^{2} = -2g^{\tau\tau}(x)$ from the temporal co-metric component. Their equivalence was stated in LCII (Remark 3.2, Section 7.3) as an open analytic bridge: it could only be verified through the Einstein equations imported from the Gravity branch.

The present paper closes this bridge purely spectrally, from local admissibility data alone. The proof proceeds through two steps: (i) a local admissibility-weight reduction that follows from $\mathrm{SU}(2)$-equivariance via Schur's lemma; (ii) the local principal-symbol reconstruction of the effective metric. The chain $\epsilon(x) \mapsto A_\tau(x) \mapsto g^{\tau\tau}(x)$ is closed without importing the Einstein equation.

Scope statement. The bridge identity $\epsilon(x) = 1 + 2g^{\tau\tau}(x)$ is a spectral admissibility identity, not a phenomenological identification between matter density and metric curvature. It depends essentially on $\epsilon(x)$ being a quadratic occupancy in $\mathrm{Sym}^{2}(V_{\rho})$ -- not an arbitrary additive scalar density.

Main result

The local capacity radius and the metric radius coincide:

$R(x)^{2} = 1 - \epsilon(x) = -2\,g^{\tau\tau}(x)$,

equivalently $\epsilon(x) = 1 + 2\,g^{\tau\tau}(x)$.

The proof proceeds through two independent identifications.

Step 1 -- Local admissibility-weight reduction (Lemma)

Define $\epsilon(x)$ as a quadratic occupancy on the rank-three sector $H_{\mathrm{eff}}(x) \simeq \mathrm{Sym}^{2}(V_{\rho})$, equipped with its $\mathrm{SU}(2)$-invariant quadratic norm. The occupied and free parts decompose orthogonally:

$C_{\mathrm{max}}(x) = C_{\mathrm{occ}}(x) \oplus C_{\mathrm{free}}(x)$,

so $\|C_{\mathrm{free}}\|^{2}/\|C_{\mathrm{max}}\|^{2} = 1 - \epsilon(x)$. Since $\mathrm{Sym}^{2}(V_{\rho})$ is irreducible under $\mathrm{SU}(2)$ and the admissibility weight $\mathrm{Ad}$ is $\mathrm{SU}(2)$-equivariant, Schur's lemma forces $\mathrm{Ad} = \lambda\cdot\mathrm{Id}$, so partial traces are proportional to subspace norms. Hence:

$A_\tau(x) = A_\tau^{\infty} \cdot (1 - \epsilon(x)) = 2\,(1 - \epsilon(x))$.

Step 2 -- Principal-symbol identification

The effective metric is reconstructed locally from the principal symbol of $L_{\mathrm{eff}}$. With the flat-space normalisation $A_\tau^{\infty} = 2$ and $g^{\tau\tau,\infty} = -\tfrac{1}{2}$ (from Q11):

$g^{\tau\tau}(x) = -\,A_\tau(x)/4$.

Combining: $-2g^{\tau\tau}(x) = A_\tau(x)/2 = 1 - \epsilon(x) = R(x)^{2}$. The bridge is closed.

Key contributions

Closes LC-O2-O1: upgrades Remark 3.2 of LCII from open postulate to theorem.
No Einstein equation: the capacity-metric identity is derived purely from local admissibility data.
Quadratic occupancy is essential: Schur's lemma applies only because $\epsilon(x)$ is a quadratic norm on $\mathrm{Sym}^{2}(V_{\rho})$, not an additive scalar.
Gravitational time dilation reinterpreted: near a mass source the capacity budget is more occupied, the residual radius $R(x) = \sqrt{1-\epsilon(x)} < 1$ slows local clocks, consistently with the standard gravitational redshift.

Structure

Introduction -- statement of the open problem from LCII.
Quadratic local occupancy -- definition of $\epsilon(x)$ as a quadratic norm on $\mathrm{Sym}^{2}(V_{\rho})$.
Local admissibility-weight reduction -- Lemma: $A_\tau(x) = 2(1-\epsilon(x))$ via $\mathrm{SU}(2)$-Schur reduction.
Principal-symbol identification -- local identification $g^{\tau\tau}(x) = -A_\tau(x)/4$.
Spectral closure of the capacity-metric bridge -- Theorem and Corollary (upgrade of LCII Remark 3.2).
Discussion -- role of the quadratic definition; Schwarzschild consistency check in the weak-field regime.

Dependencies

LCII: source of the open problem (Remark 3.2, Section 7.3); fibrewise capacity lapse framework.
W1: admissibility-weight reduction argument; local version applied here.
Q5b: principal-symbol reconstruction of the effective metric.
Q11: flat-space normalisation $A_\tau^{\infty} = 2$ and $g^{\tau\tau,\infty} = -\tfrac{1}{2}$.

References

Jérôme Beau. Local Spectral Closure of the Capacity-Metric Bridge. Preprint, 2026. DOI to appear.

Relation to the Cosmochrony program

LCIIo1 is the fourth paper of the Lorentz Capacity sub-programme. It closes the analytic bridge left open by LCII, completing the chain LorentzCapacity → TempProj → LCII → LCIIo1. The result shows that $\epsilon(x)$ and $g^{\tau\tau}(x)$ are not independent fields: they are two readings of the same local admissibility datum. Gravitational time dilation thereby becomes a derived consequence of local spectral occupancy, not an imported metric structure.