Spectral Mobility and the Lorentz Factor from Projective Capacity Saturation

The O-series saturation parameter $\eta_n^{\mathrm{O7}}$ is defined in as the ratio of cumulative occupied projective capacity to total residual frontier novelty. The analytic factorization proved below explains why this total ratio satisfies a linear filling law $B_n + \Sigma_n^{\mathrm{tot}} = 1$ and does not by itself yield the Lorentz factor: $\omega_q(a,b,c) = e^{2\pi ic/q}\omega_q(a,b,0)$, so the central coordinate $c$ contributes only a global phase and is invisible to the fingerprint span. We identify the conditional mechanism by which a temporally projected residual $\Sigma_n^\tau := \mathcal{P}_\tau^{\mathrm{Q5b}}\,\Sigma_n^{\mathrm{tot}}$ can produce the relativistic mobility ratio: under the Born–Infeld admissibility budget established in , a quadratic normalization of $B_n$ and $\Sigma_n^\tau$ implies \[ \eta_n^\tau := \frac{B_n}{\Sigma_n^\tau} \longrightarrow \beta\gamma, \] where $\beta\gamma$ is the relativistic mobility and $\gamma$ is the Lorentz factor. Consequently, $\Phi(\eta_n^\tau) = 1/\sqrt{1+(\eta_n^\tau)^2}$ converges to $1/\gamma$. The temporal projection operator $\mathcal{P}_\tau^{\mathrm{Q5b}}$ is induced by the Carnot–Carath\'eodory geometry and co-metric closure of . Constructing this operator and verifying $\mathcal{P}_\tau^{\mathrm{Q5b}}\Sigma_n^{\mathrm{tot}} \to \sqrt{1-B_n^2}$ is the remaining technical task.