Fibrewise Born-Infeld Capacity Lapse and Gravitational Time Dilation (LC-O2)

This repository contains the source of the paper Fibrewise Born-Infeld Capacity Lapse and Gravitational Time Dilation (LC-O2 of the Lorentz Capacity Programme) (Beau2026lco2).

Overview

The paper TempProj (Beau2026tp) constructs the temporal residual map $\mathcal{P}_\tau^{\mathrm{Q5b}}$ in the homogeneous (flat-space) regime and proves uniqueness. The present paper (LC-O2) extends this construction to non-homogeneous fibrewise regimes, where a localised spectral occupancy $\epsilon(x)$ reduces the local BI capacity budget.

Main Result

In a non-homogeneous configuration, the local BI capacity radius is $R(x)^2 = 1 - \epsilon(x)$. The fibrewise temporal residual map factorises canonically as:

$$\mathcal{P}_{\tau,x}^{\mathrm{Q5b}},\Sigma^{\mathrm{tot}}(x,n) = R(x),\sqrt{f(x,n)\bigl(2 - f(x,n)\bigr)}$$

where $f(x,n) = 1 - \beta(x,n)$ and $\beta(x,n)$ is the local normalised spatial mobility.

Gravitational time dilation emerges as:

$$\frac{d\tau}{dn}\bigg|_x = \frac{R(x)}{\gamma}$$

The local capacity radius is linked to the metric by:

$$R(x)^2 = -2,g_{\tau\tau}(x)$$

In the weak-field regime, $R(x) \approx 1 + \Phi_N(x)/c^2$, recovering the standard gravitational time dilation formula.

Key features

• The homogeneous case ($R(x) \equiv 1$) reduces to TempProj as a corollary.
• The local capacity radius $R(x)$ is identified metrically from Q5b/Q11; the open problem LC-O2-O1 (derive $R(x)$ directly from $\epsilon(x)$ without importing the Einstein equations) is closed by the companion paper LCIIo1 (Beau2026lco2o1).
• The factorisation $R(x) \times \sqrt{f(2-f)}$ cleanly separates local physics from the universal normalised Lorentzian structure.

Status

• Fibrewise map construction: structural
• Gravitational time dilation formula: structural (imports $g_{\tau\tau}$ from Gravity paper)
• LC-O2-O1 (derivation of $R(x)$ from $\epsilon(x)$ without Einstein equation): closed by LCIIo1 (Beau2026lco2o1) via $\mathrm{SU}(2)$-Schur reduction on $\mathrm{Sym}^{2}(V_{\rho})$ and the principal-symbol identification

Position in the sub-programme

LorCap → TempProj → LCII

Dependencies

Paper	Role
TempProj (Beau2026tp)	Flat-space baseline (homogeneous case)
LorCap (Beau2026n)	Origin of LC-O2 open problem
Q5b (Beau2026q5b)	Effective operator, co-metric, Carnot geometry
Q11 (Beau2026q11)	Co-metric completion $g^{\mu\nu} = 2\eta^{\mu\nu}$
Gravity/H (Beau2026h)	IR Einstein response — provides $g_{\tau\tau}(x)$
BI (Beau2026c)	Born-Infeld admissibility budget
Foundation (Beau2026m)	Structural axioms

Repository Contents

lc-o2/
├── out/              # Compiled PDF
├── tex/
│   ├── lc-o2.tex
│   └── cosmochrony-bibliography.bib
├── compile.sh
├── zenodo.json
└── README.md

Citation

J. Beau, Fibrewise Born-Infeld Capacity Lapse and Gravitational Time Dilation, Preprint, 2026.

Acknowledgements

Portions of the editorial refinement benefited from iterative interactions with large language models used as analytical assistants. All claims, interpretations, and final formulations remain the sole responsibility of the author.