Holography Horizon Clock Rate Turn Ratio Unity At B2 Period

The turn ratio at the forced period

In the Recognition Science framework, a recognition clock, a discrete record of events near a horizon, advances an angle as time passes. The theorem turnRatio_unity_at_b2_period states a conditional fact: if the time for one full turn equals the Euclidean period, then the turn ratio equals one. The turn ratio is the number of full rotations the clock completes in that time. The period in question, written as 2π/κ, comes from a separate part of the framework called B2, not from the rate law itself.

The rate law this builds on is simple. Near a horizon with surface gravity κ, the angle θ equals κ times the Euclidean time τ_E. The derivative, dθ/dτ_E, is exactly κ. This is a pure calculus result: the angle advances at a constant rate. The theorem turnRatio_unity_at_b2_period takes that rate and adds the B2 condition that the full-turn time is 2π/κ. Under that condition, the turn ratio collapses to 1. The proof is a direct application of an earlier equivalence, turnRatio_eq_one_iff, which links the ratio to the period.

What the theorem does not claim is just as important. It does not assert that the period 2π/κ is forced or derived here. That period is the output of the B2 carrier, a separate module, and the rate law alone cannot pin it down. The framework includes an explicit fence theorem, period_is_b2_output_not_b3_input, stating that the full-turn time is definitionally the B2 output, not an input to this rate-only result. The rate bundle, ClockRateBundle, carries no field about the period at all; a theorem clockRateBundle_silent_on_period records that silence as a trivial truth.

The practical upshot is a clean division of labor. B3 delivers the rate: the clock ticks at κ per unit time. B2 delivers the period: one full turn takes 2π/κ. The theorem here joins them: when B2's period holds, the turn ratio is exactly one. This is a precise, conditional statement, not a claim that the framework derives the period from the rate. The distinction matters for anyone reading the framework's chain of results: each module types only what it delivers, and this one delivers the unity of the turn ratio under a period it does not itself establish.

/-- 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

/-- **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

/-- Algebraic form of the rate law (the panel's `dθ/dτ_E = κ`). -/
theorem euclideanAngle_deriv_eq (kappa : ℝ) (hk : 0 < kappa) (tauE : ℝ) :
    deriv (euclideanAngle kappa) tauE = kappa :=
  (euclideanAngle_rate kappa hk tauE).deriv

/-- **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

What this page does not claim

The theorem does not prove that the Euclidean period 2π/κ exists or is forced; it only states a conditional equivalence. The theorem does not claim that the rate law alone determines the full-turn time. The theorem does not establish any thermality, KMS condition, or entropy-area law for the horizon.

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)

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