Projective Capacity and the Continuum Limit of Admissible Redundancy
Year: 2026
Type: preprint
Abstract: 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 \emph{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.
Author: Beau, Jérôme
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7139974552
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