Foundation Pair Kernel Scale Bearing Self Dual Posting Law Scale Two Does Not Pr

The unique scale

In Recognition Science, a posting is a discrete recorded event, the framework's basic unit of reality, and its law is a rule that fixes how such an event's cost and extent are recorded. The declaration scale_two_does_not_preserve_completePostingLaw27 is a result: if you take a posting event and double the scale of its recorded length, the complete record of that event changes, so the law does not survive the rescaling. The only positive scale that preserves the complete record is 1, the identity, as the companion theorem positiveScalePreservesCompletePostingLaw27_iff establishes.

This is a statement about the framework's own units, not about meters or seconds. The law's native extent is measured in a framework voxel standard, and the result says that standard is fixed: no external conversion to meters can be derived from the law itself. The result is a uniqueness property: among all positive rescalings, only the trivial one keeps the complete observation of a posting event unchanged. The result is verified in the machine-checked library of formal theorems, with no unproved assumptions and no empirical constants.

The result does not claim that the framework's previous recognition premises forced this law into existence. The law is a foundational model, a definitional choice authorized in 2026, and the properties derived from it, including this uniqueness of scale, are results. It also does not weaken the framework's earlier result that length is non-identifiable at the present interface; that result remains intact. What the declaration adds is a sharp constraint: within this law, the native length unit is not a free parameter.

/-- 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 theorem does not claim that the framework's previous recognition premises forced this law into existence. The theorem does not claim that length becomes identifiable in meters or any external unit. The theorem does not claim that the framework's earlier length non-identifiability result is weakened.

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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