Foundation Pair Kernel Physical Readout Selection S17 Physical Duration Readout

The duration readout

A physical clock measures time by counting something. The Recognition Science framework formalizes what that "something" could be. Its ledger, a discrete record of events, assigns each posting event a duration. The declaration physicalDurationReadout_implies_fundamentalTick proves that if a physical duration readout is faithful, then the event it measures must last exactly one fundamental tick, the smallest indivisible unit of the framework's clock.

The proof is short and direct. The theorem assumes a PhysicalPostingDurationReadout3, meaning the readout correctly reports the length of the normalized recognition interval. It concludes RealizedPostingLastsFundamentalTick3, meaning the event's duration is exactly one tick. The argument passes through the native recognition clock, which is defined to be normalized. A readout that reports a stretched duration, say twice the tick, is shown by a separate countermodel to fail the faithfulness condition. The theorem does not say what a physical clock is made of, how it counts, or which events it reads. It says only that if it reads correctly, the unit it reads is the fundamental tick.

This is a theorem in the framework's machine-checked library of formal theorems, with no unproved assumptions beyond the standard logical axioms. It is a statement about the internal consistency of the framework's model, not a statement about any laboratory apparatus. The framework models physical duration as a readout of its ledger; the theorem pins down what that readout must see.

/-- The duration readout forces one physical `tau0` per realized posting. -/
theorem physicalDurationReadout_implies_fundamentalTick
    {N : ℕ} [NeZero N]
    {kinematics : PostingEventKinematics3 N}
    (hread : PhysicalPostingDurationReadout3 kinematics) :
    RealizedPostingLastsFundamentalTick3 kinematics :=
  nativeClockReadout_implies_fundamentalDuration
    nativeRecognitionClock_is_nativeNormalized hread

/-- The doubled-duration kinematics cannot read the normalized Recognition
interval. This is the duration-square discriminator. -/
theorem stretchedDuration_not_physicalReadout :
    ¬ PhysicalPostingDurationReadout3
      (stretchedDurationKinematics3 3) := by
  intro hread
  exact stretchedDuration_countermodel.2.1
    (physicalDurationReadout_implies_fundamentalTick hread)

What this page does not claim

The theorem does not claim that any particular physical clock exists or is faithful. It does not claim that the framework's fundamental tick corresponds to any measured physical duration. It does not claim that the framework's model of duration is the only possible one.

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/PairKernelPhysicalReadoutSelectionS17.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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