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 itself an effective object reconstructed from admissible non-injective projection, so the structural question is: where does $\gamma$ come from? The sub-programme answers it 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 $g^{\mu\nu} = 2\eta^{\mu\nu}$ is supplied separately by the Q5b-Q11 closure. The Lorentz group is the isometry group of that metric, not the symmetry group of the capacity sphere. This distinction organises the whole sub-programme.

The structural chain

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

Each arrow has a distinct logical status. The bounded capacity relation fixes the residual temporal capacity $F^\tau = \sqrt{1-\beta^2}$. The Q5b-Q11 closure supplies the homogeneous co-metric with no free parameter. The Lorentz boost $\Lambda(\beta)$ is the unique proper orthochronous isometry mapping one inertial splitting to another. A local occupancy field $\epsilon(x)$ then reduces the residual temporal capacity, 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) — identifies the projective mobility parameter and the reading $F^\tau = 1/\gamma$.
TempProj (Beau2026tp) — constructs the temporal residual map $F^\tau_n = \sqrt{1-B_n^2}$; closes time dilation and the light cone.
LCII (Beau2026lco2) — extends the construction to the non-homogeneous fibrewise regime, with local capacity radius $R(x)^2 = 1-\epsilon(x)$.
LC-O1 (Beau2026lco1) — reconstructs the Lorentz boost and length contraction $\ell = \ell_0/\gamma$ as isometries of the effective metric.
LC-O2-O1 (Beau2026lco2o1) — closes the local capacity-metric bridge $R(x)^2 = 1-\epsilon(x) = -2g^{\tau\tau}(x)$ spectrally, without the Einstein equation.
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 admissibility budget supplies the unit capacity form $(F^\tau)^2 + \beta^2 = 1$; projective temporal ordering supplies a non-decreasing temporal proxy $d\tau/dn = F^\tau$; the Q5b-Q11 closure supplies the effective co-metric $g^{\mu\nu} = 2\eta^{\mu\nu}$.

Outputs. Special-relativistic kinematics (time dilation, light cone, Lorentz boosts, length contraction); a capacity-side reading of local gravitational time dilation $R(x)^2 = 1-\epsilon(x) = -2g^{\tau\tau}(x)$, derived before any use of the Einstein equation; and finite-$q$ control of the Carnot identification.

Status

No structural open problem remains inside the Lorentz-capacity sub-programme. The Lorentzian metric is inherited from Q5b-Q11 and the Born-Infeld capacity relation from the admissibility budget; both qualifications are scope, not open problems. In this sense the Lorentz factor is not postulated — it is the temporal residual of a bounded admissible projection — and the Lorentz group is not imposed on the capacity sphere but is the isometry group of the effective metric reconstructed from admissibility.