Gravity Page Curve Structural Triangle Page Curve At Zero

The starting point

The Page curve describes how the entropy of a black hole's radiation changes as the black hole evaporates. In the standard picture, a black hole emits Hawking radiation, and the entropy of that radiation rises, peaks, and then falls back to zero when the black hole is gone. The very first point of that curve, at the moment evaporation begins, is the subject of the declaration trianglePageCurve_at_zero. It states that the radiation entropy at time zero is exactly zero: S_rad(0) = 0.

This is a structural theorem, not a dynamical one. The Recognition Science framework's machine-checked library of formal theorems defines a specific piecewise-linear function, trianglePageCurve, to represent the Page curve's shape. The declaration proves that this function, when evaluated at time zero, returns zero. The proof is a direct simplification of the function's definition: the first branch of the piecewise definition applies, and the result is zero. It does not require any assumptions about the peak entropy S_max or the Page time t_Page; it holds for any real values of those parameters.

The significance of this theorem is that it anchors the entire Page curve at a known point. Without this, the curve could start anywhere. With it, the curve is fixed to begin at zero, which is the physically correct starting condition: before any radiation has been emitted, the radiation's entropy must be zero. This is a necessary boundary condition for any future dynamical derivation of the Page curve, and the theorem guarantees that the structural shape used by the framework satisfies it.

In Recognition Science, this theorem is part of a larger effort to derive the Page curve from first principles. The framework models the universe as a ledger, a discrete record of recognition events, and aims to show that the Page curve's shape is a consequence of that ledger's structure. However, this particular declaration does not perform that derivation. It only establishes the kinematic shape, the triangle, and its properties. The dynamical derivation, which would show that the actual entropy of a real black hole follows this shape, is explicitly left as future work.

What this means in practice is that the framework has a proven, consistent starting point for its Page curve model. The theorem is a small but necessary piece of a larger puzzle. It is not a claim about the dynamics of black hole evaporation, nor is it a derivation of the Page curve from the ledger's rules. It is a statement about the geometry of a curve, a fixed point from which the rest of the shape must follow.

/-- `S_rad(0) = 0`: at the start, no radiation has been emitted. -/
theorem trianglePageCurve_at_zero (S_max t_Page : ℝ) :
    trianglePageCurve S_max t_Page 0 = 0 := by
  unfold trianglePageCurve
  simp

/-- `S_rad(0) = 0`: at the start, no radiation has been emitted. -/
theorem trianglePageCurve_at_zero (S_max t_Page : ℝ) :
    trianglePageCurve S_max t_Page 0 = 0 := by
  unfold trianglePageCurve
  simp

/-- **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⟩⟩

What this page does not claim

This declaration does not derive the Page curve from the framework's first principles. It does not claim that the radiation entropy of a real black hole follows this triangle shape. It does not establish the peak entropy value or the Page time; these are parameters of the function.

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.

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