Foundation Pair Kernel Scale Bearing Self Dual Posting Law Old Free Length Decoy

The decoy that preserves

A ledger, a discrete record of recognition events, assigns each event a position and an extent. The Recognition Science framework models a posting event as a point in a three-dimensional space, and it asks how the ledger's data behaves when that extent is rescaled. The declaration old_free_length_decoy_preserves_present_data27 is a theorem in the framework's machine-checked library of formal theorems. It states that when a posting event's length scale is doubled, the observable recognition data at the present moment is unchanged. The name "decoy" is exact: the old, free length is a decoy because it has no observable effect on the present record.

The theorem is a formal consequence of a broader principle proved in the same library: any positive rescaling of the length scale preserves the present recognition data if and only if the scale factor is exactly one. Doubling the scale, therefore, is not a symmetry of the full law; it is a symmetry of the present data alone. The declaration isolates that fact. It does not claim that the doubled length is physically real or that the full posting law is invariant under it. The full law, which includes the cost of recognition and the field action, is not preserved by a factor of two, and the library proves that separately.

The theorem's content is best understood as a statement about what the ledger cannot see. The present recognition data, the record a hypothetical observer can read at a single moment, does not encode the absolute length scale. A length and its double leave the same present trace. The framework calls this the present-interface length non-identifiability, and the declaration is a precise instance of it. The old free length, the scale that existed before the law was fixed, is not recoverable from the present data. The theorem does not say that length is meaningless; it says that the present data alone cannot distinguish one scale from another.

In Recognition Science, this result is a deliberate boundary on what the framework claims. The module that contains the theorem states plainly that the physical selection of the posting law is a foundational model, not a derived consequence. The theorem itself is proved from that model, and it does not weaken the non-identifiability result. The declaration is a theorem, not a model choice, and it is proved without any new axioms or empirical constants. What it establishes is a sharp fact about the ledger's present interface: the present data is blind to the old free length. What it does not establish is any claim that the old length is real, that the full law is scale-invariant, or that the framework derives the meter from first principles.

/-- The known-wrong old interface still admits the rescaling that the new
complete interface rejects. -/
theorem old_free_length_decoy_preserves_present_data27
    {event : RealizedPostingEvent3 3}
    (carrier : RealizedPostingWeylLengthCarrier27 event) :
    SamePresentWeylRecognitionData27 carrier
      (scaleRealizedPostingWeylLengthCarrier27
        2 (by norm_num) carrier) :=
  scale_preserves_presentWeylRecognitionData27
    2 (by norm_num) carrier

/-- Anti-rescaling certificate: preserving the complete new law forces the
identity rechart. -/
theorem positiveScalePreservesCompletePostingLaw27_iff
    (event : RealizedPostingEvent3 3)
    (ψ : WeylEventAmplitude27)
    (scale : ℝ) :
    PositiveScalePreservesCompletePostingLaw27
        event ψ scale ↔
      scale = 1 := by
  constructor
  · rintro ⟨_, hpreserves⟩
    have hlength := congrArg
      (fun observation : WeylEventSite27 × ℝ × ℝ =>
        observation.2.2) hpreserves
    simpa [rechartCompletePostingObservation27,
      completeScaleBearingPostingObservation27,
      nativePostingExtentResponse27,
      nativePostingExtentQuantum27,
      unitPostingDilation,
      Constants.RSNativeUnits.voxel] using hlength
  · intro hscale
    subst scale
    simp [PositiveScalePreservesCompletePostingLaw27,
      rechartCompletePostingObservation27]

theorem scale_two_does_not_preserve_completePostingLaw27
    (event : RealizedPostingEvent3 3)
    (ψ : WeylEventAmplitude27) :
    ¬ PositiveScalePreservesCompletePostingLaw27
      event ψ 2 := by
  rw [positiveScalePreservesCompletePostingLaw27_iff]
  norm_num

/-- FOUNDATIONAL MODEL: one stable dilation cost plus the one-weight Weyl
sector.  No numerical coefficient or length field is accepted. -/
def scaleBearingSelfDualPostingAction27
    (event : RealizedPostingEvent3 3)
    (dilation : PostingDilationCoord)
    (ψ : WeylEventAmplitude27) : ℝ :=
  Cost.Jcost dilation.1 + selfDualPostingWeylCost27 event ψ

What this page does not claim

The old free length is physically real or recoverable from the present data. The full posting law is invariant under a scale factor of two. The framework derives the meter from first principles without an external conversion.

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

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