Physicists know how to get the Page curve — the time history of entanglement entropy for an evaporating black hole that is consistent with quantum mechanics — from gravitational formulas. What has been missing is a clear account of which microscopic black hole degrees of freedom actually carry that entanglement. This paper argues that those degrees of freedom live in the theory’s Hamiltonian phase space and are distinguished by surface charges at the black hole’s bifurcation surface.

The authors build on a recently proven generalized Ryu–Takayanagi formula that keeps explicit contact with the Hamiltonian phase space of the theory. The main claim is concrete: in any diffeomorphism‑invariant field theory that contains stationary black holes with a bifurcate Killing horizon (a symmetric kind of horizon with a crossing surface), the phase‑space states that can be told apart by their Hamiltonian surface charges over that crossing surface must provide the degrees of freedom responsible for the black hole’s Wald entropy. Wald entropy is a version of black hole entropy that generalizes the familiar Bekenstein–Hawking formula to broad classes of gravity theories.

At a high level the paper is part of a larger program that rewrites quantum predictions as weighted sums over paths in phase space rather than sums over spacetime field configurations. The idea is that a different organization of these sums can make physical structure obvious that is hidden in the usual spacetime picture and in standard Feynman‑diagram approaches. The generalized Ryu–Takayanagi prescription used here is one output of that program, and the same framework has also led the authors to statements like a gravitational entropy bound.

Why this matters: it offers a concrete, gauge‑invariant place to look for the black hole microstates that underlie both the Page curve and black hole entropy. If the argument holds, then diffeomorphism invariance (the symmetry underlying general relativity) forces the gravitational theory itself to contain these phase‑space degrees of freedom. In everyday terms, the paper claims to uncover a form of “black hole hair” — extra, measurable data at the horizon — that was overlooked by naive spacetime reasoning.