Shell-Level Saturation and the Conditional Fibre-to-Rate Prescription

Abstract

The spectral admissibility sub-programme of Cosmochrony has identified the canonical fibre-level observable $\sigma_{\mathrm{pair}}^{\mathrm{can}}(n)$ as the relevant quantity for the lepton mass hierarchy. However, existing saturation criteria rely on the power-law fit $\sigma(n) \sim C\,n^{-\delta}$, whose amplitude $C$ and exponent $\delta$ depend on the fitting window and are not intrinsic. We construct a fully observable saturation rank $n_3^{\mathrm{obs}}(n_0, n_1, q)$ by eliminating both $C$ and $\delta$ using two local measurements of $\sigma_{\mathrm{pair}}^{\mathrm{can}}$. This rank is invariant under global rescaling and all pipeline-dependent normalisations, and depends only on directly measurable quantities. We conjecture that physical admissibility requires $n_3^{\mathrm{obs}}$ to align with a BFS shell of the Cayley graph $\mathrm{Cay}(G_q, S_q)$, providing a geometric interpretation of saturation. Under this shell-alignment hypothesis and an additional cross-substrate identification, the O7 relation $\beta^* = 1/(\delta_{\mathrm{pair}} + \tfrac{1}{2})$ defines a conditional phenomenological prescription. The fibre construction does not supply a native Heisenberg growth equation carrying the pair observable. O22 and O24 discharge the shell-alignment and rank-three conditions on the fibre side of the chain. Those results do not supply the independent Heisenberg growth carrier required by the capacity-to-rate step.