Prove the one-shot QES memory-budget bound

Prove, for semiclassical states with a compressible finite-dimensional predictive sector, that the conditional predictive memory cost of the late Hawking-radiation process is bounded by the generalized-entropy quantum-extremal-surface memory budget, up to the stated smoothing and finite-block correction; equivalently, establish the one-shot identification between the conditional max-entropy of the minimal predictive memory and the generalized-entropy budget.

Background

The paper defines the conditional memory cost m_ε(L|R) as the smallest auxiliary predictive register that, together with the early radiation R, renders the late radiation process Markovian to tolerance ε. For semiclassical states with a finite-dimensional predictive sector, it proposes relating this operational cost to the conditional smooth max-entropy of the minimal predictive memory M_min given R.

A heuristic conjecture identifies this one-shot information quantity with a generalized-entropy budget constructed from the quantum-extremal-surface area contribution and the entropy of the bulk degrees of freedom carrying the predictive memory. Establishing this relation would connect the collapse of the operational memory cost across the Page transition to a resource-theoretic refinement of the Page curve and clarify whether the area term controls all predictive memory or only the portion visible to a specified class of late-time operations.

References

Conjecture (heuristic). In this regime, \begin{equation} m_\epsilon(L\,|\,R_t) \ \lesssim\ \frac{\mathcal B_{\mathrm{QES}(t)}}{\ln 2} +\Delta_{\mathrm{1shot}(\epsilon,\psi), \end{equation} where \Delta_{\mathrm{1shot} is a smoothing and finite-block correction that vanishes in the asymptotic limit. The finite-dimensional evaporation comb fixes the bulk normalization: before recovery its cost is $\log_2\dim M_{\min}$, and after recovery it is zero at the selected tolerance. The JT calculation supplies the QES exchange and the $\sqrt{\epsilon(C)}$ reconstruction correction. The additional step in~eq:memory-budget-conjecture, identifying the one-shot predictive entropy with the generalized-entropy budget, is the holographic input. Its proof would turn the collapse of $m_\epsilon$ into a resource-theoretic refinement of the Page curve, with the dilaton and bulk-memory terms providing the expected semiclassical scaling.

— The Page Transition in the Multi-Time Correlations of Hawking Radiation  (2609.20456 - Waghmare, 17 Sep 2026) in Section 7.2, “A semiclassical memory budget,” especially the paragraph following Eq. (7.11) and Section 7.4, “The one-shot QES bridge”