The Lorentz-Capacity Sub-Programme

Current Zenodo release: version 2.3 (2026-10-26). Official title: The Lorentz-Capacity Sub-Programme.

Overview

The Lorentz factor is usually introduced as part of the kinematics of special relativity, taken as foundational. In Cosmochrony the Lorentzian metric is meant to be an effective object of admissible non-injective projection; it is not yet reconstructed. The structural question is: where does $\gamma$ come from? The sub-programme answers it conditionally, by separating two roles fused in the standard presentation.

Capacity sphere vs Lorentzian metric. The Born–Infeld capacity relation $(F^\tau)^2 + \beta^2 = 1$ 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 Lorentzian interval is a separate, supplied input: a 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 (Q5b version 2.3, Q8 version 2.0 and Q10 version 2.0.1 establish no coefficient value). After a rescaling of $\tau$ it is proportional to $\eta$, and the normalisation $g^{\mu\nu} = 2\eta^{\mu\nu}$ is a choice of units. The two structures are linked by the hypothesis [H-cone]: the light-cone speed of the co-metric equals the Born–Infeld bound per ordering step. This is the coupling datum that Q11 leaves open, and no source derives it.

The structural chain

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

The co-metric step is supplied, under the hypotheses of Q11 Corollary 6.1, and the arrow to $\Lambda(\beta)$ uses [H-cone]; the sub-programme uses only the signature, the decoupled temporal block and the isometry group of the co-metric, never a coefficient value.

Each arrow has a distinct logical status. The quadratic capacity relation, a hypothesis expressing the capacity bound, fixes the residual temporal capacity $F^\tau = \sqrt{1-\beta^2}$. Under [H-cone] the Lorentz boost $\Lambda(\beta)$ is, on the tangent space, the unique proper orthochronous isometry fixing the transverse directions and mapping one inertial splitting to another at load $\beta$. A local occupancy field $\epsilon(x)$ then reduces the residual temporal capacity, its identification with the metric lapse being a declared postulate, and the finite-$q$ correction bounds the domain in which continuum Carnot estimates of the spatial load are valid.

Papers of the sub-programme

LorCap (Beau2026n): the linear filling law, and the conditional limit $F^\tau = 1/\gamma$ under the quadratic budget.
TempProj (Beau2026tp): constructs the temporal residual map $F^\tau_n = \sqrt{1-B_n^2}$ under a supplied co-metric: the Born–Infeld sphere and the Weil filling law fix its values at the attained values of the total residual, and the splitting identifies them as temporal; time dilation and the light cone under [H-cone] and the Lorentzian signature, the light cone $B = 1$ being reached by the exact torus load of LC-O3.
LCII (Beau2026lco2): extends the construction to the non-homogeneous fibrewise regime, with local capacity radius $R(x)^2 = 1-\epsilon(x)$.
LC-O1 (Beau2026lco1): the Lorentz boost and length contraction $\ell = \ell_0/\gamma$ as isometries of $\eta$, fixed by the capacity load under [H-cone].
LC-O2-O1 (Beau2026lco2o1): proves that the scalar free-fraction law does not close the local capacity-metric bridge, for any postulated flat value; the bridge remains open.
LC-O3 (Beau2026lco3): identifies the exact finite-$q$ domain of validity of the Carnot parabolic identification, with correction $\Delta_q(n) = 4m^2/q^2$.

Inputs and outputs

Upstream inputs. The Born–Infeld capacity bound, expressed by the quadratic budget hypothesis, supplies the unit capacity form $(F^\tau)^2 + \beta^2 = 1$; projective temporal ordering supplies a candidate temporal proxy along a trajectory, non-decreasing only within a numerical tolerance (exact monotonicity is conjectural, and an order on states requires [H-acyc]), with no step and no coefficient, the clock-rate reading of $F^\tau$ being that of TempProj under [H-cone]; the co-metric is supplied in the conditional form of Q11, with two independent coefficients; [H-cone] is assumed.

Outputs. Special-relativistic kinematics (time dilation, light cone, Lorentz boosts, length contraction), conditional on the supplied co-metric and on [H-cone]; the parallel structural definitions $R(x)^2 = 1-\epsilon(x)$ and $R(x)^2 = g_{\tau\tau}(x)/g_{\tau\tau}^{\infty}$, whose equivalence is open; and finite-$q$ control of the Carnot identification.

Status

The inertial kinematic chain is complete conditionally on a supplied co-metric and on [H-cone], while the non-homogeneous capacity-metric bridge remains open. In this sense, under the stated hypotheses, the Lorentz factor is read as the temporal residual of a bounded admissible projection rather than postulated, and the Lorentz group is not imposed on the capacity sphere but is the isometry group of the supplied effective co-metric. Deriving [H-cone] is the open coupling problem Q11-O1.