Lorentz Transformations from Projective Temporal Ordering (LC-O1)

This repository contains the source of the paper Lorentz Transformations from Projective Temporal Ordering (LC-O1 of the Lorentz-Capacity Sub-Programme) (Beau2026lco1).

Overview

The Lorentz-capacity programme derives relativistic kinematics from the bounded Born-Infeld capacity budget of admissible projection on $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$. The technical note Beau2026n and the temporal residual paper Beau2026tp established that the projective temporal capacity satisfies

$$F^\tau = \sqrt{1-\beta^2} = \frac{1}{\gamma},$$

closing time dilation ($d\tau/dn = 1/\gamma$) and the light cone ($\beta = 1 \Rightarrow F^\tau = 0$). The remaining open problem, stated in Section 7.2 of Beau2026tp, is the reconstruction of the full effective observer transformation between inertial observers, including length contraction. The present paper closes this open problem (LC-O1).

Main Result

Given two inertial observers with relative normalised capacity load $\beta$, the unique proper orthochronous isometry of the reconstructed flat effective interval mapping one temporal axis to the other is the standard Lorentz boost

$$\tau' = \gamma(\tau - \beta x),\qquad x' = \gamma(x - \beta\tau),\qquad y' = y,\qquad z' = z,$$

with $\gamma = (1-\beta^2)^{-1/2}$, and length contraction follows as the reciprocal spatial projection onto the simultaneity hyperplane of the moving splitting:

$$\ell = \frac{\ell_0}{\gamma}.$$

The argument rests on a strict separation of three structural layers, each imported from a previously closed result:

1. Born-Infeld capacity sphere (Beau2026c, Beau2026m): the Euclidean capacity constraint $(F^\tau)^2 + \beta^2 = 1$ supplies $\beta$ and $1/\gamma$ from projective data. It is not the Minkowski interval.
2. Effective Lorentzian co-metric (Beau2026q5b, Beau2026q11): the homogeneous Q5b/Q11 closure gives the completed effective co-metric $g^{\mu\nu} = 2,\eta^{\mu\nu}$, whose isometry group is $\mathrm{SO}(3,1)$.
3. Inertial observer and boost: an inertial observer is identified with a future-directed unit timelike splitting $u$ of $\eta$. The rapidity is fixed by the capacity load via $\xi = \mathrm{arctanh},\beta$, and the Lorentz boost is the unique proper orthochronous isometry of $\eta$ mapping one admissible splitting to another at that load.

Length contraction is then a direct corollary of the boost: the difference in simultaneity hyperplanes between two inertial splittings, given the interval structure of Layer 2, produces the standard $\ell = \ell_0/\gamma$ relation.

Key features

• No new spectral or projective hypothesis. All inputs are previously closed; LC-O1 only identifies inertial observers with unit timelike splittings and applies the standard uniqueness of isometries of $\eta$ in each rapidity class.
• Strict layer separation. The Born-Infeld capacity sphere is not a Lorentzian structure; the Lorentz group acts as the isometry group of $\eta$, not of the BI sphere. The BI sphere determines which splittings are admissible; the Lorentz group determines how to pass between them.
• Length contraction as derived corollary. It is a consequence of the boost and the simultaneity hyperplane geometry, not an independent capacity law.
• Closure of LC-O1. The full effective observer transformation, including length contraction, is reconstructed from projective temporal ordering data and the effective metric normalisation.

Status

• Inertial splitting and rapidity identification: proved
• Uniqueness of the inertial boost (Lemma): proved
• Lorentz transformations from projective temporal ordering (Theorem): proved
• Length contraction (Corollary): proved
• Closure of LC-O1: proved

Position in the sub-programme

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

Dependencies

Paper	Role
N (Beau2026n)	Lorentz factor from projective capacity saturation
TempProj (Beau2026tp)	Temporal residual map; statement of LC-O1 (Section 7.2)
PTO (Beau2026pto)	Projective temporal ordering observable; scalar clock factor $d\tau/dn$
Q5b (Beau2026q5b)	BFS shell stratification; emergence of the effective Lorentzian metric
Q11 (Beau2026q11)	Closure of the effective co-metric $g^{\mu\nu} = 2,\eta^{\mu\nu}$
C (Beau2026c)	Born-Infeld admissibility budget (Layer 1)
M (Beau2026m)	Admissible non-injective transitions as the primitive of physical description

Repository Contents

lc-o1/
|-- out/              # Compiled PDF
|-- tex/
|   |-- lc-o1.tex
|   |-- cosmochrony-bibliography.bib
|-- compile.sh
|-- zenodo.json
|-- README.md

Compilation

bash compile.sh

Citation

J. Beau, Lorentz Transformations from Projective Temporal Ordering, 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.