Gravity Page Curve Dynamical Recognition Tick Capacity Transfer Prop Holds

The derived Page curve

The Page curve describes how the entropy, or hidden information, of a black hole and its emitted radiation changes as the hole evaporates. A simple triangular shape, rising to a peak at half-evaporation and falling back to zero, has long been a standard expectation in quantum gravity. The Recognition Science framework's library, a machine-checked collection of formal theorems, now derives this triangular shape from two principles instead of assuming it.

The first principle is a linear transfer of capacity. If the black hole starts with a total capacity for information, S_BH, then at a fraction t of its evaporation, the bulk capacity is S_BH · (1 - t) and the radiation capacity is S_BH · t. The second principle is Schmidt purification: the joint quantum state of the black hole and its radiation remains pure, which forces the entropy on both sides to be equal and bounded by the smaller of the two capacities. The entropy therefore equals the minimum of the two linear functions, which is exactly the triangular curve.

This result is a structural theorem with zero unproved assumptions and zero framework-specific axioms. It proves that the peak of the curve at t = 1/2 is forced, not chosen, and that the entropy returns to zero at full evaporation, which is information preservation. The declaration recognition_tick_capacity_transfer_prop_holds packages this derivation as a certified property of the framework's ledger dynamics.

In Recognition Science, the framework models the evaporation as a discrete process of recognition ticks, where each tick transfers a fixed fraction of capacity. The continuous linear functions are the limit of this discrete process. The theorem establishes that under these two principles, no other curve is possible.

What this does not claim is that the capacity transfer itself is derived from the fundamental recognition update. That step, modeling the explicit bulk-to-radiation transfer rate at each tick, remains open and is estimated to take several more sessions of work. The theorem also does not claim that the triangular curve is the only possible curve under different, non-linear transfer rates; those would give different shapes.

/-- 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

theorem pageCurveDynamicalCert_inhabited :
    Nonempty PageCurveDynamicalCert :=
  ⟨pageCurveDynamicalCert⟩

What this page does not claim

The capacity transfer rate itself is not derived from the fundamental recognition update; that step remains open. The triangular curve is not the only possible curve under non-linear capacity evolutions. The theorem does not prove the physical existence of black holes or Hawking radiation in the conventional sense.

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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