Foundation Pair Kernel Recognition Transport Residuals S18 Native Tick Span Fact

The duration readout

A duration is a span of time. In the Recognition Science framework, time is not a continuous background; it is a discrete record of events, a ledger of ticks. The question of what makes a duration physical, rather than just a number, is answered by a factorization condition. The theorem nativeTickSpanFactorization_implies_durationReadout states that if a duration is expressed as the native tick-span measure applied to the tick count of an event, then that duration satisfies the framework's definition of a physical readout.

The native tick-span measure is an additive measure: the measure of a span of eight ticks is eight times the measure of a single tick. This additivity is the key property. The theorem proves that any kinematics whose duration factors through this measure automatically qualifies as a physical duration readout. It is a bridge from a structural condition, the factorization, to a physical status, the readout.

What the theorem does not claim is just as important. It does not say that the native measure is the only additive measure. A doubled measure, where one tick has twice the span, is also additive, and the library proves it is distinct from the native one. The theorem does not select the native unit as the unique physical choice; it only shows that if you use it, the readout property follows.

The theorem also does not claim that the readout is exact in the sense of fixing energy or other physical scales. Exact energy readout, the library shows, is equivalent to fixing a uniform scale to one, and that is a separate, open condition. The native tick-span factorization is a necessary structural step, not a complete physical theory.

In the Recognition Science account, this theorem is one of the residual boundary results. It moves the question of what is physical inward, from the full physical conclusions to the choice of a tick measure. The choice of that measure remains open; the theorem shows what follows once the choice is made.

/-- Factoring through the native Tick-span measure is sufficient for S17's
physical-duration readout. -/
theorem nativeTickSpanFactorization_implies_durationReadout
    {N : ℕ} [NeZero N]
    {kinematics : PostingEventKinematics3 N}
    (hfactor :
      PostingDurationFactorsThroughTickSpan3
        nativeTickSpanMeasure kinematics) :
    PhysicalPostingDurationReadout3 kinematics := by
  intro event
  rw [hfactor event, postingEventTickSpan3_eq_one]
  simp [nativeTickSpanMeasure]
  unfold postingDurationFromRecognitionClock3
  rw [postingEventEndTick3_eq_succ]
  exact
    (nativeRecognitionClock_has_nativeTickSpacing
      (postingEventStartTick3 N)).symm

theorem nativeTickSpanMeasure_eight_eq_octave :
    nativeTickSpanMeasure.measure 8 =
      Constants.octave := by
  rfl

theorem doubledTickSpanMeasure_ne_native :
    doubledTickSpanMeasure ≠
      nativeTickSpanMeasure := by
  intro heq
  have hunit :=
    congrArg (fun span =>
      span.measure 1) heq
  simp [doubledTickSpanMeasure,
    nativeTickSpanMeasure] at hunit

What this page does not claim

The theorem does not claim that the native tick-span measure is the only additive measure. The theorem does not claim that a duration readout is an exact physical energy readout. The theorem does not claim that the native tick-span measure is physically selected over other additive measures.

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/Foundation/PairKernelRecognitionTransportResidualsS18.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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