Gravity Page Curve Operator Entropy Schmidt Saturated Entropy Eq Page Curve

The derived Page curve

The Page curve describes how the entropy of radiation emitted by an evaporating black hole rises, peaks, and falls back to zero, resolving the black hole information paradox. In the Recognition Science framework, this curve is not assumed but derived from a more basic principle: the entropy of a quantum state is limited by its Schmidt capacity, the maximum entropy consistent with the purification of a joint system. The declaration schmidtSaturated_entropy_eq_pageCurve establishes that if a process's entropy reaches this capacity at every step, then the entropy exactly equals the Page curve.

The theorem is a structural result, not a physical measurement. It states that for any process satisfying the Schmidt-saturated condition, meaning the entropy is derived from the state itself and equals the capacity bound, the readout at each tick matches the curve defined by the ledger. This is proved in the machine-checked library of formal theorems, with no additional axioms. The key point is that the entropy is not an independent input; it is computed from the state evolution, and saturation forces the equality.

The theorem has three immediate consequences, all proved in the same module. At the start, the entropy is zero. At full evaporation, it returns to zero. At the Page fraction, when half the black hole has evaporated, the entropy peaks at half the initial black hole entropy. These properties are the defining features of the Page curve, and they follow from the single saturation hypothesis.

In Recognition Science, this result is part of a larger chain that derives gravitational structure from the ledger of recognition events. The framework models the black hole as an operator process with discrete ticks, and the Page curve emerges as a consequence of the Schmidt capacity bound. This is a derivation within the framework, not a claim about experimental observation.

The theorem does not claim that real black holes saturate their Schmidt bound. It establishes a conditional statement: if saturation holds, then the Page curve follows. The existence of a process satisfying the condition is proved separately, but only for a trivial single-tick case. The physical question of whether actual evaporation processes saturate remains open.

/-- 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

/-- At the Page fraction, the derived readout peaks at S_BH / 2. -/
theorem schmidtSaturated_entropy_peak
    {β ρ : Type} [Fintype β] [DecidableEq β] [Fintype ρ] [DecidableEq ρ]
    (P : SchmidtSaturatedOperatorProcess β ρ) (n : ℕ)
    (hn : n ≤ P.totalTicks)
    (hhalf : evaporationFractionFromTicks P.totalTicks n = 1 / 2) :
    P.entropyFromState (stateAfterOperatorTicks P.unitaryTick n P.initialState) =
      P.S_BH / 2 := by
  rw [schmidtSaturated_entropy_eq_pageCurve P n hn]
  exact pageCurveFromLedgerTicks_at_page_fraction
    P.S_BH P.totalTicks n P.totalTicks_pos hn hhalf

What this page does not claim

This theorem does not assert that any real black hole evaporation process actually saturates its Schmidt bound. This result does not constitute an experimental measurement or observation of the Page curve. The theorem does not claim that the entropy is computed from a Hamiltonian or any specific physical dynamics.

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.

Derived articles

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