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

Current Zenodo release: version 2.1 (2026-10-24). Official title: Failure of the Scalar Free-Fraction Closure of the Local Capacity-Metric Bridge.

Overview

The paper LCII defines 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, $R(x)^{2} = g_{\tau\tau}(x)/g_{\tau\tau}^{\infty}$ from the temporal component of a supplied co-metric, whose flat-space reference is postulated. Their equivalence is a declared postulate of LCII (Remark 3.2); its derivation from admissibility data alone is the open problem LC-O2-O1.

This note shows that 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. The flat-space value $A_\tau^{\infty}$ is postulated: Q11 shows that spatial $\mathrm{SU}(2)$ invariance leaves the temporal coefficient free, and the normalisation $g^{\mu\nu} = 2\eta^{\mu\nu}$, in which $A_\tau^{\infty} = 2$, is a choice of units. The no-go does not depend on that value.

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 $A_\tau(x) = A_\tau^{\infty}\bigl(1-\epsilon(x)\bigr)$ with any postulated $A_\tau^{\infty} > 0$, the principal-symbol identification $g^{\tau\tau} = -A_\tau$ (flat-space value $g^{\tau\tau,\infty} = -A_\tau^{\infty}$), and a diagonal temporal block $g^{\tau i} = 0$, without which $g_{\tau\tau} \neq 1/g^{\tau\tau}$ in general. Then:

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

the value of $A_\tau^{\infty}$ cancelling. 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. From $A_\tau^{\infty} = 2$, taking $\kappa = 1/4$ reproduces the flat-space value $-1/2$ used in the proposed derivation 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 free-fraction derivation

The trace form is a fresh postulate. W1 version 2.0.1 retains a conditional partial-sum proxy for admissibility weights of a filtered Dirichlet form; it does not establish Q5a's actual weights or supply a partial trace, and has no dependence on any occupancy field. Q11 fixes no value of $A_\tau$: spatial $\mathrm{SU}(2)$ invariance leaves it free, and its Casimir value $2$ is a choice of units, 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 is missing is any such rule, without which the homogeneous form of Q11 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 $g_{\tau\tau}/g_{\tau\tau}^{\infty} = 1/(1-\epsilon)$ for any postulated flat value, 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 open content: LC-O2-O1 is open; candidate routes.

Dependencies

LCII: source of the open problem (Remark 3.2, Section 7.3); fibrewise capacity lapse framework.
W1: a conditional partial-sum proxy for admissibility weights of a Dirichlet form; W1 supplies neither Q5a's actual weight limit nor a partial trace.
Q5b: principal symbol $\sigma_{2}(L_{\mathrm{eff}}) = g^{\mu\nu}k_\mu k_\nu$, under [H-L] and [H-lift].
Q11: the conditional homogeneous form (Corollary 6.1); $A_\tau$ is free, so the flat-space value $A_\tau^{\infty}$ is postulated.

References

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

Relation to the Cosmochrony program

LCIIo1 is the paper of the Lorentz Capacity sub-programme on the capacity–metric bridge. It shows that the analytic bridge of LCII is not closed by the free-fraction law: the chain LorentzCapacity → TempProj → LCII → LCIIo1 does not close at its final step. The sub-programme is complete for inertial kinematics, conditionally on a supplied co-metric and on the cone–capacity identification [H-cone]; 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.