Overview
The O-series saturation parameter $\eta_n^{\mathrm{O7}}$ is defined in O7 as the ratio of cumulative occupied projective capacity to residual frontier novelty. A key analytic factorization shows that the central coordinate $c$ of the Heisenberg group contributes only a global phase to Weil representations and is invisible to the fingerprint span. This forces the total O7 residual $\Sigma_n^{\mathrm{tot}}$ to satisfy a linear filling law $B_n + \Sigma_n^{\mathrm{tot}} = 1$, confirmed numerically for $q \in \{7, 11, 13\}$ with mean deviations of $0.02$–$0.04$.
The linear law does not by itself yield the Lorentz factor. Under the Born–Infeld admissibility budget $dI_\tau^2 + dI_{\mathrm{space}}^2 = c_{\mathrm{BI}}^2\,dn^2$, the required structure is a temporally projected residual $\Sigma_n^\tau := \mathcal{P}_\tau^{\mathrm{Q5b}}\,\Sigma_n^{\mathrm{tot}}$ satisfying the quadratic normalization $B_n^2 + (\Sigma_n^\tau)^2 \to 1$. Under this condition, $\eta_n^\tau := B_n/\Sigma_n^\tau \to \beta\gamma$ and $\Phi(\eta_n^\tau) \to 1/\gamma$.
Main result
Under the Born–Infeld admissibility budget and a quadratic normalization of the O7 observables,
$\eta_n^{\tau} \longrightarrow \beta\gamma, \qquad \Phi\!\left(\eta_n^{\tau}\right) = \dfrac{1}{\sqrt{1+(\eta_n^{\tau})^2}} \longrightarrow \dfrac{1}{\gamma}.$
The inverse Lorentz factor is the residual-capacity profile of the admissible spectral cascade. The Lorentz factor is not an external kinematical postulate; it emerges from the occupied-over-residual capacity ratio under a bounded projective budget.
Key contributions
- Central-phase factorization: $\omega_q(a,b,c) = e^{2\pi ic/q}\omega_q(a,b,0)$, so the central coordinate $c$ contributes only a global phase; the total O7 residual satisfies $B_n + \Sigma_n^{\mathrm{tot}} = 1$ (linear law, proved analytically and confirmed numerically).
- Algebraic BI target: purely algebraic derivation that if $\eta = dI_{\mathrm{space}}/dI_\tau$ under the BI budget, then $\Phi(\eta) = 1/\gamma$ (proved).
- Insufficiency of the linear law: $\Sigma_n^{\mathrm{tot}}$ is too large by a factor $1/\sqrt{1-B_n^2}$ to be the Lorentzian temporal residual; the required upgrade is quadratic normalization (proved).
- Lorentz convergence: if the Q5b temporal projection operator $\mathcal{P}_\tau^{\mathrm{Q5b}}$ exists satisfying $\mathcal{P}_\tau^{\mathrm{Q5b}}\Sigma_n^{\mathrm{tot}} \to \sqrt{1-B_n^2}$, then $\eta_n^\tau \to \beta\gamma$ (proved conditionally; construction of $\mathcal{P}_\tau^{\mathrm{Q5b}}$ closed in TempProj).
Structure
- O7 occupation and residual novelty — definitions of $B_n$ and $\Sigma_n$ and their ratio.
- Algebraic Born–Infeld target — purely algebraic derivation showing $\Phi(\eta) = 1/\gamma$.
- Why the reduced filling model is insufficient — the linear residual law does not produce the Lorentzian square root; numerical confirmation for $q \in \{7, 11, 13\}$.
- From linear filling to quadratic normalization via Q5b — the Q5b/Q11 geometric upgrade and Lemma 1 (closed in TempProj).
- Consequence for the Lorentz factor — Theorem: the O7 profile converges to $1/\gamma$, yielding time dilation and the light cone as corollaries.
Dependencies
- O7: definition of $\eta_n^{\mathrm{O7}}$, $B_n$, $\Sigma_n$, and the reduced filling model.
- Born–Infeld paper: mobility budget $c_{\mathrm{BI}}$ and ordinal cascade depth.
- Q5b: BFS-to-Carnot–Carathéodory convergence in the large-$q$ pre-saturation regime.
- Q11: co-metric closure fixing the Lorentzian signature $g^{\mu\nu} = 2\eta^{\mu\nu}$.
References
Jérôme Beau. Spectral Mobility and the Lorentz Factor from Projective Capacity Saturation. Preprint. 10.5281/zenodo.20299014
Relation to the Cosmochrony program
This technical note provides the missing bridge between the O-series spectral admissibility programme and the emergence of special-relativistic kinematics. It establishes that the Lorentz factor is not an independent postulate but follows structurally from the same projective capacity framework that governs quantum mechanics and spacetime geometry. The temporal projection operator $\mathcal{P}_\tau^{\mathrm{Q5b}}$ is constructed explicitly in TempProj, closing the Lorentz-capacity sub-programme.