Energy Records and Relational Clocks: An Analytic Decomposition in a Black-Hole–Radiation Toy Model

Abstract

We study a stationary finite-dimensional bipartite quantum system composed of mass levels, labelled “black hole,” and energy levels, labelled “radiation.” Matched levels satisfy a fixed total-energy constraint. Analytically and across 128 paired numerical realizations, the diagonal part of the global state provides a perfect record between remaining mass and radiated energy, whereas off-diagonal coherence across energy branches supports Page–Wootters conditional evolution. Complete dephasing preserves stationarity, the energy shell, both imposed thermal local density matrices, and the classical energy record, while eliminating all nontrivial clock variation. Relational clocks are detected in 128/128 coherent states and 0/384 mixed controls. Within this finite family, diagonal correlations encode the record of change, whereas coherence supports relational change. This construction is a controlled toy model. It does not reproduce black-hole geometry, Hawking pair creation, backreaction, or the Page curve, and it does not claim to resolve the black-hole information paradox.