The Lorentz-Capacity Sub-Programme (LC-Synthesis)

This repository contains the source of the synthesis paper The Lorentz-Capacity Sub-Programme: Projective Origin of the Lorentz Factor in the Cosmochrony Programme (Beau2026lcsyn).

Overview

The Lorentz-capacity (LC) sub-programme answers one specific structural question of the Cosmochrony framework: where does the Lorentz factor $\gamma = (1 - \beta^2)^{-1/2}$ come from, if Lorentzian geometry is not assumed as fundamental?

The answer is that $\gamma$ is not a kinematic postulate. It is reconstructed from the bounded Born-Infeld capacity budget of admissible projection, then lifted to the observer-transform structure of special relativity through the effective Lorentzian co-metric established in the geometric branch of the programme. This paper synthesises the six constituent papers and exhibits the single completed chain that links them.

The Structural Chain

$$\text{BI budget} ;\to; F^\tau = \tfrac{1}{\gamma} ;\to; g^{\mu\nu} = 2\eta^{\mu\nu} ;\to; \Lambda(\beta) ;\to; R(x)^2 = 1 - \epsilon(x) ;\to; \text{finite-}q\text{ domain.}$$

A central organising distinction runs through the whole sub-programme:

• The Born-Infeld capacity sphere $(F^\tau)^2 + \beta^2 = 1$ is a Euclidean norm constraint in capacity space. It supplies the projective load $\beta$ and the residual clock factor $F^\tau = \sqrt{1 - \beta^2} = 1/\gamma$. It is not the Minkowski interval.
• The effective Lorentzian metric $g^{\mu\nu} = 2\eta^{\mu\nu}$ is supplied separately by the Q5b-Q11 geometric closure. The Lorentz group is the isometry group of this metric, not the symmetry group of the capacity sphere.

Constituent Papers

Label	Key	Main result
LorCap	Beau2026n	Lorentz mobility from projective capacity; $F^\tau = 1/\gamma$
TempProj	Beau2026tp	Temporal residual map $F^\tau_n = \sqrt{1 - B_n^2}$; time dilation, light cone
LCII	Beau2026lco2	Fibrewise local lapse $R(x)^2 = 1 - \epsilon(x)$
LC-O1	Beau2026lco1	Lorentz boost $\Lambda(\beta)$; length contraction $\ell = \ell_0/\gamma$
LC-O2-O1	Beau2026lco2o1	No-go: the free-fraction law does not close the bridge; LC-O2-O1 reopened
LC-O3	Beau2026lco3	Finite-$q$ correction $\Delta_q(n) = 4m^2/q^2$ to the Carnot identification

Outputs

• Special-relativistic kinematics: time dilation ($d\tau/dn = 1/\gamma$), the light cone ($B = 1 \Rightarrow F^\tau = 0$), Lorentz boosts, and length contraction, all reconstructed from projective capacity and the effective metric closure.
• Local gravitational capacity lapse: LCII supplies the parallel structural definitions $R(x)^2 = 1 - \epsilon(x)$ and $R(x)^2 = -2g_{\tau\tau}(x)$. Their equivalence gives a capacity-side reading of gravitational time dilation in the Einstein regime, but its derivation from admissibility remains open.
• Finite-$q$ control: the exact correction $\Delta_q(n) = 4m^2/q^2$ bounds the domain in which continuum Carnot estimates of the spatial load are valid.

Status

The inertial kinematic chain is complete. The non-homogeneous capacity-metric identification remains open. The Lorentzian metric is inherited from Q5b-Q11, while the Born-Infeld capacity relation comes from the admissibility budget.

Label	Status
LorCap	proved (conditional on $[U]$)
TempProj	proved (conditional on $[U]$)
LCII	structural; capacity-metric bridge open (declared postulate, Remark 3.2)
LC-O1	proved
LC-O2-O1	proved no-go; bridge reopened
LC-O3	proved

Upstream Dependencies

Paper	Role
Born-Infeld admissibility (Beau2026c, Beau2026m)	Unit capacity form $(F^\tau)^2 + \beta^2 = 1$
Projective temporal ordering (Beau2026pto)	Non-decreasing temporal proxy $d\tau/dn = F^\tau$
Q5b-Q11 metric closure (Beau2026q5b, Beau2026q11)	Effective co-metric $g^{\mu\nu} = 2\eta^{\mu\nu}$

Repository Contents

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

Compilation

bash compile.sh

Citation

J. Beau, The Lorentz-Capacity Sub-Programme: Projective Origin of the Lorentz Factor in the Cosmochrony Programme, 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.