Purity, thermodynamics, and leakage

The complete prepared evolution preserves the purity of a pure diary–reference input with its retained preparation purifiers. This is not a theorem of energy conservation or heat production.

Readout-and-forgetting equality does not determine heat, work, or cyclic memory-reset cost. As a direct example, SWAP with a blank qubit implements the same reset channel for every common excitation gap ${ϵ}_{\mathrm{gap}}>0$, while transferring energy ${ϵ}_{\mathrm{gap}}p$ for input excitation probability $p$. The blank resource is consumed and the information moves to its unreturned carrier; a cyclic reset would also have to restore that resource CheungEn07A.

Complete purity, thermodynamic reset, and input-dependent residual leakage are therefore separate ledgers. No $k_{\mathrm{B}}T\ln 2$ conclusion follows from channel notation alone.

An externally supplied luminosity

would give

This elementary integration does not identify the cell clock with mass evolution, establish an energy condition, or prove a below-cutoff endpoint CheungHawk04D. No such luminosity or temperature schedule is needed here.