Projection as Conditional Expectation and the Capacity Ceiling

Short structural note of the Cosmochrony non-injective foundations sub-programme.

Under four graded hypotheses --- fibre-invariance (forced by the emergence criterion of ENI), fixity on resolved content, linearity, and non-amplification --- the amplitude-level action of the non-injective projection is derived to be the conditional expectation onto the sigma-algebra generated by the distinguishable-history record. Two independent classical characterisations (contractive projections on Hilbert space; Moy--Rota--Douglas averaging operators) give the same kernel, and the degraded minimum-distortion selection gives the same operator.

Two consequences follow as mathematics. The law of total variance converts the kernel into strict ensemble-power suppression (the suppression is the erased intra-fibre power), answering the cosmic-variance objection structurally. A one-shot rate--distortion argument gives the parameter-free capacity ceiling (N-1)/N for a complex Gaussian mode supported on N distinguishable histories --- identically the Bessel envelope found empirically preferred, with no fitting, against the Planck low-multipole TT spectrum by the distinguishable-history capacity note.

Consumers: the low-ell capacity note, the cosmological branch, and the extraction of an evolving effective equation of state from full-history compression.

Build

bash compile.sh

Output: out/ProjectionConditionalExpectation.pdf.

Status

Deposited on Zenodo, version 1.0.0. Concept DOI: 10.5281/zenodo.21227547.