Projective Information Loss and Fibre Erasure in Admissible Non-Injective Projection

Abstract

We establish a fibre-erasure theorem for the admissible coarse-graining hierarchy of the Cosmochrony spectral programme. The non-injective projection $\Pi: {\Omega} \to {\mathcal{O}}$ defines a residual fibre-information functional $I(c;\sigma(\ell))$, where $c$ is a Weil-block fibre label and $\sigma_c(\ell)$ is the BFS capacity profile at coarse-graining depth $\ell$. The structural sufficiency hypothesis [H-suff] — that $\sigma(\ell)$ is a sufficient statistic for the hierarchy beyond $\ell$ — is proved at the level of the BFS rank observable (Proposition prop:rank-suff), where the fibre label is erased identically; for the full Born–Infeld capacity it remains conditional on a single preservation step, numerically supported for $q \in \{61, 151, 211\}$ (Section sec:hsuff-test). Granting [H-suff], the data-processing inequality implies that $I(c;\sigma(\ell))$ is non-increasing in $\ell$: fibre information is erased, not created, under admissible coarse-graining. This result is not a $c$-theorem; it does not count effective degrees of freedom. It is a quantitative realisation of the ENI no-go theorem: non-injectivity of $\Pi$ forces projective information loss, and the data-processing inequality makes this loss monotone and measurable. The inter-sector variance $\mathrm{Var}_c(\sigma_c(\ell))$ serves as a computable proxy; its restriction to $\mathrm{SU}(3)$ colour-triplets connects directly to hypothesis [H-color].