Overview
This article continues the spectral hierarchy programme after the obstruction proved in O6: no fingerprint based on a fixed finite-dimensional representation can sustain the power-law redundancy decay needed to explain the cascade exponent.
O7 does not attempt to evade that no-go theorem by introducing another static encoding. Instead, it changes the relevant observable. Exact redundancy is replaced by a coarse-grained notion of projective capacity, built from the residual novelty of admissible path fingerprints beyond exact span growth.
This shift yields a discrete notion of projective occupancy and saturation pressure, providing the first explicit bridge between the discrete admissible redundancy programme and the nonlinear Gross--Pitaevskii-type dynamics derived in Appendix B.14 of the white paper.
Core contributions
- Discrete projective capacity: introduction of a coarse-grained field $\Sigma_n(x)$ built from the effective novelty of admissible $k$-step path fingerprints.
- State law in a reduced filling model: exact derivation of $R_n^{(k)}(x)\approx \Phi(\eta_n(x))$ with a monotone occupancy-dependent state function.
- Capacity reinterpretation of the cascade exponent: $\beta$ is no longer read as a static representation-theoretic quantity, but as an exponent governing capacity growth and exhaustion along the cascade.
- Projective capacity saturation mechanism: local depletion of $\Sigma_n(x)$ increases the effective coupling $g_n=\gamma/\Sigma_n$, generating a discrete saturation pressure.
- Continuum bridge: formal identification of the B.14 nonlinear term $g|\psi|^2\psi$ as the continuum image of microscopic capacity exhaustion.
Interpretation
The article shows that the obstruction identified in O6 is not the end of the programme, but the signal that the wrong observable was being tracked. Exact admissible redundancy saturates too abruptly to carry a meaningful scaling law. The correct object is the residual novelty that remains after projection onto the already-spanned admissible subspace.
- Fixed-dimensional fingerprints saturate at bounded depth and cannot explain the cascade exponent.
- Coarse-grained projective capacity captures how much admissible novelty remains locally available.
- Projective occupancy measures the filling fraction of this residual capacity.
- Nonlinear projected dynamics emerges when finite capacity feeds back on the effective coupling.
In this picture, the nonlinear term is not an additional phenomenological ingredient. It is the macroscopic manifestation of finite projective capacity.
Relation to the Cosmochrony program
O7 follows the obstruction proved in O6. Spectral admissibility, capacity, rigidity, and stratigraphy define the spectral backbone of the theory. O1 restores ordering through projective dynamics, O3 amplifies the hierarchy via valence growth, O5 isolates admissible redundancy structurally, and O6 proves that no fixed finite-dimensional representation can generate the required power-law regime.
The present paper performs the next necessary step: it reinterprets that obstruction as the microscopic signature of projective capacity exhaustion and identifies the first consistent discrete-to-continuum bridge toward the nonlinear projected dynamics of the white paper.