Lorentz Transformations from Projective Temporal Ordering

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 N and the temporal residual paper TempProj established that the projective temporal capacity satisfies $F^\tau = \sqrt{1-\beta^2} = 1/\gamma$, 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 TempProj, was the reconstruction of the full effective observer transformation between inertial observers, including length contraction. The present paper closes this problem.

Layer separation. The argument rests on a strict separation of three structural layers: (i) the Born-Infeld capacity sphere $(F^\tau)^2 + \beta^2 = 1$ (not a Minkowski interval); (ii) the effective Lorentzian co-metric $g^{\mu\nu} = 2\eta^{\mu\nu}$ from the homogeneous Q5b / Q11 closure; (iii) the identification of an inertial observer with a future-directed unit timelike splitting of $\eta$. The BI sphere selects which splittings are admissible; the Lorentz group acts as the isometry group of $\eta$ and determines how to pass between them.

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 $\gamma = (1-\beta^2)^{-1/2}$. Length contraction follows directly:

$\ell = \ell_0/\gamma.$

The rapidity of the boost is fixed by the capacity load via $\xi = \mathrm{arctanh}\,\beta$; the Lorentz boost is the unique proper orthochronous isometry of $\eta$ mapping one admissible timelike splitting to another at that rapidity class.

Key features

No new spectral or projective hypothesis. All inputs are previously closed results; LC-O1 only identifies inertial observers with unit timelike splittings and applies the standard uniqueness of $\eta$-isometries 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.
Length contraction as derived corollary. Consequence of the boost and the simultaneity hyperplane geometry, not an independent capacity law.
Closure of LC-O1. Full effective observer transformation, including length contraction, reconstructed from projective temporal ordering data and the effective metric normalisation.

Dependencies

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.

References

Jérôme Beau. Lorentz Transformations from Projective Temporal Ordering. Preprint, 2026. 10.5281/zenodo.20334069

Relation to the Cosmochrony program

LCo1 is one of the three closure papers of the Lorentz Capacity sub-programme, alongside LCIIo1 (LC-O2-O1, capacity-metric bridge) and LCo3 (LC-O3, finite-$q$ corrections at saturation). With these three closures, the chain LorentzCapacity → TempProj → LCII reaches a complete sub-programme: kinematic time dilation, gravitational time dilation, full Lorentz transformations, and finite-$q$ corrections are all derived from the projective capacity framework.