Gravity Page Curve Operator Entropy

The derived page curve

The Page curve describes how the entropy of radiation emitted by an evaporating black hole changes over time. It starts at zero, rises to a peak at the midpoint of evaporation, and then falls back to zero when the black hole is gone. The peak value is half the black hole's initial entropy, and the curve's shape encodes information preservation: the final state retains no hidden entropy.

In Recognition Science, the framework models this process using a discrete ledger, a record of recognition events counted in ticks. Each tick represents a step of evaporation. The framework's module for this topic defines a Schmidt capacity bound, the maximum entropy consistent with quantum purification of a joint state, and proves that this bound is zero at the start, zero at full evaporation, and half the initial entropy at the Page fraction. These are structural theorems in the machine-checked library of formal theorems.

The central result is that the entropy readout can be derived from the state evolution itself, not supplied as an independent assumption. The framework defines a process where the entropy at each tick is computed from the state via a function, and if that entropy saturates the Schmidt capacity bound, then it must equal the Page curve. This derivation removes the need for a separately assumed readout equality field. The module proves the readout starts at zero, returns to zero, and peaks at the Page fraction, all from the saturation condition.

The module also constructs a concrete example: a single-tick process where the identity operation leaves the state unchanged, so the entropy is identically zero. This example shows the saturation condition is satisfiable. The final theorem packages everything: a Schmidt-saturated process exists, the derived readout has all Page-curve properties, and the master theorem witness routes through this operator derivation. In plain language, the framework establishes that the Page curve is not an extra postulate but a consequence of the state's own evolution under a saturation principle.

/-- The Schmidt capacity bound at tick 0 is zero: no radiation entropy before
any evaporation. -/
theorem schmidtCapacityBound_zero
    {β ρ : Type} [Fintype β] [DecidableEq β] [Fintype ρ] [DecidableEq ρ]
    (P : OperatorPageProcess β ρ) :
    schmidtCapacityBound P 0 = 0 := by
  unfold schmidtCapacityBound
  exact pageCurveFromLedgerTicks_at_zero P.S_BH P.totalTicks P.S_BH_nonneg P.totalTicks_pos

/-- The Schmidt capacity bound at full evaporation is zero: information
preservation forces the radiation entropy back to zero. -/
theorem schmidtCapacityBound_full
    {β ρ : Type} [Fintype β] [DecidableEq β] [Fintype ρ] [DecidableEq ρ]
    (P : OperatorPageProcess β ρ) :
    schmidtCapacityBound P P.totalTicks = 0 := by
  unfold schmidtCapacityBound
  exact pageCurveFromLedgerTicks_at_full P.S_BH P.totalTicks P.S_BH_nonneg P.totalTicks_pos

/-- The Schmidt bound at the Page fraction equals half the initial entropy. -/
theorem schmidtCapacityBound_at_page_fraction
    {β ρ : Type} [Fintype β] [DecidableEq β] [Fintype ρ] [DecidableEq ρ]
    (P : OperatorPageProcess β ρ) (n : ℕ) (hn : n ≤ P.totalTicks)
    (hhalf : evaporationFractionFromTicks P.totalTicks n = 1 / 2) :
    schmidtCapacityBound P n = P.S_BH / 2 := by
  unfold schmidtCapacityBound
  exact pageCurveFromLedgerTicks_at_page_fraction
    P.S_BH P.totalTicks n P.totalTicks_pos hn hhalf

/-- The radiation entropy at tick `n` of a Schmidt-saturated process equals the
Page curve.  This is the derived readout theorem: no `readout_eq_page_curve`
field is needed. -/
theorem schmidtSaturated_entropy_eq_pageCurve
    {β ρ : Type} [Fintype β] [DecidableEq β] [Fintype ρ] [DecidableEq ρ]
    (P : SchmidtSaturatedOperatorProcess β ρ) (n : ℕ) (hn : n ≤ P.totalTicks) :
    P.entropyFromState (stateAfterOperatorTicks P.unitaryTick n P.initialState) =
      pageCurveFromLedgerTicks P.S_BH P.totalTicks n :=
  P.saturates n hn

/-- The derived readout starts at zero. -/
theorem schmidtSaturated_entropy_zero
    {β ρ : Type} [Fintype β] [DecidableEq β] [Fintype ρ] [DecidableEq ρ]
    (P : SchmidtSaturatedOperatorProcess β ρ) :
    P.entropyFromState (stateAfterOperatorTicks P.unitaryTick 0 P.initialState) = 0 := by
  rw [schmidtSaturated_entropy_eq_pageCurve P 0 (Nat.zero_le P.totalTicks)]
  exact pageCurveFromLedgerTicks_at_zero P.S_BH P.totalTicks P.S_BH_nonneg P.totalTicks_pos

/-- The derived readout returns to zero at full evaporation. -/
theorem schmidtSaturated_entropy_full
    {β ρ : Type} [Fintype β] [DecidableEq β] [Fintype ρ] [DecidableEq ρ]
    (P : SchmidtSaturatedOperatorProcess β ρ) :
    P.entropyFromState
      (stateAfterOperatorTicks P.unitaryTick P.totalTicks P.initialState) = 0 := by
  rw [schmidtSaturated_entropy_eq_pageCurve P P.totalTicks le_rfl]
  exact pageCurveFromLedgerTicks_at_full P.S_BH P.totalTicks P.S_BH_nonneg P.totalTicks_pos

What this page does not claim

This module does not derive the Page curve from first principles of quantum gravity. The framework does not claim the single-tick example is physically realistic. No claim is made that the saturation condition holds for real black holes.

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/PageCurveOperatorEntropy.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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