Projective Capacity and the Continuum Limit of Admissible Redundancy

The Cosmochrony spectral cascade programme has progressively constrained the exponent $\beta$ governing the growth $p(n) \sim n^\beta$ of the effective relational valence. O4 established $\beta \leq 1$ 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 $R_n \sim p(n)^{-\alpha}$ required to close on $\beta^* \in (0.09, 0.13)$. 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 $\Sigma(x,t)$ and the projective occupancy $\eta = |\psi|^2/\Sigma$ give rise to the effective Gross–Pitaevskii equation with coupling $g = \gamma/\Sigma$ encoding projective saturation pressure. We construct the discrete projective capacity $\Sigma_n(x)$ 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 $\eta_n(x) = |\psi_n(x)|^2/\Sigma_n(x)$ and establish, exactly in a reduced filling model, the state law $R_n^{(k)}(x) \approx \Phi(\eta_n(x))$, where $\Phi(\eta) = 1/\sqrt{1+\eta^2}$ satisfies $\Phi(0)=1$, $\Phi(\eta)\to 0$ as $\eta \to \infty$, and $\Phi' < 0$. Numerical support is provided by reduced-model simulations for ambient dimensions $D \in \{9,27,81\}$ (corresponding to $k=2,3,4$ adjoint fingerprints), with the LPS graph $X_{5,13}$ included as a reference confirming the bounded-depth pre-saturation predicted by O6. We show that $\beta$ can be reread as the exponent governing the growth of $\Sigma_n$ along the cascade. The Gross–Pitaevskii coupling of B.14 is then identified as the continuum image of the discrete saturation pressure $\gamma\eta_n\psi_n$. The central identification $R_n^{(k)} \approx \Phi(\eta_n)$ 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.