The Page curve describes how entanglement between a black hole and its radiation evolves during evaporation: entropy rises as radiation escapes, reaches a maximum at the “Page time” when the black hole has lost half its degrees of freedom, then falls as the remaining black hole shrinks. Reproducing this curve from microscopic physics was a central achievement of the gravitational entropy program.
Two coupled Ising spin chains in a pure bipartite state exhibit the same curve. One chain shrinks while the other grows, mimicking the transfer of degrees of freedom from black hole to radiation. The entanglement entropy traces the Page curve faithfully.
The shock: it persists when the boundary coupling is zero.
At zero coupling between the chains, there is no interaction, no energy transfer, no dynamical mechanism to generate or destroy entanglement. The Page curve appears purely from the shrinking of one subsystem's Hilbert space dimension. As the effective number of degrees of freedom on one side decreases, the accessible entanglement between the two sides first increases (because the growing side has more room to be entangled with the shrinking side) and then decreases (because the shrinking side runs out of states to entangle). The rise-peak-fall profile is kinematic — determined by dimension counting, not by any physical process.
This does not mean the black hole information problem is trivial. The dynamical question — whether information actually escapes and how — remains real. But the Page curve itself, the shape that has driven decades of theoretical effort, turns out to be largely a property of Hilbert space geometry. Any system in which one subsystem loses dimensions while its complement gains them will produce this shape. The curve is not a signature of gravitational physics. It is a signature of shrinking.