Projective Capacity and the Continuum Limit of Admissible Redundancy

The Cosmochrony spectral cascade programme has progressively constrained the exponent~ governing the growth p (n) n^ of the effective relational valence. O4 established 1 from Born--Infeld saturation and the Cheeger isoperimetric inequality. O5 and O6 proved that vertex-level and fixed-representation fingerprints saturate within bounded BFS depth, independent of~q, and O6, Proposition~6. 2 formalised this as a universal no-go: no fingerprint based on a fixed finite-dimensional representation can produce the power-law decay Rₙ p (n) ^- required to close on ^* (0. 09, 0. 13). The present paper introduces the bridge between this discrete obstruction and the nonlinear projected dynamics derived in Appendix~B. 14 of the Cosmochrony white paper, where the finite local spectral admissible capacity (x, t) and the projective occupancy = ||²/ give rise to the effective Gross--Pitaevskii equation with coupling g = / encoding projective saturation pressure. We construct the discrete projective capacity ₙ (x) from path fingerprints of depth~k via the effective novelty of each path relative to the already-spanned fingerprint subspace. We define the discrete projective occupancy ₙ (x) = |ₙ (x) |²/ₙ (x) and establish, exactly in a reduced filling model, the state law Rₙ^ (k) (x) (ₙ (x) ), where () = 1/1+² satisfies (0) =1, () 0 as, and ' < 0. Numerical support is provided by reduced-model simulations for ambient dimensions D \9, 27, 81\ (corresponding to k=2, 3, 4 adjoint fingerprints), with the LPS graph X₅, ₁₃ included as a reference confirming the bounded-depth pre-saturation predicted by O6. We show that can be reread as the exponent governing the growth of~ₙ along the cascade. The Gross--Pitaevskii coupling of B. 14 is then identified as the continuum image of the discrete saturation pressure ₙₙ. The central identification Rₙ^ (k) (ₙ) is established exactly in the reduced model and supported numerically; its full graph-theoretic derivation is left as an open problem. This reframes the sequence of no-go results of O5--O6 as a positive convergence statement: admissible redundancy is not merely a combinatorial obstruction count, but the microscopic precursor of a continuous projective dynamics whose nonlinear structure is already determined by the framework.