Gravity Page Curve Structural Page Curve Derived Structural Prop Holds

The triangular shape

The Page curve describes how the entropy of radiation emitted by an evaporating black hole changes over time. In the standard account, the curve rises as the hole radiates, peaks when half the hole has evaporated, then falls back to zero when the hole is gone. The Recognition Science library contains a machine-checked theorem, ledger (a discrete record of events), that proves a specific triangular version of this curve has the required shape properties.

The theorem establishes that a piecewise-linear function, called the triangular Page curve, starts at zero, rises linearly to a peak value at the Page time, falls linearly back to zero at twice the Page time, and stays zero afterward. It also proves the curve is never negative. These are the kinematic facts: the shape itself, independent of any underlying dynamics.

In Recognition Science, the declaration page_curve_derived_structural_prop_holds provides a concrete witness to the master theorem hypothesis. This means it supplies an example of a curve with the required properties, retiring the hypothesis that such a shape exists. The witness uses the simplest possible parameters: a peak value of one and a Page time of one.

What the theorem does not do is derive this curve from the ledger structure of quantum gravity. The full derivation, involving replica wormholes and quantum extremal surfaces, remains future work. The module explicitly states this: it ships the kinematic shape, not the dynamical origin. The theorem is a structural result, not a physical derivation.

The practical consequence is that any future dynamical derivation must reproduce this triangular shape. The theorem sets a clear target. It proves the shape is consistent and well-defined, so the remaining problem is to show that the ledger's dynamics produce it.

/-- **TRACK 3.C ONE-STATEMENT** (structural form). The triangular Page
curve is theorem-grade in its kinematic content: starts at zero,
peaks at `S_max` at the Page time `t_Page`, returns to zero at full
evaporation `2·t_Page`, is non-negative throughout, and vanishes
after full evaporation. The master theorem hypothesis input
`PageCurveDerived` is inhabited by `pageCurveDerivedWitness`. The
**dynamical derivation** from RS substrate first principles (replica
wormholes, QES, ledger-side back-reaction) remains future multi-session
work. -/
theorem page_curve_one_statement :
    (∀ S t, trianglePageCurve S t 0 = 0) ∧
    (∀ S t, 0 < t → trianglePageCurve S t t = S) ∧
    (∀ S t, 0 < t → trianglePageCurve S t (2 * t) = 0) ∧
    (∀ S t r, 0 ≤ S → 0 < t → 0 ≤ trianglePageCurve S t r) ∧
    (Nonempty Gravity.MasterTheorem.PageCurveDerived) :=
  ⟨trianglePageCurve_at_zero,
   trianglePageCurve_at_peak,
   trianglePageCurve_at_end,
   trianglePageCurve_nonneg,
   ⟨pageCurveDerivedWitness⟩⟩

/-- The Page curve is non-negative everywhere, assuming `0 ≤ S_max`
and `0 < t_Page`. -/
theorem trianglePageCurve_nonneg
    (S_max t_Page t : ℝ) (h_S : 0 ≤ S_max) (h_t_Page : 0 < t_Page) :
    0 ≤ trianglePageCurve S_max t_Page t := by
  unfold trianglePageCurve
  by_cases h0 : t ≤ 0
  · simp [h0]
  · simp [h0]
    by_cases h1 : t ≤ t_Page
    · simp [h1]
      have : 0 ≤ t := by push_neg at h0; linarith
      have h_slope : 0 ≤ S_max / t_Page := div_nonneg h_S (le_of_lt h_t_Page)
      exact mul_nonneg h_slope this
    · simp [h1]
      by_cases h2 : t ≤ 2 * t_Page
      · simp [h2]
        have h_decline : S_max / t_Page * (t - t_Page) ≤ S_max := by
          have h_tail : t - t_Page ≤ t_Page := by linarith
          have h_slope : 0 ≤ S_max / t_Page :=
            div_nonneg h_S (le_of_lt h_t_Page)
          have h_t_pos : 0 ≤ t - t_Page := by
            push_neg at h1; linarith
          calc S_max / t_Page * (t - t_Page)
              ≤ S_max / t_Page * t_Page :=
                mul_le_mul_of_nonneg_left h_tail h_slope
            _ = S_max := by field_simp
        linarith
      · simp [h2]

theorem page_curve_derived_structural_prop_holds :
    page_curve_derived_structural_prop := by
  refine ⟨1, 1, by norm_num, by norm_num, ?_, ?_, ?_, ?_⟩
  · exact trianglePageCurve_at_zero 1 1
  · exact trianglePageCurve_at_peak 1 1 (by norm_num)
  · exact trianglePageCurve_at_end 1 1 (by norm_num)
  · intro t; exact trianglePageCurve_nonneg 1 1 t (by norm_num) (by norm_num)

What this page does not claim

The theorem does not derive the Page curve from ledger dynamics or quantum gravity first principles. The theorem does not prove that the actual radiation entropy of a real black hole follows this curve. The theorem does not identify the peak entropy S_max with any specific physical quantity.

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/PageCurveStructural.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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