Overview
This short structural note answers a question left open by the low-multipole capacity note: why does a mode supported on only $N$ distinguishable projective histories carry less ensemble power, rather than merely a larger cosmic variance?
The answer is a projection-kernel theorem. Under four hypotheses graded individually against the corpus — fibre-invariance (forced by the ENI emergence criterion), fixity on already-resolved content, linearity (the load-bearing structural transfer), and non-amplification — the action of the non-injective projection on mode amplitudes is exactly the conditional expectation onto the sigma-algebra generated by the distinguishable-history record. Two independent classical characterisations (contractive projections on Hilbert space; averaging operators) give the same kernel, and the degraded minimum-distortion selection gives the same operator.
Core contributions
- Projection kernel: under the four graded hypotheses, the amplitude assignment is the conditional expectation onto the distinguishable-record sigma-algebra, by two independent classical routes (contractive projections; Moy–Rota–Douglas averaging operators).
- Robustness: the fibre median shows the linearity hypothesis is genuinely load-bearing, and the minimum-distortion fallback leaves the operator unchanged — only the epistemic status of the selection principle can be downgraded.
- Entropy-to-power theorem: by the law of total variance, the projected ensemble power is the substrate power minus the erased intra-fibre variance — strict suppression whenever fibres are non-trivial, answering the cosmic-variance objection structurally.
- Capacity ceiling: a one-shot rate–distortion argument bounds the retained power of a complex Gaussian mode on $N$ distinguishable histories by $(N-1)/N$ — with no adjustable parameter and no resolution scale.
- Identification: the ceiling coincides identically with the Bessel envelope that the capacity note found empirically preferred, with no fitting, against the Planck low-multipole TT spectrum — upgrading it from estimator heuristic to information-theoretic bound.
What is proved and what is interpretive
The kernel theorem, the suppression theorem, and the ceiling are mathematics, given their hypotheses. The hypotheses themselves are graded: fibre-invariance is forced by the emergence criterion, fixity is forced-adjacent, and linearity and non-amplification are structural, with the linearity transfer identified as the one load-bearing step. The cosmological identifications — per-mode support, independence across modes, the rank–time dictionary — remain conditional and belong to the consumer papers. The reading of the observed proximity of the Planck data to the ceiling as capacity saturation is explicitly labelled interpretation.
Relation to the Cosmochrony program
The note belongs to the non-injective foundations sub-programme: it takes the emergence criterion of ENI and the proved distinguishable-history counts of the trajectory-branching note as inputs, and supplies the counting-to-power converter consumed by the low-multipole capacity note, by the cosmological branch, and by any extraction of an evolving equation of state from full-history compression.
References
Jérôme Beau. Projection as Conditional Expectation and the Capacity Ceiling. Result note, Zenodo. 10.5281/zenodo.21227547