Gravity Page Curve Dynamical Radiation Capacity From Ticks Eq Radiation Capacity

The tick-based capacity identity

When a black hole evaporates, physicists describe the process with continuous variables: time t runs from 0 to 1, and the radiation's Hilbert-space capacity grows linearly as S_BH * t. The Recognition Science framework instead models the evaporation as a discrete ledger of recognition events, each tick transferring a fixed fraction of capacity from the bulk to the radiation. The declaration radiationCapacityFromTicks_eq_radiationCapacity proves these two descriptions agree: the capacity computed from N total ticks after n ticks, namely S_BH * (n/N), equals the continuous formula S_BH * t evaluated at the evaporation fraction t = n/N.

The proof is a definitional equality, meaning the tick-based definition unfolds by computation to the continuous one. This is not a deep theorem about physics; it is a bridge between two ways of writing the same quantity. The framework's library of formal theorems records this bridge as a machine-checked fact, so the discrete ledger picture and the smooth curve picture are provably consistent at the level of definitions.

The identity matters because it lets the framework move between worlds. The continuous Page curve, with its triangular shape peaking at half-evaporation, is derived from the min of bulk and radiation capacities. The discrete version, pageCurveFromLedgerTicks, uses the same min but with tick-based capacities. The theorem ensures that as the number of ticks N grows, the discrete curve approaches the continuous one, tick by tick.

What the declaration does not claim is just as important. It does not derive the linear capacity evolution itself; that remains a modeling choice, not a theorem. It does not prove that Hawking radiation actually follows this schedule, nor does it establish the physical recognition-to-linking bridge that would make the ledger picture a statement about spacetime. The identity is a consistency result, not a dynamical law.

/-- Tick radiation capacity is exactly the linear radiation capacity at the
tick-induced evaporation fraction. -/
theorem radiationCapacityFromTicks_eq_radiationCapacity (S_BH : ℝ) (N n : ℕ) :
    radiationCapacityFromTicks S_BH N n =
      radiationCapacity S_BH (evaporationFractionFromTicks N n) := by
  rfl

/-- Tick radiation capacity is exactly the linear radiation capacity at the
tick-induced evaporation fraction. -/
theorem radiationCapacityFromTicks_eq_radiationCapacity (S_BH : ℝ) (N n : ℕ) :
    radiationCapacityFromTicks S_BH N n =
      radiationCapacity S_BH (evaporationFractionFromTicks N n) := by
  rfl

What this page does not claim

The linear capacity evolution is a modeling choice, not a derived theorem. The identity does not prove that Hawking radiation follows this schedule in physical spacetime. The physical recognition-to-linking bridge that would connect the ledger to gravity remains open.

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/PageCurveDynamical.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: