Gravity Page Curve Dynamical Page Curve From Unitarity Anti Mono Phase2

The derived Page curve

The Page curve describes how the entropy of a black hole and its radiation changes as the hole evaporates. In the 1990s, Don Page argued on general grounds that the curve should rise, peak, and then fall back to zero, so that information is preserved. That triangular shape was a physical expectation, not a derivation. The Recognition Science framework now derives the same shape from a single principle: the joint state of the black hole and its radiation stays pure under unitary evolution.

The framework's machine-checked library of formal theorems defines the evaporation as a parameter t from 0 to 1. The black hole's capacity for entropy falls linearly, and the radiation's capacity rises linearly, so their sum is constant. A theorem of Schmidt says that for a pure joint state, the entropy of each part is equal and bounded by the smaller capacity. The library proves that the radiation entropy saturates this bound, giving the formula pageCurveFromUnitarity(S_BH, t) = min(S_BH·(1−t), S_BH·t). This is the triangular curve, with its peak forced at t = 1/2 and a return to zero at t = 1, which is information preservation.

This is a structural theorem: it says that if the capacity transfer is linear and the joint state is pure, then the curve must be that triangle. It is not a postulate. The library also shows that the same shape arises from discrete ticks of a ledger, where each tick transfers a fixed amount of capacity. The peak at half-evaporation is not chosen; it is forced by the two linear capacities.

What this does not claim is equally important. The derivation assumes linear capacity transfer; a non-uniform Hawking emission rate would give a different curve. The capacity evolution itself is not derived from the recognition update; that remains open. The theorem does not prove that real black holes follow this curve, only that within the model, unitarity plus linear transfer forces the triangle.

/-- 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)

/-- **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 returned at full evaporation.** -/
theorem S_rad_information_returned : P.S_rad 1 = 0 := P.S_rad_at_one

/-- Capacity-sum invariant: bulk + radiation = S_BH at every `t`.
Reflects conservation of Hilbert-space capacity under linear transfer. -/
theorem capacity_sum_invariant (S_BH t : ℝ) :
    bulkCapacity S_BH t + radiationCapacity S_BH t = S_BH := by
  unfold bulkCapacity radiationCapacity
  ring

What this page does not claim

The capacity evolution is not derived from the recognition update. The theorem does not apply to non-linear capacity transfer. Real black hole evaporation is not proven to follow this curve.

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)

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