Holography Horizon Clock Rate Legacy Horizon Rate Is Separate

The separation of two rates

In general relativity, a black hole horizon has a surface gravity, often written κ, which governs how fast a clock hovering near the horizon ticks compared with one at infinity. In the Recognition Science framework, a machine-checked library of formal theorems records a precise claim about this rate. The declaration legacy_horizonRate_is_separate states that the legacy normalization κ = 1/R, where R is a Schwarzschild radius, is logically equivalent to the definition of the legacy HorizonRate predicate. This is a theorem, proved by reflexivity, meaning the two statements are definitionally the same.

The theorem's purpose is to mark a boundary. It explicitly separates the legacy normalization socket, which is a definitional choice (MODEL), from the newer B3 rate-only delivery. The B3 delivery, in the same module, proves that the Euclidean angular coordinate θ advances at rate κ per unit Euclidean time: dθ/dτ_E = κ. This is a theorem about the rate itself, derived from the near-horizon Rindler geometry. The legacy socket, by contrast, is a separate bridge for the Clausius connection, and the theorem ensures no one conflates the two.

What the theorem does not claim is as important as what it does. It does not assert the 2π closure period of the Euclidean time circle; that is the output of a different module, B2, which defines the full-turn time as 2π/κ. The rate law alone, as a separate theorem in the same file states, does not pin down a closure time. The theorem also does not import any thermality, KMS condition, or entropy-area law. It is purely a statement about the equivalence of two definitions, a fence to keep the B3 rate-only result clean.

In plain terms, this declaration is a bookkeeping rule. It tells anyone reading the framework's library that the legacy rate normalization is a separate socket, not part of the newer rate-only proof. This separation matters because it prevents a reader from thinking the newer theorem depends on the older normalization, when in fact the newer theorem is self-contained. The consequence is a cleaner logical structure, where each definition and theorem knows exactly what it does and does not rely upon.

/-- The Clausius-bridge normalization `κ = 1/R` is an independent MODEL socket; it is
not part of the B3 rate-only delivery above. -/
theorem legacy_horizonRate_is_separate (kappa R : ℝ) :
    DeficitFreePeriod.HorizonRate kappa R ↔ kappa = 1 / R :=
  Iff.rfl

/-- **B3 rate (THEOREM).** The continued recognition clock advances Euclidean angle at
rate `κ`: `dθ/dτ_E = κ` for all `τ_E`. Pure calculus on `θ = κ τ_E`; no period input. -/
theorem euclideanAngle_rate (kappa : ℝ) (hk : 0 < kappa) (tauE : ℝ) :
    HasDerivAt (euclideanAngle kappa) kappa tauE := by
  unfold euclideanAngle
  simpa using (hasDerivAt_id (x := tauE)).const_mul kappa

/-- The Clausius-bridge normalization `κ = 1/R` is an independent MODEL socket; it is
not part of the B3 rate-only delivery above. -/
theorem legacy_horizonRate_is_separate (kappa R : ℝ) :
    DeficitFreePeriod.HorizonRate kappa R ↔ kappa = 1 / R :=
  Iff.rfl

What this page does not claim

This theorem does not assert the 2π closure period of the Euclidean time circle. This theorem does not import thermality, KMS condition, or entropy-area law. This theorem does not claim the legacy normalization is derived from the B3 rate law.

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/Holography/HorizonClockRate.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: