From Record-Relay Boundary Accounting to Black-Hole Ringdown Response: Closed-Form Tests and a Two-Layer Resolution of the Page-Transition Width

This paper addresses a neglected dynamical question in black-hole information physics: if black-hole evaporation is unitary and information is not destroyed, can the recovery of conserved information be described not only as boundary accounting, but as a closed time-dependent recovery law? The paper develops the closed-form consequences of the record-relay framework for evaporating black holes. In this framework, a black hole is not treated as a destruction device, but as an information relay system: information is first recorded at the horizon and then redistributed outward through a boundary-mediated recovery process. The framework is built on four established theoretical anchors. Page-curve analysis requires information to return under unitary evaporation. Hayden–Preskill recoverability explains when scrambled information becomes operationally accessible. The island and quantum-extremal-surface constructions provide the gravitational saddle structure that realizes the Page curve. Fast-scrambling physics supplies the natural delay scale connecting horizon recording to delayed relay response. The paper first derives the recovery law from boundary-flux conservation. Conservation of reference information turns recovery into a flux-balance problem. The empty-reservoir condition at the beginning of recovery is derived from Hayden–Preskill recoverability rather than assumed. Together with saturation, analyticity, generic endpoint behavior, and the large-system limit, this selects a logistic recovery law and excludes stretched-exponential alternatives within the declared analytic class. The central contribution is the resolution of the Page-transition width left open by the companion paper. The paper shows that what looked like a single width is actually two different quantities: a microscopic saddle-switching rate that becomes infinitely sharp in the thermodynamic limit, and a finite coarse-grained projection width that describes how the Page transition appears at benchmark resolution. These two quantities are shown to arise from the same saddle partition structure, as two different layers of the same transition. For the four-dimensional Schwarzschild case, the benchmark projection width is fixed at 6.47 within the declared Page-branch projection protocol. This value is not claimed to be a universal constant or the physical microscopic width of a real black hole. The microscopic transition remains a sharp first-order-type kink; 6.47 is the finite coarse-grained image of that transition under the stated projection rule. The paper also derives ringdown-facing consequences of the record-relay structure. The relay stage is predicted to occur after a scrambling-time delay, with a delayed secondary burst whose relative scale is of order one percent after Kerr ringdown correction and is controlled mainly by remnant spin. This is not presented as a detected signal. It is a future falsification handle: a target for next-generation high-signal-to-noise black-hole spectroscopy and multi-event stacking. The conclusion is deliberately limited. This paper does not claim that a record-relay burst has been observed, nor that the benchmark width is a universal physical constant, nor that the relay amplitude is fixed from first principles. It argues that, under standard black-hole information inputs, the record-relay recovery law is closed at the level of shape and benchmark width, and that this closed structure generates a delayed ringdown target scale. In the broader Information Physics Series, the paper functions as the black-hole dynamics closure: the companion paper supplies the conserved boundary budget, Paper 24 supplies the recovery dynamics and benchmark width, and the detector-facing sequel tests the resulting delayed relay coordinates against gravitational-wave data. Keywords: black hole information, record-relay framework, Page curve, island formula, Hayden–Preskill recovery, quantum extremal surface, boundary flux, logistic recovery, scrambling time, quasinormal modes, ringdown, Page-transition width, saddle partition function, projection width, information conservation.