Projection as Conditional Expectation and the Capacity Ceiling

Within the Cosmochrony framework, this short structural note closes, at the level of a conditional theorem chain, the step left open by the distinguishable-history capacity note: why a small number of projectively distinguishable histories suppresses ensemble power rather than merely enlarging cosmic variance. The load-bearing object is identified and derived: under four graded hypotheses — fibre-invariance (forced by the emergence criterion), fixity on resolved content, linearity, 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) yield the same kernel, and a degraded selection principle (minimum distortion) yields the same operator, so the result is robust to the epistemic downgrading of any single hypothesis. Two consequences follow as mathematics. First, the law of total variance converts the kernel into strict ensemble-power suppression: the suppression equals the erased intra-fibre power, answering the cosmic-variance objection. Second, a one-shot rate–distortion argument gives a parameter-free capacity ceiling for a complex Gaussian mode supported on N distinguishable histories: the retained power fraction is at most (N-1) /N. This ceiling coincides identically, as a function of N, with the Bessel envelope that the capacity note found to be the empirically preferred no-fit cell against the Planck low-multipole spectrum, upgrading that envelope from an estimator heuristic to an information-theoretic bound. Structural reading (interpretation, not a derived result): on this reading, the large-angle CMB deficit is the visible cost of projecting an early universe that admitted only a handful of distinguishable histories, and the data sit near the maximal power that such a projection permits.