Foundation Pair Kernel Weyl Event Length Non Identifiability Present Weyl Recogn

The unselected radius

A physical radius usually comes with a unit attached: 3 meters, not just 3. The Recognition Science framework asks whether the data it records at a single recognition event can pin down such a dimensioned radius. The answer, proved in the framework's machine-checked library of formal theorems, is no. The theorem presentWeylRecognitionData_does_not_select_dimensionedRadius27 establishes that the recognition data available at one realized posting cannot select a unique physical radius.

To see why, consider what the framework calls a length carrier: a record that pairs a dimensionless radius with a positive physical event-length unit, such as meters. The framework's library defines the dimensioned radius as the product of these two numbers. It then asks whether two different length carriers that share the same recognition data must report the same dimensioned radius. The theorem answers no. A positive rescaling of only the event-length unit, say multiplying it by 2, changes the dimensioned radius while leaving the dimensionless radius and the recognized center untouched. Two carriers can therefore agree on all present recognition data yet disagree on the physical radius.

The theorem is a no-go result for the present-data interface, not a claim that length can never be derived. It says that the surviving theory must add a scale-bearing physical carrier whose length response is part of the object from birth. The library states this plainly: no future construction can extract length from the current data alone. The proof is axiom-clean, with no sorry and no new axioms, and it introduces no empirical constant.

What the theorem does not claim is equally important. It does not prove that no future Recognition Science construction can derive length from richer data. It does not say that length is unreal or that all radii are meaningless. It establishes only that the specific data recorded at a single present posting, the center and the dimensionless radius, are insufficient to select a dimensioned radius. The constructive countermodel theorem in the same library makes this concrete: it exhibits two carriers with identical recognition data and different dimensioned radii.

/-- Two positive length units remain compatible with exactly the same present
Weyl Recognition data and yield different physical radii. -/
theorem presentWeylRecognitionData_does_not_select_dimensionedRadius27
    {event : RealizedPostingEvent3 3}
    (carrier : RealizedPostingWeylLengthCarrier27 event) :
    ¬ PresentWeylRecognitionDataSelectsDimensionedRadius27
      carrier := by
  intro hselect
  have hsame :=
    scale_preserves_presentWeylRecognitionData27
      2 (by norm_num) carrier
  have heq := hselect
    (scaleRealizedPostingWeylLengthCarrier27
      2 (by norm_num) carrier) hsame
  rw [dimensionedWeylEventRadius_scale27] at heq
  have hradius_pos :
      0 < dimensionedWeylEventRadius27 carrier :=
    mul_pos carrier.eventLengthUnit_pos
      carrier.dimensionlessRadius_pos
  linarith

theorem dimensionedWeylEventRadius_scale27
    {event : RealizedPostingEvent3 3}
    (scale : ℝ)
    (hscale : 0 < scale)
    (carrier : RealizedPostingWeylLengthCarrier27 event) :
    dimensionedWeylEventRadius27
        (scaleRealizedPostingWeylLengthCarrier27
          scale hscale carrier) =
      scale * dimensionedWeylEventRadius27 carrier := by
  simp [dimensionedWeylEventRadius27,
    scaleRealizedPostingWeylLengthCarrier27]
  ring

theorem scale_preserves_presentWeylRecognitionData27
    {event : RealizedPostingEvent3 3}
    (scale : ℝ)
    (hscale : 0 < scale)
    (carrier : RealizedPostingWeylLengthCarrier27 event) :
    SamePresentWeylRecognitionData27 carrier
      (scaleRealizedPostingWeylLengthCarrier27
        scale hscale carrier) := by
  exact ⟨rfl, rfl⟩

/-- Constructive countermodel pair for external consumers. -/
theorem presentWeylLengthScale_countermodel27
    {event : RealizedPostingEvent3 3}
    (carrier : RealizedPostingWeylLengthCarrier27 event) :
    ∃ other : RealizedPostingWeylLengthCarrier27 event,
      SamePresentWeylRecognitionData27 carrier other ∧
        dimensionedWeylEventRadius27 carrier ≠
          dimensionedWeylEventRadius27 other := by
  refine
    ⟨scaleRealizedPostingWeylLengthCarrier27
        2 (by norm_num) carrier,
      scale_preserves_presentWeylRecognitionData27
        2 (by norm_num) carrier, ?_⟩
  intro heq
  rw [dimensionedWeylEventRadius_scale27] at heq
  have hradius_pos :
      0 < dimensionedWeylEventRadius27 carrier :=
    mul_pos carrier.eventLengthUnit_pos
      carrier.dimensionlessRadius_pos
  linarith

What this page does not claim

The theorem does not prove that no future Recognition Science construction can derive length from richer data. The theorem does not claim that length is unreal or that all radii are meaningless. The theorem does not itself introduce a physical unit of length or an empirical constant.

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

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