Finite-$q$ Corrections to the Carnot Capacity Identification (LC-O3)

This repository contains the source of the paper Finite-$q$ Corrections to the Carnot Capacity Identification (LC-O3 of the Lorentz-Capacity Sub-Programme) (Beau2026lco3).

Overview

The Lorentz-capacity sub-programme derives the temporal residual $F^\tau_n = \sqrt{1 - B_n^2}$ from the bounded Born-Infeld capacity budget of the admissibility cascade on $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$. The technical note Beau2026n and the temporal residual paper Beau2026tp established the quadratic identity

$$B_n^2 + (F^\tau_n)^2 = 1,$$

closing time dilation and the Lorentz factor identification. The remaining open problem, stated in Section 7.2 of Beau2026tp, is the status of the quadratic law and of the parabolic Carnot capacity identification near the finite-$q$ saturation boundary $n \simeq q/2$. The present paper closes this open problem (LC-O3).

Main Result

LC-O3 is closed by a precise separation between two statements that were conflated in the open problem:

1. The quadratic Lorentz-capacity identity is exact and receives no correction. For every $n$, with $B_n$ the exact finite-$q$ torus load,

   $$B_n^2 + (F^\tau_n)^2 = 1, \qquad F^\tau_n := \sqrt{1 - B_n^2}.$$

2. The parabolic Carnot--Carath'eodory formula is corrected outside the pre-saturation regime. Setting $R := (q-1)/2$ and $n = R + m$, the exact torus count is

   $$B_n^{\mathrm{torus}} = \frac{q(2m+1) + 2R^2 - 2m^2}{q^2},$$

   and the deviation from the unfolded parabolic formula $B_n^{\mathrm{parab}} = (2n^2 + 2n + 1)/q^2$ is

   $$\Delta_q(n) := B_n^{\mathrm{parab}} - B_n^{\mathrm{torus}} = \frac{4 m^2}{q^2}.$$

In the macroscopic folded regime $n/q \to \alpha > 1/2$, the correction converges to $\Delta_q(n) \to 4(\alpha - 1/2)^2 > 0$ and the parabolic identification is not asymptotically valid. In the pre-saturation regime ($n \leq R$) and in the sublinear boundary layer ($m/q \to 0$), the correction vanishes and the Carnot identification is exactly recovered.

Thus LC-O3 is a correction to the Carnot approximation, not to the capacity law.

Key features

• The quadratic capacity identity is unconditional. It holds for every $n$ once $B_n$ is the exact torus load, by the very definition $F^\tau_n = \sqrt{1 - B_n^2}$ supplied by the temporal residual map of Beau2026tp.
• The Carnot identification has a precise domain of validity. It is exact for $n \leq (q-1)/2$ (no cyclic folding) and asymptotically valid in any sublinear neighbourhood of the folding boundary.
• Folded-regime correction is finite and explicit. $\Delta_q(n) = 4m^2/q^2$ is closed-form, and the parabolic formula even exceeds $1$ (becomes unphysical) for sufficiently large $m$ at fixed $q$.
• Closure of the Lorentz-capacity sub-programme. With LC-O1, LC-O2-O1, and LC-O3 now resolved, time dilation, the light cone, Lorentz transformations and length contraction, the gravitational capacity-metric bridge, and the finite-$q$ saturation domain of validity are all established.

Status

• Pre-saturation parabolic formula (Lemma): proved
• Exact torus count after cyclic folding (Lemma): proved
• Finite-$q$ correction theorem $\Delta_q(n) = 4m^2/q^2$: proved
• Domain of validity of the Carnot identification (three regimes): proved
• Asymptotic Carnot recovery (Corollary): proved
• Closure of LC-O3: proved

Position in the sub-programme

LorCap -> TempProj -> LC-O2 -> LC-O2-O1
                            \-> LC-O1
                            \-> LC-O3

Dependencies

Paper	Role
N (Beau2026n)	Weil factorisation $\Sigma^{\mathrm{tot}}_n = 1 - B_n$
TempProj (Beau2026tp)	Temporal residual map $F^\tau_n = \sqrt{1 - B_n^2}$; statement of LC-O3 (Section 7.2)

Repository Contents

lc-o3/
|-- out/              # Compiled PDF
|-- tex/
|   |-- lc-o3.tex
|   |-- cosmochrony-bibliography.bib
|-- compile.sh
|-- zenodo.json
|-- README.md

Compilation

bash compile.sh

Citation

J. Beau, Finite-$q$ Corrections to the Carnot Capacity Identification, 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.