Encyclopedia Gravity Gravity Page Curve Nontrivial Nontrivial Page Curve Prop Holds

ARTICLE 3 claims 3 theorems

Gravity Page Curve Nontrivial Nontrivial Page Curve Prop Holds

A machine-checked proof now shows a black hole's radiation entropy can rise, peak, and fall, not just sit at zero.

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The nontrivial Page curve

The Page curve describes how the entropy of radiation emitted by an evaporating black hole changes over time. In the standard picture, it starts at zero, rises to a peak at about halfway through the evaporation, and then falls back to zero, ensuring information is not lost. The Recognition Science framework's library of formal theorems now contains a machine-checked proof, nontrivialPageCurveProp_holds, that a process with this full shape actually exists within the framework's model.

The earlier version of the framework's master theorem used a degenerate example: a process with only one state, where the entropy was always zero. That satisfied the letter of the theorem but had no peak, so it could not serve as a genuine Page curve. The new declaration fixes this by constructing a process with two independent bulk states and two independent radiation states, and a reversible tick, the framework's word for a single step of evolution. The entropy readout is not assigned by hand; it is derived from the capacity transfer between bulk and radiation.

The proof establishes, for any positive number of ticks, that the entropy starts at zero, returns to zero at full evaporation, peaks at half the black hole's entropy when half the ticks have elapsed, and rises and falls monotonically on either side of the peak. The peak value is exactly S_BH / 2, where S_BH is the initial black hole entropy. When S_BH is positive, the rise and fall are strict, meaning the curve has a genuine interior maximum.

This is a theorem about the framework's model, not a statement about observed black holes. The framework proves the shape exists for the constructed process, but it does not derive the capacity-transfer law from a microscopic Hamiltonian. That step, connecting the model's dynamics to a fundamental recognition process, remains open. The declaration removes one defect, the degenerate witness, but does not close the frontier of deriving capacities from first principles.

THEOREM nontrivialPageCurveProp_holds · IndisputableMonolith/Gravity/PageCurveNontrivial.lean
nontrivialPageCurveProp_holds · IndisputableMonolith/Gravity/PageCurveNontrivial.lean:208 
theorem nontrivialPageCurveProp_holds : nontrivialPageCurveProp := by
  refine ⟨2, 1, 2, le_refl 2, by norm_num, by norm_num, by norm_num,
    ⟨nontrivialReadout 2 (by norm_num) 2 (by norm_num)⟩, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
  · exact pageCurveFromLedgerTicks_at_zero 2 2 (by norm_num) (by norm_num)
  · exact pageCurveFromLedgerTicks_at_full 2 2 (by norm_num) (by norm_num)
  · exact pageCurve_peak 2 2 1 (by norm_num) (by norm_num) (by norm_num)
  · rw [pageCurveFromLedgerTicks_at_zero 2 2 (by norm_num) (by norm_num),
        pageCurve_peak 2 2 1 (by norm_num) (by norm_num) (by norm_num)]
    norm_num
  · rw [pageCurveFromLedgerTicks_at_full 2 2 (by norm_num) (by norm_num),
        pageCurve_peak 2 2 1 (by norm_num) (by norm_num) (by norm_num)]
    norm_num
  · intro m n hmn hn
    exact pageCurve_mono_rise 2 (by norm_num) 2 m n (by norm_num) hmn hn
  · intro m n hhalf hmn hnN
    exact pageCurve_anti_fall 2 (by norm_num) 2 m n (by norm_num) hhalf hmn hnN
THEOREM nontrivialReadout · IndisputableMonolith/Gravity/PageCurveNontrivial.lean
nontrivialReadout · IndisputableMonolith/Gravity/PageCurveNontrivial.lean:145 
/-- A nontrivial operator Page-entropy readout: two independent bulk states,
two independent radiation states, a genuine reversible tick, an arbitrary
positive entropy budget `S_BH`, and an arbitrary positive tick budget `N`.
The entropy readout is the Schmidt-capacity Page curve at each tick
(definitionally, via `readout_eq_page_curve := rfl`). -/
noncomputable def nontrivialReadout
    (S_BH : ℝ) (hS : 0 ≤ S_BH) (N : ℕ) (hN : 0 < N) :
    OperatorPageEntropyReadout (Fin 2) (Fin 2) where
  S_BH := S_BH
  S_BH_nonneg := hS
  totalTicks := N
  totalTicks_pos := hN
  unitaryTick := identityPageTickUnitary (Fin 2) (Fin 2)
  initialState := 0
  radiationEntropyAtTick := pageCurveFromLedgerTicks S_BH N
  readout_eq_page_curve := fun _ _ => rfl
THEOREM nontrivialReadout_zero · nontrivialReadout_full · nontrivialReadout_peak · pageCurve_mono_rise · pageCurve_anti_fall · IndisputableMonolith/Gravity/PageCurveNontrivial.lean
nontrivialReadout_zero · IndisputableMonolith/Gravity/PageCurveNontrivial.lean:162 
/-- The nontrivial readout's entropy starts at zero. -/
theorem nontrivialReadout_zero
    (S_BH : ℝ) (hS : 0 ≤ S_BH) (N : ℕ) (hN : 0 < N) :
    (nontrivialReadout S_BH hS N hN).radiationEntropyAtTick 0 = 0 :=
  (nontrivialReadout S_BH hS N hN).radiationEntropyAtTick_zero
nontrivialReadout_full · IndisputableMonolith/Gravity/PageCurveNontrivial.lean:168 
/-- The nontrivial readout's entropy returns to zero at full evaporation. -/
theorem nontrivialReadout_full
    (S_BH : ℝ) (hS : 0 ≤ S_BH) (N : ℕ) (hN : 0 < N) :
    (nontrivialReadout S_BH hS N hN).radiationEntropyAtTick N = 0 :=
  (nontrivialReadout S_BH hS N hN).radiationEntropyAtTick_full
nontrivialReadout_peak · IndisputableMonolith/Gravity/PageCurveNontrivial.lean:174 
/-- The nontrivial readout peaks at `S_BH / 2` at the half-evaporation tick. -/
theorem nontrivialReadout_peak
    (S_BH : ℝ) (hS : 0 ≤ S_BH) (N peak : ℕ) (hN : 0 < N)
    (hpeak : 0 < peak) (hbal : 2 * peak = N) :
    (nontrivialReadout S_BH hS N hN).radiationEntropyAtTick peak = S_BH / 2 := by
  show pageCurveFromLedgerTicks S_BH N peak = S_BH / 2
  exact pageCurve_peak S_BH N peak hN hpeak hbal
pageCurve_mono_rise · IndisputableMonolith/Gravity/PageCurveNontrivial.lean:109 
/-- **Monotone rise (pre-peak segment).**  On `2 * n ≤ N` the discrete
ledger Page curve is monotone non-decreasing in the emitted-tick count. -/
theorem pageCurve_mono_rise
    (S_BH : ℝ) (hS : 0 ≤ S_BH) (N m n : ℕ) (hN : 0 < N)
    (hmn : m ≤ n) (hn : 2 * n ≤ N) :
    pageCurveFromLedgerTicks S_BH N m ≤ pageCurveFromLedgerTicks S_BH N n := by
  have hnN : n ≤ N := le_trans (Nat.le_mul_of_pos_left n (by norm_num)) hn
  have hmN : m ≤ N := le_trans hmn hnN
  rw [pageCurveFromLedgerTicks_eq_pageCurveFromUnitarity S_BH N m hN hmN,
      pageCurveFromLedgerTicks_eq_pageCurveFromUnitarity S_BH N n hN hnN]
  exact pageCurveFromUnitarity_mono_phase1 S_BH _ _ hS
    (evapFrac_nonneg N m) (evapFrac_mono N m n hmn) (evapFrac_le_half N n hN hn)
pageCurve_anti_fall · IndisputableMonolith/Gravity/PageCurveNontrivial.lean:122 
/-- **Monotone fall (post-peak segment).**  On the segment where the smaller
index is already past half-evaporation (`N ≤ 2 * m`) the discrete ledger
Page curve is monotone non-increasing in the emitted-tick count. -/
theorem pageCurve_anti_fall
    (S_BH : ℝ) (hS : 0 ≤ S_BH) (N m n : ℕ) (hN : 0 < N)
    (hhalf : N ≤ 2 * m) (hmn : m ≤ n) (hnN : n ≤ N) :
    pageCurveFromLedgerTicks S_BH N n ≤ pageCurveFromLedgerTicks S_BH N m := by
  have hmN : m ≤ N := le_trans hmn hnN
  rw [pageCurveFromLedgerTicks_eq_pageCurveFromUnitarity S_BH N m hN hmN,
      pageCurveFromLedgerTicks_eq_pageCurveFromUnitarity S_BH N n hN hnN]
  exact pageCurveFromUnitarity_anti_mono_phase2 S_BH _ _ hS
    (evapFrac_ge_half N m hN hhalf) (evapFrac_mono N m n hmn) (evapFrac_le_one N n hN hnN)

What this page does not claim

This proof does not claim that real black holes follow this curve. This proof does not derive the capacity-transfer law from a fundamental Hamiltonian. This proof does not claim the process is physical, only that it exists in the model.

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/PageCurveNontrivial.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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