Spectral Mobility and the Lorentz Factor from Projective Capacity Saturation

The O-series saturation parameter ηₙO7 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 Bₙ + Σₙᵗᵒᵗ = 1 and does not by itself yield the Lorentz factor: ωq(a,b,c) = e2π ic/qωq(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 Σₙ^τ := P_τQ5b\,Σₙᵗᵒᵗ can produce the relativistic mobility ratio: under the Born–Infeld admissibility budget established in , a quadratic normalization of Bₙ and Σₙ^τ implies \[ η_n^τ := B_n/Σ_n^τ βγ, \] where βγ is the relativistic mobility and γ is the Lorentz factor. Consequently, Φ(ηₙ^τ) = 1/√1+(ηₙ^τ)² converges to 1/γ. The temporal projection operator P_τQ5b is induced by the Carnot–Carath\'eodory geometry and co-metric closure of . Constructing this operator and verifying P_τQ5bΣₙᵗᵒᵗ → √1-Bₙ² is the remaining technical task.