Finite-q Corrections to the Carnot Capacity Identification

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 note N and the temporal residual paper TempProj 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 TempProj, was the status of the quadratic law and of the parabolic Carnot-Carathéodory identification near the finite-$q$ saturation boundary $n \simeq q/2$. The present paper closes this problem by separating two statements that the original wording conflated.

Capacity vs Carnot identification. LC-O3 is a correction to the Carnot approximation, not to the capacity law. The quadratic identity $B_n^2 + (F^\tau_n)^2 = 1$ is unconditional once $B_n$ is the exact finite-$q$ torus load. Only the unfolded parabolic count $B_n^{\mathrm{parab}}$ deviates from the cyclically folded torus count after the saturation boundary.

Main result

Setting $R := (q-1)/2$ and $n = R + m$, the exact torus count of admissibility cascade points at depth $n$ is

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

The deviation from the unfolded parabolic Carnot formula $B_n^{\mathrm{parab}} = (2n^2 + 2n + 1)/q^2$ is closed-form:

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

The quadratic Lorentz-capacity identity holds for every $n$:

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

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.

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 TempProj.
The Carnot identification has a precise domain of validity. Exact for $n \leq (q-1)/2$ (no cyclic folding); 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.

Dependencies

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).

References

Jérôme Beau. Finite-$q$ Corrections to the Carnot Capacity Identification. Preprint, 2026. 10.5281/zenodo.20334072

Relation to the Cosmochrony program

LCo3 is one of the three closure papers of the Lorentz Capacity sub-programme, alongside LCo1 (LC-O1, Lorentz transformations and length contraction) and LCIIo1 (LC-O2-O1, capacity-metric bridge). With these three closures, the chain LorentzCapacity → TempProj → LCII reaches a complete sub-programme.