Gravity Page Curve Dynamical Page Curve Derived From Recognition Ticks Prop Hold

The derived Page curve

The Page curve describes how the entropy of radiation emitted by an evaporating black hole changes over time. In the standard account, it rises, peaks, and falls back to zero, forming a triangle. The peak marks the moment when the black hole has lost half its mass, and the return to zero signals that all information initially trapped in the hole has been released. This curve is central to the black hole information paradox, because it shows how information can escape without violating quantum mechanics.

The derivation works by tracking two quantities: the capacity of the black hole's interior to hold information, and the capacity of the surrounding radiation to absorb it. The interior capacity decreases steadily as the hole shrinks, while the radiation capacity grows at the same rate. Their sum stays constant, equal to the initial entropy of the hole. The entropy of the radiation is then the smaller of the two capacities at any moment. This minimum of a falling line and a rising line is exactly the triangular shape, with its forced peak at the halfway point.

In Recognition Science, this curve is not an assumption. The framework models the evaporation as a sequence of discrete updates, called ticks, on a ledger that records the state of the hole and its radiation. Each tick transfers a fixed fraction of the remaining capacity from the hole to the radiation. The machine-checked library of formal theorems proves that this process yields the same triangular curve, with the peak at half evaporation and a return to zero at the end. The declaration page_curve_derived_from_recognition_ticks_prop bundles this result: the capacity transfer follows from the tick dynamics, and the curve follows from the transfer.

The proof relies on two ingredients. First, the joint state of the hole and the radiation remains pure throughout, because the evolution is unitary. Second, Schmidt purification forces the entropy of the two subsystems to be equal and bounded by the smaller capacity. The radiation entropy saturates this bound, giving the minimum form. The theorem S_rad_at_page_time confirms the peak value is exactly half the initial entropy, and S_rad_information_returned confirms the entropy vanishes at the end, meaning information is fully recovered.

What this result does not do is explain why the capacity transfer itself is linear. That step, deriving the transfer rate from the underlying recognition update at each tick, remains open. The current work takes the linear transfer as a model, not as a consequence. Different transfer rates would produce different curves, so the triangular shape is the canonical case under a specific assumption about how the hole sheds its capacity.

/-- 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 linear capacity transfer rate is not derived from recognition dynamics; it is assumed as a model. The result does not prove that real black holes follow this exact curve, since the physical bridge to recognition ticks is open. The derivation does not address the full information paradox beyond showing entropy returns to zero.

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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