Foundation Pair Kernel Recognition Transport Residuals S18 Duration Readout Impl

Duration readout and its measure

In Recognition Science, a ledger is a discrete record of posting events, and each event carries a duration. The theorem nativeTickSpanFactorization_implies_durationReadout states that if an event's duration equals the value assigned by the native tick-span measure to the event's tick span, then that duration is a valid physical readout. The native measure assigns to any tick count n the value n times a base unit tau0, so a duration of, say, eight ticks is exactly eight tau0. This is a structural bridge: it connects the abstract ledger to a concrete notion of elapsed time.

The theorem does not claim that the native measure is the only possible one. The framework's own library shows that a doubled measure, assigning n times two tau0, is also additive and also factors durations, yet it is distinct from the native measure. Additivity alone fixes the measure's shape from its one-successor unit, but it does not select that unit. The theorem therefore establishes a sufficiency, not a uniqueness: if the factorization holds with the native measure, then readout follows, but nothing in this declaration forces the native measure to be the physical one.

In Recognition Science, the framework models physical duration through such factorization, and its machine-checked library of formal theorems contains this result. The theorem is a positive implication, proved from the stated hypothesis. What remains open is the source arrow: which posting dynamics actually supply the native factorization. The library documents this gap explicitly, listing the canonical source-side objects as witnesses that the factorization is possible, while the arrows attaching them to physical duration remain an open target.

The practical consequence is a clean separation. The theorem gives a sufficient condition for duration readout, and the library shows the condition is not vacuous: the native measure exists and is additive. But because the doubled measure also satisfies additivity, the framework cannot yet claim that physical duration is uniquely the native one. The reader can see exactly where the framework stands: one direction is proved, the converse and the selection of the unit are not.

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

theorem stretchedDuration_factors_doubledTickSpan :
    PostingDurationFactorsThroughTickSpan3
      doubledTickSpanMeasure
      (stretchedDurationKinematics3 3) := by
  intro event
  rw [postingEventTickSpan3_eq_one]
  simp [doubledTickSpanMeasure,
    stretchedDurationKinematics3]

/-- Additivity over `ℕ` fixes the whole measure from its one-successor unit. -/
theorem additiveTickSpanMeasure_eq_nat_mul_unit
    (span : AdditiveTickSpanMeasure)
    (n : ℕ) :
    span.measure n =
      (n : ℝ) * span.measure 1 := by
  induction n with
  | zero =>
      simp [span.measure_zero]
  | succ n ih =>
      calc
        span.measure (n + 1) =
            span.measure n + span.measure 1 :=
          span.measure_add n 1
        _ = (n : ℝ) * span.measure 1 +
            span.measure 1 := by
          rw [ih]
        _ = ((n + 1 : ℕ) : ℝ) *
            span.measure 1 := by
          push_cast
          ring

What this page does not claim

The theorem does not claim that the native measure is the only additive measure that factors durations. It does not claim that physical duration is uniquely the native one, only that the native factorization is sufficient for readout. It does not claim that the source arrows attaching the canonical objects to physical duration are supplied by committed posting dynamics.

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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