Foundation Pair Kernel Recognition Transport Residuals S18 Duration Readout Iff

The duration readout

A ledger, a discrete record of events, assigns each posting event a duration. The declaration in question, durationReadout_iff_nativeTickSpanFactorization, states a precise equivalence: a duration function is a physical readout if and only if it factors through the native tick-span measure. That means the duration of any event equals the native measure applied to the event's tick span. The native measure is additive: the measure of a combined span is the sum of the measures of its parts. Additivity forces the entire measure to be determined by its value on a single tick, but it does not select that value.

The theorem proves that additivity alone fixes the measure up to a positive scalar multiple. The native measure uses the constant tau0 as its unit; the doubled measure uses twice tau0. Both are additive, both satisfy the same structural axioms, and they are distinct. The equivalence therefore does not claim that the native unit is the only possible one. It claims only that if a duration readout exists, it must be of the native form. The framework's library proves this with a machine-checked collection of formal theorems, with no unproved assumptions and no new axioms.

The theorem also separates the completeness of a physical carrier into two independent halves: injectivity, meaning no two distinct events share the same classification, and surjectivity, meaning every classification is realized by some event. Each half has its own cardinal bound and its own countermodel. A carrier that is complete in one sense but not the other exists. The equivalence shows that full completeness is exactly the conjunction of the two halves.

What the declaration does not claim is that the native unit tau0 is physically forced. The unit remains an open choice within the framework. The theorem also does not claim that any particular physical system realizes the native measure. It establishes a conditional: if a duration readout exists, it has the native form. The source arrows that attach the native measure to actual physical duration remain open targets, not proved results.

/-- 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

/-- 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

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

theorem carrierComplete_iff_injective_and_surjective
    {N : ℕ} [NeZero N]
    (physical : PostingPhysicalChannelCarrier3 N) :
    PostingCarrierCoherenceComplete3 physical ↔
      PostingCarrierParentInjective3 physical ∧
        PostingCarrierParentSurjective3 physical := by
  constructor
  · intro hcomplete
    exact
      ⟨fun event => (hcomplete event).1,
        fun event => (hcomplete event).2⟩
  · rintro ⟨hinjective, hsurjective⟩ event
    exact
      ⟨hinjective event, hsurjective event⟩

What this page does not claim

The native tick-span unit tau0 is physically forced. Any particular physical system realizes the native measure. The source arrows attaching the native measure to physical duration are proved.

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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