The O-series saturation parameter ₙ^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 Bₙ + ₙ^tot = 1 and does not by itself yield the Lorentz factor: q (a, b, c) = e^2 ic/qq (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 ₙ^: = P_^Q5b\, ₙ^tot can produce the relativistic mobility ratio: under the Born–Infeld admissibility budget established in, a quadratic normalization of Bₙ and ₙ^ implies \ ₙ^: = Bₙₙ^, \ where is the relativistic mobility and is the Lorentz factor. Consequently, (ₙ^) = 1/1+ (ₙ^) ² converges to 1/. The temporal projection operator P_^Q5b is induced by the Carnot–Carath\'eodory geometry and co-metric closure of. Constructing this operator and verifying P_^Q5bₙ^tot 1-Bₙ² is the remaining technical task.
Jérôme Beau (Sat,) studied this question.