Foundation Pair Kernel Weyl Event Length Non Identifiability Dimensioned Weyl Ev

The scale wall

A physical length is a number with a unit attached. The declaration dimensionedWeylEventRadius_scale27 concerns a structure that carries a radius this way: it pairs a dimensionless number with a positive length unit, and the reported radius is their product. The theorem states a simple scaling law: if you multiply the length unit by a positive factor, the reported radius multiplies by that same factor. This is a formal, machine-checked fact in the framework's library of theorems.

The deeper point is what this scaling law implies. The framework's recognition data, the discrete record of events, includes the center of an event and its dimensionless radius, but no unit. The theorem presentWeylRecognitionData_does_not_select_dimensionedRadius27 proves that two carriers with the same recognition data can report different physical radii. A countermodel theorem constructs such a pair explicitly. This is a no-go result: the present data cannot single out a length scale.

In Recognition Science, this is a precise boundary. The framework proves that a cost function must take a specific form, and from that derives constants and dimensions. But this theorem shows that a physical length is not among the things the present data can force. The surviving theory must add a scale-bearing carrier whose length response is part of the object from the start. The theorem does not claim this is impossible; it claims the current interface cannot do it.

This result is a theorem in the formal sense: it is proved in the machine-checked library with no gaps and no new axioms. It is also a theorem with a precise scope. It is not a proof that no future Recognition construction can derive length. It is a statement about the present data interface, and it names exactly what that interface lacks.

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

/-- 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

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

What this page does not claim

The theorem does not prove that no future Recognition construction can derive length. The theorem does not establish that the framework's derived constants are physical lengths. The theorem does not claim that the scaling law itself is a physical principle.

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.

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