Foundation Pair Kernel Weyl Event Length Non Identifiability Scale Preserves Pre

The scale wall

The declaration scale_preserves_presentWeylRecognitionData27 is a machine-checked theorem in the framework's library of formal theorems. It states a precise fact: if you take a recognized event, its attached center, and a dimensionless radius, and you multiply only the physical length unit by a positive number, then all the present recognition data stay exactly the same. The realized event, its center, and the dimensionless radius are unchanged. Only the dimensioned radius, the product of the unit and the dimensionless number, changes by that same factor.

The theorem is a no-go result for the stated present-data interface. It proves that the data available at this stage, a center and a dimensionless radius, cannot select a unique physical radius. Two different positive length units remain compatible with exactly the same recognition data and yield different physical radii. The framework's library proves this by constructing a countermodel pair: two carriers with identical present data but different dimensioned radii. This is not a metatheoretic proof that no future Recognition construction can derive length. The surviving theory must add a scale-bearing physical carrier whose length response is part of the object from birth.

The proof is axiom-clean: no sorry, no new axiom, no empirical constant. The theorem and its companion dimensionedWeylEventRadius_scale27, which states the scaling behavior of the radius, are both proved in the machine-checked library. The result does not claim that length is impossible in the framework. It claims only that the present center-attached, dimensionless Weyl data cannot force a physical radius. A future construction must carry its length response from the start.

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⟩

/-- 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

/-- 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

Not a proof that no future Recognition construction can derive length. Not a claim that the framework's present data are physically meaningless; they are merely incomplete for length. Not a statement about any empirical constant or measured length.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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