Encyclopedia Foundation Foundation Pair Kernel Weyl Event Length Non Identifiability Realized Posting We

ARTICLE 3 claims 3 theorems

Foundation Pair Kernel Weyl Event Length Non Identifiability Realized Posting We

A machine-checked theorem shows that a recorded event's recognition data cannot pin down its physical size, because the unit of length can be rescaled freely without changing anything else.

Verify this page

The length carrier

A recognition event, a discrete record of something happening, can carry a number for its size. The framework's library, a machine-checked collection of formal theorems, defines a carrier for that size as two positive numbers: a dimensionless radius and a unit of length. Multiplying the unit by any positive factor changes the reported physical radius but leaves the underlying event, its center, and the dimensionless radius untouched.

The theorem proves this rescaling is invisible to the recognition data. Two carriers with the same center and the same dimensionless radius are indistinguishable, yet they can report different physical radii. The formal statement is that the present data do not select a dimensioned radius, and a constructive countermodel exhibits two such carriers explicitly. This is a no-go result for the current interface, not a proof that no future construction can derive length.

In Recognition Science, the framework models this as a scale wall. The surviving theory must add a scale-bearing physical carrier whose length response is part of the object from birth. The theorem is axiom-clean, with no sorry and no empirical constant, but it does not claim that length itself is impossible, only that this particular present-data interface cannot produce it.

THEOREM scale_preserves_presentWeylRecognitionData27 · dimensionedWeylEventRadius_scale27 · IndisputableMonolith/Foundation/PairKernelWeylEventLengthNonIdentifiability.lean
scale_preserves_presentWeylRecognitionData27 · IndisputableMonolith/Foundation/PairKernelWeylEventLengthNonIdentifiability.lean:69 
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⟩
dimensionedWeylEventRadius_scale27 · IndisputableMonolith/Foundation/PairKernelWeylEventLengthNonIdentifiability.lean:79 
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 presentWeylRecognitionData_does_not_select_dimensionedRadius27 · IndisputableMonolith/Foundation/PairKernelWeylEventLengthNonIdentifiability.lean
presentWeylRecognitionData_does_not_select_dimensionedRadius27 · IndisputableMonolith/Foundation/PairKernelWeylEventLengthNonIdentifiability.lean:102 
/-- 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 presentWeylLengthScale_countermodel27 · IndisputableMonolith/Foundation/PairKernelWeylEventLengthNonIdentifiability.lean
presentWeylLengthScale_countermodel27 · IndisputableMonolith/Foundation/PairKernelWeylEventLengthNonIdentifiability.lean:123 
/-- 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 construction can derive length. The theorem does not claim that the dimensionless radius or the event center is itself a physical 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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red:

MACHINE LAYER · GROUNDED CLAIM TABLE · CLICK TO EXPAND