Failure of the Scalar Free-Fraction Closure of the Local 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.

Versions 1.0 and 1.0.1 of this note claimed to close that bridge from local admissibility data alone. That claim is withdrawn. The present version establishes instead a precise negative result: the natural scalar reduction moves the temporal coefficient in the wrong direction, so the chain $\epsilon(x) \mapsto A_\tau(x) \mapsto g_{\tau\tau}(x)$ does not close.

Scope statement. The result is deliberately narrow. A reciprocal scalar law $A_\tau = A_\tau^{\infty}/(1-\epsilon)$ would close the bridge algebraically; it is not derived here, and none is available in the corpus. What fails is therefore the free-fraction law, not scalar closure in general. The paper does not claim that an operator-valued datum is necessary, nor that a covariantly transforming family of local subspaces is impossible.

Main result (negative)

Theorem. Assume the free-fraction law $A_\tau(x) = A_\tau^{\infty}\bigl(1-\epsilon(x)\bigr)$, the Q11 principal-symbol identification $g^{\tau\tau} = -A_\tau$ (flat-space value $g^{\tau\tau,\infty} = -2$, from $g^{\mu\nu} = 2\eta^{\mu\nu}$), and a diagonal temporal block $g^{\tau i} = 0$ -- without which $g_{\tau\tau} \neq 1/g^{\tau\tau}$ in general. Then:

$R_{\mathrm{metric}}(x)^{2} = g_{\tau\tau}(x)/g_{\tau\tau}^{\infty} = 1/\bigl(1-\epsilon(x)\bigr)$,

so agreement with the capacity definition $R_{\mathrm{capacity}}^{2} = 1-\epsilon$ requires $(1-\epsilon)^{2} = 1$, hence $\epsilon = 0$ in the physical range $0 \le \epsilon < 1$. The free-fraction law closes the homogeneous vacuum and nothing else.

The failure is one of direction, not of normalisation: a free-fraction law makes the temporal co-metric coefficient decrease with occupancy, whereas a Schwarzschild exterior requires it to increase.

No renormalisation repairs it

Constant rescaling is inert. For $\widetilde g^{\mu\nu} = \kappa g^{\mu\nu}$ the factor cancels in both normalised ratios. Taking $\kappa = 1/4$ reproduces the flat-space value $-1/2$ used in v1.0 while leaving the no-go intact.
A local conformal factor would beg the question. Agreement would require $\Omega^{2}(x) \propto (1-\epsilon(x))^{-2}$ -- a law read off the desired answer.
No densitised co-metric is available. Q5b identifies the principal symbol directly as $\sigma_{2}(L_{\mathrm{eff}}) = g^{\mu\nu}k_\mu k_\nu$, leaving only a global scale freedom.

What survives, and what remains open

$g^{\tau\tau}/g^{\tau\tau}_{\infty} = 1-\epsilon$ (admissible ansatz) $g_{\tau\tau}/g_{\tau\tau}^{\infty} = 1-\epsilon$ (does not follow -- the open bridge)

The co-metric statement can be posed as an ansatz, but it is the metric inverse of the capacity lapse and does not govern proper-time dilation.

Three defects of the withdrawn derivation

The trace form is a fresh postulate. W1 supplies cumulative admissibility weights of a filtered Dirichlet form -- a coefficient multiplying squared increments, not a partial trace, with no dependence on any occupancy field. Q11 supplies a Casimir eigenvalue, not a normalised trace over a variable subspace.
Type mismatch: vector versus subspace. The occupancy is defined by the norms of vectors, but the proof treats them as subspaces carrying a partial trace. For a genuine subspace, $\mathrm{tr}_C(\lambda I) = \lambda \dim C$ is quantised in $\{0,\tfrac13,\tfrac23,1\}$, whereas the argument requires $\lambda\|C\|^{2}$, continuous on $[0,1)$.
No local covariance law is stated. A proper subspace of an irreducible module is not invariant. This is an absence, not an impossibility: a family $C_{\mathrm{free}}(x)$ may transform covariantly. What v1.0 lacked was any such rule, without which Q11's flat-space anchor cannot be transported pointwise.

The last two defects are independent of the inversion: they would persist even if the monotonicity were correct.

Structure

Introduction -- statement of the open problem from LCII.
Quadratic local occupancy -- definition of $\epsilon(x)$ as a quadratic norm on $\mathrm{Sym}^{2}(V_{\rho})$.
The no-go -- Theorem: the free-fraction law yields $g_{\tau\tau}/g_{\tau\tau}^{\infty} = 1/(1-\epsilon)$, agreeing only at $\epsilon = 0$.
No renormalisation repairs the inversion -- constant rescaling is inert; local conformal factors beg the question; no densitised co-metric is defined.
Provenance and typing of the trace reduction -- the partial-trace form is a fresh postulate, and it is mistyped.
The missing local covariance law -- an absence, not an impossibility.
Status and reopened content -- LC-O2-O1 reopened; candidate routes.

Dependencies

LCII: source of the open problem (Remark 3.2, Section 7.3); fibrewise capacity lapse framework.
W1: admissibility weights of a Dirichlet form -- attribution corrected: W1 does not supply a partial trace.
Q5b: principal-symbol reconstruction of the effective metric.
Q11: co-metric normalisation $g^{\mu\nu} = 2\eta^{\mu\nu}$, giving $A_\tau^{\infty} = 2$ and $g^{\tau\tau,\infty} = -2$.

References

Jérôme Beau. Failure of the Scalar Free-Fraction Closure of the Local Capacity-Metric Bridge. Preprint, version 2.0, 2026. DOI: 10.5281/zenodo.20332042.

Relation to the Cosmochrony program

LCIIo1 is the fourth paper of the Lorentz Capacity sub-programme. Version 2.0 reopens the analytic bridge left open by LCII: the chain LorentzCapacity → TempProj → LCII → LCIIo1 does not close at its final step. The sub-programme remains complete for inertial kinematics; its non-homogeneous extension is not. Gravitational time dilation in the capacity language therefore rests on the declared postulate of LCII (Remark 3.2), not on a derivation from local spectral occupancy. The negative result is nonetheless informative: it identifies exactly why the most natural scalar reduction fails, and what an alternative construction would have to achieve.