Spectral Mobility and the Lorentz Factor from Projective Capacity Saturation

Current Zenodo release: version 1.4 (2026-10-24). Official title: Spectral Mobility and the Lorentz Factor from Projective Capacity Saturation.

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 the exact 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 quadratic Born–Infeld mobility budget $dI_\tau^2 + dI_{\mathrm{space}}^2 = c_{\mathrm{BI}}^2\,dn^2$, a hypothesis of the note expressing the capacity bound [A-cap], the required structure is a temporally projected residual $\Sigma_n^\tau := \mathcal{P}_\tau^{\mathrm{Q5b}}\,\Sigma_n^{\mathrm{tot}}$ satisfying the quadratic normalization $B_n^2 + (\Sigma_n^\tau)^2 \to 1$; then $\eta_n^\tau := B_n/\Sigma_n^\tau \to \beta\gamma$ and $\Phi(\eta_n^\tau) \to 1/\gamma$. The projection is induced by a supplied 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. It uses only the signature and the decoupled temporal block; the temporal direction is the ordering factor, not the central direction of the Heisenberg group, which is spatial.

Scope statement. The central-phase factorization and the linear filling law are proved. The Lorentz-factor limit is conditional on the quadratic budget hypothesis and on the convergence lemma (Lemma 1), which TempProj establishes under its own hypotheses. Reading $\beta$ as a velocity relative to the light cone of the co-metric requires [H-cone]: the light-cone speed equals the Born–Infeld bound per ordering step. This is the coupling datum that Q11 leaves open, and no source derives it.

Main result

Under the quadratic Born–Infeld budget hypothesis and the convergence lemma,

$\eta_n^{\tau} \longrightarrow \beta\gamma, \qquad \Phi\!\left(\eta_n^{\tau}\right) = \dfrac{1}{\sqrt{1+(\eta_n^{\tau})^2}} \longrightarrow \dfrac{1}{\gamma}.$

The inverse Lorentz factor is then the residual-capacity profile of the admissible spectral cascade. Under [H-cone] this reads as time dilation, with the light cone at $\beta = 1$.

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 $B_n + \Sigma_n^{\mathrm{tot}} = 1$ (proved analytically and confirmed numerically).
Algebraic BI target: if $\eta = dI_{\mathrm{space}}/dI_\tau$ under the quadratic budget, then $\Phi(\eta) = 1/\gamma$ (proved).
Insufficiency of the linear law: $\Sigma_n^{\mathrm{tot}} = 1 - B_n$ is smaller than the Lorentzian temporal residual $\sqrt{1-B_n^2}$; the required upgrade is quadratic normalization.
Lorentz convergence: if $\mathcal{P}_\tau^{\mathrm{Q5b}}$ satisfies $\mathcal{P}_\tau^{\mathrm{Q5b}}\Sigma_n^{\mathrm{tot}} \to \sqrt{1-B_n^2}$, then $\eta_n^\tau \to \beta\gamma$ (conditional; the projection is constructed in TempProj under its hypotheses).
[H-cone] named: the identification of the light-cone speed of the co-metric with the Born–Infeld bound, on which the kinematic reading rests.

Structure

O7 occupation and residual novelty — definitions of $B_n$ and $\Sigma_n$ and their ratio.
Algebraic Born–Infeld target — algebraic derivation of $\Phi(\eta) = 1/\gamma$ under the quadratic budget.
Why the reduced filling model is insufficient — the linear law does not produce the Lorentzian square root; numerical confirmation for $q \in \{7, 11, 13\}$.
From linear filling to quadratic normalization — the projection induced by a supplied co-metric and Lemma 1, established in TempProj.
Weil factorization — the central coordinate is invisible to the fingerprint span; the linear law is exact.
Numerical tests — three candidate observables fail the quadratic law.
Consequence for the Lorentz factor — conditional theorem: the projected profile converges to $1/\gamma$; time dilation and the light cone under [H-cone].

Dependencies

O7: definition of $\eta_n^{\mathrm{O7}}$, $B_n$, $\Sigma_n$, and the reduced filling model.
Born–Infeld paper: the capacity bound [A-cap] with constant $c_{\mathrm{BI}}$; the quadratic budget is a hypothesis of this note.
Q5b: BFS-to-Carnot–Carathéodory convergence; a co-metric of rank three under [H-L] and [H-lift].
Q8: $A_H = A_Z = 2\lambda$ with $\lambda$ free for a supplied invariant form; no value of $A_Z$.
Q10: no coefficient value is derived.
Q11: the conditional rank-four 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).
TempProj: construction of $\mathcal{P}_\tau^{\mathrm{Q5b}}$ under the hypotheses stated there.

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 connects the O-series spectral admissibility programme to special-relativistic kinematics: under the quadratic budget hypothesis, a supplied co-metric and [H-cone], the Lorentz factor is read as the temporal residual of the same projective capacity framework rather than as an independent postulate. The temporal projection $\mathcal{P}_\tau^{\mathrm{Q5b}}$ is constructed in TempProj.