Lorentz Transformations from Projective Temporal Ordering

Current Zenodo release: version 1.2 (2026-10-26). Official title: Lorentz Transformations from Projective Temporal Ordering.

Overview

The Lorentz-capacity programme studies relativistic kinematics from the bounded Born–Infeld capacity budget of admissible projection on $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$. LorentzCapacity and TempProj give, under a quadratic budget hypothesis and the identifications stated there, the projective temporal capacity $F^\tau = \sqrt{1-\beta^2} = 1/\gamma$.

The problem LC-O1, stated in Section 7.2 of TempProj, is the reconstruction of the full effective observer transformation between inertial observers, including length contraction. No source constructs the Lorentzian co-metric needed for it: the paper takes a supplied rank-four co-metric of the conditional form of Q11, $\mathrm{diag}(-A_\tau, 2\lambda, 2\lambda, 2\lambda)$, whose two coefficients are independent and fixed by no source. After a rescaling of $\tau$ it is proportional to $\eta$, whose isometry group is $\mathrm{SO}(3,1)$; the normalisation $g^{\mu\nu} = 2\eta^{\mu\nu}$ is a choice of units.

Layer separation. The argument separates three layers: (i) the Born–Infeld capacity sphere $(F^\tau)^2 + \beta^2 = 1$, which is not a Minkowski interval; (ii) the supplied co-metric, proportional to $\eta$ after rescaling $\tau$; (iii) the identification of an inertial observer with a future-directed unit timelike splitting of $\eta$. Layers (i) and (ii) are linked by one named hypothesis, [H-cone]: the light-cone speed of the co-metric equals the Born–Infeld bound per ordering step, so that the capacity load is the velocity of a splitting. This is the coupling datum that Q11 leaves open (Q11-O1), and no source derives it.

Main result

Assume a supplied co-metric of the form of Q11 Corollary 6.1, written in the rescaled coordinate $\tau$, and [H-cone]. Given two inertial observers with relative capacity load $\beta$, the unique proper orthochronous isometry of the tangent interval mapping one temporal axis to the other and fixing the transverse directions is the standard Lorentz boost, a statement on the tangent space:

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

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

$\ell = \ell_0/\gamma.$

The uniqueness of the boost is a statement about $\eta$ and holds whatever the values of $A_\tau$ and $\lambda$; [H-cone] is what fixes its rapidity at $\xi = \mathrm{arctanh}\,\beta$ from the capacity data.

Key features

One named kinematic hypothesis. The inputs are conditional (supplied co-metric, quadratic budget, identifications of TempProj); LC-O1 identifies inertial observers with unit timelike splittings, names [H-cone], 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 a corollary. Consequence of the boost and the simultaneity hyperplane geometry, not an independent capacity law.
Conditional closure of LC-O1. The full effective observer transformation, including length contraction, is reconstructed from projective temporal ordering data, a supplied co-metric and [H-cone].

Dependencies

LorentzCapacity (Beau2026n): Lorentz factor from projective capacity saturation, under the quadratic budget hypothesis.
TempProj (Beau2026tp): temporal residual map; statement of LC-O1 (Section 7.2).
PTO (Beau2026pto): projective temporal ordering observable; a monotone ordering proxy, with no step and no coefficient.
Q5b (Beau2026q5b): a co-metric of rank three under [H-L] and [H-lift]; its rank-four extension is open.
Q8 (BeauQ8): $A_H = A_Z = 2\lambda$ with $\lambda$ free for a supplied invariant form.
Q10 (Beau2026q10): no coefficient value is derived.
Q11 (Beau2026q11): the conditional form $\mathrm{diag}(-A_\tau, 2\lambda, 2\lambda, 2\lambda)$ (Corollary 6.1), with $A_\tau$ free; the coupling datum is open (Q11-O1).
C (Beau2026c): the capacity bound [A-cap] (Layer 1).

References

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

Relation to the Cosmochrony program

LCo1 is one of three papers that address the open problems of the Lorentz Capacity sub-programme, alongside LCIIo1 (LC-O2-O1, capacity–metric bridge) and LCo3 (LC-O3, finite-$q$ corrections at saturation). The inertial chain LorentzCapacity → TempProj → LCo1 gives time dilation, Lorentz transformations and length contraction, conditionally on a supplied co-metric and on [H-cone]. The non-homogeneous bridge LC-O2-O1 is open: LCIIo1 proves that the scalar free-fraction law does not close it.