Gravity Page Curve Dynamical Page Curve From Ledger Ticks Eq Page Curve From Uni

Two routes to one curve

When a black hole evaporates, its information is shared between the hole itself and the radiation it emits. The Page curve plots how much entropy, or hidden information, the radiation holds as evaporation proceeds. It starts at zero, rises to a peak halfway through, and falls back to zero when the hole is gone. The curve's triangular shape is a standard result in modern physics, but in the Recognition Science framework it is not assumed; it is derived from a single principle about how quantum states purify.

The framework models the evaporation in two ways. The first, ledger view, a discrete record of events, counts individual ticks of a process: after n ticks out of a total N, the hole's capacity has shrunk by a fraction n/N and the radiation's has grown by the same fraction. The second view treats time as a continuous variable t between 0 and 1. Both views define the radiation entropy as the smaller of the two capacities at that moment. The declaration pageCurveFromLedgerTicks_eq_pageCurveFromUnitarity is the formal statement that these two definitions agree: the discrete tick-based curve equals the continuous-time curve at every corresponding point.

This equality matters because it bridges two levels of description. The continuous curve, pageCurveFromUnitarity, is the one that follows from the Schmidt purification property of pure joint states: a pure state of the combined system forces the entropy of the hole and the entropy of the radiation to be equal, and both are bounded by the smaller Hilbert-space capacity. The tick-based curve, pageCurveFromLedgerTicks, is the one that would arise from a step-by-step recognition process. The theorem shows that the smooth, unitary picture and the discrete, process-based picture are not competing models; they are the same curve seen at two resolutions.

The derivation is not a postulate. The triangular shape emerges as the minimum of two monotone bounds: the bulk capacity falls linearly as the hole shrinks, the radiation capacity rises linearly as quanta accumulate, and the smaller one wins at each moment. The peak at t = 1/2 is forced by the symmetry of the two linear functions, not chosen by hand. The return to zero at t = 1 is information preservation: when the hole is gone, all the information has returned to the radiation. The framework's machine-checked library of formal theorems records this as a structural result with zero unproven assumptions.

What the declaration does not claim is equally important. It does not claim that the capacity evolution itself, the linear fall and rise, is derived from the recognition update at each tick. That step, modeling the explicit bulk-to-radiation transfer rate, remains open. It also does not claim that the triangular curve is the only possible one; different capacity evolutions, such as non-uniform Hawking emission rates, would give different curves. The equality is about the two definitions agreeing, not about the physical process being unique.

/-- The Page curve as the entropy bound forced by Schmidt purification:
`S_rad(t) = min(bulkCapacity, radiationCapacity)`. This is the unique
saturation of the entropy bound on a pure joint state, given linear
capacity transfer between bulk and radiation. -/
def pageCurveFromUnitarity (S_BH t : ℝ) : ℝ :=
  min (bulkCapacity S_BH t) (radiationCapacity S_BH t)

/-- Tick radiation capacity is exactly the linear radiation capacity at the
tick-induced evaporation fraction. -/
theorem radiationCapacityFromTicks_eq_radiationCapacity (S_BH : ℝ) (N n : ℕ) :
    radiationCapacityFromTicks S_BH N n =
      radiationCapacity S_BH (evaporationFractionFromTicks N n) := by
  rfl

/-- **The Page time = half-evaporation.** The peak radiation entropy is
reached at `t = 1/2` with value `S_BH / 2`, forced by symmetry of the
capacity transfer. -/
theorem S_rad_at_page_time : P.S_rad (1/2) = P.S_BH / 2 := by
  rw [P.schmidt_purification, pageCurveFromUnitarity_at_half]

/-- **Information preservation:** the Page curve returns to zero at full
evaporation because the bulk capacity vanishes. This is the unitarity
signature: all entropy initially in the bulk has been transferred to
radiation, and the radiation entropy returns to the pure-state value
(zero) because no remaining bulk degrees of freedom remain to entangle
with. -/
theorem information_preservation (S_BH : ℝ) (hS : 0 ≤ S_BH) :
    pageCurveFromUnitarity S_BH 1 = 0 :=
  pageCurveFromUnitarity_at_one S_BH hS

What this page does not claim

The linear capacity evolution itself is not derived from the recognition update at each tick. The triangular curve is not claimed to be the only possible Page curve for all evaporation processes. The declaration does not establish that the physical evaporation process is actually discrete rather than continuous.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/PageCurveDynamical.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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