This repository contains the source of the O7 Cosmochrony paper
Projective Capacity and the Continuum Limit of Admissible Redundancy.

This work completes the spectral redundancy programme by resolving the obstruction identified in O6: the impossibility of extracting the cascade exponent from fixed finite-dimensional representations due to bounded saturation depth.

While O5–O6 establish that all static fingerprints necessarily saturate, they leave open a fundamental question: how can redundancy still produce a non-trivial scaling law along the cascade?

The present work introduces a coarse-grained notion of projective capacity that captures the residual novelty of admissible paths beyond exact linear independence.

Core Idea

Instead of measuring redundancy through exact span growth, we define a mesoscopic quantity:

• Projective capacity $\Sigma_n(x)$: accumulated residual novelty of $k$-step path fingerprints within coarse-grained cells

This leads to a natural notion of projective occupancy:

• $\eta_n(x) = |\psi_n(x)|^2 / \Sigma_n(x)$

The admissible frontier is then governed by a state equation:

• $R_n^{(k)}(x) \approx \Phi(\eta_n(x))$

where $\Phi$ is a monotone decreasing function with:

• $\Phi(0) = 1$ (unsaturated regime)
• $\Phi(1) = 0$ (full saturation)

Main Results

• Introduction of a coarse-grained capacity field $\Sigma_n(x)$ derived from path fingerprints
• Establishment of a monotone state law linking redundancy to projective occupancy
• Reinterpretation of the cascade exponent $\beta$ as the growth exponent of projective capacity
• Structural bridge between discrete redundancy (O5–O6) and the continuous nonlinear dynamics derived in §B.14 of the white paper

Physical Interpretation

The nonlinear term appearing in the effective Schrödinger equation:

$$ i\hbar_{\mathrm{eff}},\partial_t \psi = \hat{H}_{\mathrm{eff}} \psi + \frac{\gamma}{\Sigma(x,t)} |\psi|^2 \psi $$

is interpreted as the macroscopic manifestation of finite projective capacity.

In this picture:

• redundancy → capacity depletion
• capacity depletion → increased effective coupling
• increased coupling → nonlinear saturation dynamics

Position in the Programme

• O4: upper bound on $\beta$ from admissibility constraints
• O5: vertex-level saturation obstruction
• O6: universal representation-theoretic saturation (no-go)
• O7: emergence of a capacity-driven continuum description

O7 shows that the obstruction is not a dead end, but a signal that the correct observable is not exact redundancy, but its coarse-grained capacity.

Outlook

The framework opens several directions:

• Rigorous discrete-to-continuum limit $\Sigma_n(x) \to \Sigma(x,t)$
• Determination of the state function $\Phi$ for $k \geq 2$
• Extension to higher-order fingerprints ($k=3$ and beyond)
• Numerical extraction of capacity growth exponents

O7 establishes the first consistent bridge between the discrete spectral programme and emergent nonlinear quantum dynamics.