Foundation Pair Kernel Scale Covariant Observables S20 Boundary Unit Rescaling C

The boundary unit theorem

In ordinary measurement, choosing a unit is a convenience, not a discovery. A stopwatch that counts seconds and one that counts minutes report different numbers for the same physical interval, yet they agree on every ratio of intervals. The Recognition Science declaration boundaryUnit_rescaling_changes_absolute_not_ratio makes this familiar fact into a formal theorem about its own model of time.

The framework models elapsed time as a ledger, a discrete record of events. A positive elapsed-time observable assigns a positive real number to every tick count, and the theorem states that rescaling the boundary unit, the fixed positive number that converts tick counts into durations, multiplies every absolute duration by the same factor while leaving every normalized ratio of tick counts unchanged. The proof is machine-checked in the framework's library of formal theorems.

The theorem does not claim that the framework selects an absolute unit of time. It proves the opposite: the physical content of elapsed time is carried entirely by ratios, not by any single absolute number. The framework's S20 stage explicitly removes the boundary-unit convention, retaining only what is invariant under rescaling.

This result matters because it sharpens what the framework can and cannot say about physical time. It can say that ratios of elapsed times are unit-free and therefore meaningful. It cannot say that one particular unit, such as the second or the Planck time, is forced by the mathematics. The choice of an absolute unit remains a convention, exactly as it is in ordinary physics.

/-- A scale-covariant physical elapsed-time observable. Its only dimensional
datum is the positive image of one Tick. -/
structure PositiveElapsedTimeObservable where
  span : AdditiveTickSpanMeasure
  unit_pos : 0 < span.measure 1

/-- Positive unit rescaling changes absolute elapsed time while preserving
every normalized Tick count. -/
theorem boundaryUnit_rescaling_changes_absolute_not_ratio
    (observable : PositiveElapsedTimeObservable) :
    let doubled :=
      elapsedTimeFromBoundaryUnit
        ⟨2 * observable.span.measure 1,
          mul_pos (by norm_num) observable.unit_pos⟩
    doubled.span.measure 1 ≠ observable.span.measure 1 ∧
      ∀ n,
        normalizedElapsedTickCount doubled n =
          normalizedElapsedTickCount observable n := by
  dsimp
  constructor
  · intro h
    norm_num [elapsedTimeFromBoundaryUnit] at h
    nlinarith [observable.unit_pos]
  · intro n
    exact normalizedElapsedTickCount_invariant _ _ n

/-- All dimensionless time ratios are invariant under the choice of positive
boundary unit. -/
theorem normalizedElapsedTickCount_invariant
    (left right : PositiveElapsedTimeObservable)
    (n : ℕ) :
    normalizedElapsedTickCount left n =
      normalizedElapsedTickCount right n := by
  rw [normalizedElapsedTickCount_eq_count,
    normalizedElapsedTickCount_eq_count]

/-- Normalization removes the boundary unit exactly. -/
theorem normalizedElapsedTickCount_eq_count
    (observable : PositiveElapsedTimeObservable)
    (n : ℕ) :
    normalizedElapsedTickCount observable n = n := by
  rw [normalizedElapsedTickCount,
    elapsedTime_factors_through_Tick_count]
  field_simp [observable.unit_pos.ne']

What this page does not claim

The theorem does not claim that the framework selects an absolute unit of time. The theorem does not claim that the framework derives the second or any other SI unit. The theorem does not claim that all physical observables are invariant under unit rescaling; it applies specifically to elapsed-time observables.

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/PairKernelScaleCovariantObservablesS20.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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