Holography Horizon Clock Rate Period Is B2 Output Not B3 Input

The rate and the period

In general relativity, a clock held just outside a black hole's horizon runs slow as seen from far away. The strength of that slowing is set by the surface gravity, usually written κ. In the Euclidean picture used for thermal calculations, this shows up as a simple angular rate: the clock's phase advances by κ for each unit of imaginary time. The framework's machine-checked library of formal theorems records this as a pure calculus fact, derivative of the angle with respect to Euclidean time, and nothing more.

The declaration period_is_b2_output_not_b3_input draws a line between two different facts. The rate, dθ/dτ_E = κ, is one thing. The period, the time needed for the angle to complete a full 2π turn, is another. The theorem states that this period equals 2π/κ by definition, as a rewriting rule. It does not derive that period from the rate law. The period is the output of an earlier stage of the framework's forcing chain, labeled B2, which establishes the full-turn closure. The rate law from stage B3 merely accepts that period as an input.

Why draw this line? The distinction keeps the framework's logical dependencies honest. The rate law is a local statement about how fast the angle advances. It holds for any positive κ, and it says nothing about when the angle returns to its starting value. The period is a global statement about completing a circuit. The framework proves that at the B2-forced period, the turn ratio, the number of full turns per period, is exactly one. That unity is a B2 result, not a consequence of the B3 rate.

The declaration also fences off a separate legacy definition. An older normalization socket sets κ = 1/R for Schwarzschild geometry, a bridge to the Clausius entropy argument. The new B3 delivery does not use that socket. It records only the existence of a positive rate parameter, with no thermality, no KMS condition, and no entropy-area law imported. The bundle of facts that B3 types carries no field about the period at all, a silence the library makes explicit.

What this buys a reader is a clean separation of concerns. The rate is forced by near-horizon geometry. The period is forced by the recognition cycle's closure. Mixing the two would blur which theorem supplies which fact. The framework's library keeps them apart, so each step in the chain can be audited on its own.

/-- **Fence theorem (THEOREM).** B3's rate law alone does not pin a closure time; the
full-turn time `2π/κ` is definitionally the B2 carrier output, not an input here. -/
theorem period_is_b2_output_not_b3_input (kappa : ℝ) :
    euclideanPeriod kappa = 2 * Real.pi / kappa := 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

/-- At the B2-forced period, the turn ratio is unity (B1/B2 linkage; B3 not used). -/
theorem turnRatio_unity_at_b2_period (kappa T : ℝ) (hk : 0 < kappa) (hT : 0 < T) :
    T = euclideanPeriod kappa → turnRatio kappa T = 1 :=
  (turnRatio_eq_one_iff kappa T hk).mpr

/-- **Explicit non-claim:** `ClockRateBundle` carries no field about `T = 2π/κ`. -/
theorem clockRateBundle_silent_on_period {kappa : ℝ} (_h : ClockRateBundle kappa) :
    True := trivial

What this page does not claim

The declaration does not prove that the period is 2π/κ from the rate law alone. It does not assert that surface gravity κ equals 1/R; that is a separate legacy normalization socket. It does not import thermality, KMS condition, or entropy-area law into the B3 rate bundle.

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.

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