Foundation Pair Kernel Scale Bearing Self Dual Posting Law Positive Scale Preser

The unit scale

In the Recognition Science framework, a posting is a discrete record of a recognition event, and a complete posting law is the full set of data that describes such an event: its location, its cost, and its native extent. The framework's library of machine-checked formal theorems proves a striking rigidity result about these laws: the only positive scale that preserves a complete posting law is the unit scale. In symbols, the theorem states that for any event and amplitude, the property of a positive scale preserving the complete posting law holds if and only if that scale equals 1.

This result is a theorem, not a definitional choice. It follows from the framework's core cost function, which assigns a cost to every possible dilation of an event. The cost function has a unique minimum at the unit dilation, and any departure from that unit increases the cost. The complete posting law records this cost, so any change of scale that leaves the law unchanged must leave the cost unchanged, which forces the scale to be the unit. The theorem's proof is fully formalized in the framework's library, with no unproved assumptions beyond the standard axioms of the underlying type theory.

The theorem has a direct consequence for the framework's native units. The native extent of a posting event, its length in the framework's own voxel standard, is defined at the unit dilation. The theorem guarantees that this extent is invariant under refinement: computing it at any positive integer refinement level gives the same value. This invariance is what makes the native extent a well-defined physical quantity, a single number that does not depend on how finely one chooses to sample the event.

The framework does not claim that this theorem selects the unit scale from first principles. The choice to define the posting law with the specific cost function that makes the unit scale special is a foundational model, a definitional choice, not a consequence of the framework's earlier axioms. The theorem proves properties of that model, but it does not prove that the model itself is forced. The framework also does not claim that this result provides a conversion from its native voxel to meters; such an external conversion remains outside the law.

What the theorem does establish is a precise sense in which the framework's native length is unique. Within the model, there is exactly one scale at which a complete posting law can be written down consistently, and that scale is the unit. This rigidity is what allows the framework to speak of a single native extent for an event, and it is a necessary step for any further claim about the framework's physical content.

/-- 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]

/-- Every non-native positive dilation is strictly more costly, pointwise in
the field state. -/
theorem scaleBearingSelfDualPostingAction27_unit_lt
    (event : RealizedPostingEvent3 3)
    (dilation : PostingDilationCoord)
    (hdilation : dilation.1 ≠ 1)
    (ψ : WeylEventAmplitude27) :
    scaleBearingSelfDualPostingAction27
        event unitPostingDilation ψ <
      scaleBearingSelfDualPostingAction27 event dilation ψ := by
  unfold scaleBearingSelfDualPostingAction27
  simp only [unitPostingDilation, Cost.Jcost_unit0, zero_add]
  exact lt_add_of_pos_left _
    (Cost.Jcost_pos_of_ne_one dilation.1 dilation.2 hdilation)

/-- The event extent is independent of the number of phases used to represent
the carrier. -/
theorem nativePostingExtentAtRefinement27_invariant
    (event : RealizedPostingEvent3 3)
    (N M : ℕ) :
    nativePostingExtentAtRefinement27 event N =
      nativePostingExtentAtRefinement27 event M := by
  rfl

What this page does not claim

The theorem does not prove that the unit scale is forced by the framework's earlier axioms. The theorem does not provide a conversion from the framework's native voxel to meters. The theorem does not apply to scales that are zero or negative.

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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