Abstract

We develop conditional Page-curve extrapolations from a finite Haar proxy while retaining the access and output distinctions of the Philosophy of Enter. Nonuniform coherent cell release gives the Page–Sen mean, whereas destructive readout gives an append-only classical record with a different exact entropy mean. Their entropy difference is basis-dependent relative-entropy coherence, not a whole-code recovery error or a thermodynamic cost.

A compatibility theorem identifies a missing condition in a proposed clock replacement: a fixed-dimensional tensor-release proxy requires the sum of its two logarithmic capacities to remain constant. Consequently, identifying radiation capacity with a coarse entropy production law larger than black-hole entropy loss is not merely a reparametrization of Hawk05-B. We give an explicit rounded, fixed-dimensional realization for the compatible case and an exact direct-sum calculation for a conserved additive-charge sector. A positive-mass inverse-square clock yields a conditional capacity crossover, its cutoff criterion, and a peak-location stability bound.

Physical entropy transfer, ensemble typicality, and quantum recovery receive separate quantitative allowances. A proposed Petz decoder is made into a deterministic channel on its entire input space and can be certified by its decoded Choi discrepancy. No occupation law, entropy slope, restricted observable algebra, or finite termination supplies the missing physical comparisons. The result is a proved finite extrapolation framework and a collection of explicitly conditional evaporation sketches, not a microscopic gravitational Page curve, an island construction, or spacetime emergence.