Foundation Pair Kernel Physical Source Scale No Go Scale Breaking Attachment Law

The source-scale obstruction

The declaration scaleBreakingAttachmentLaw_rejects_current_candidate establishes a precise obstruction in the Recognition Science framework. The framework models physical structure through a ledger, a discrete record of recognition events, and seeks to derive fundamental constants from a small set of axioms. The theorem states that any law which successfully attaches a unique physical source scale, meaning a law that selects exactly one magnitude from the currently admitted possibilities, must reject at least one of the two specific candidates that the present primitive package allows.

The two candidates are distinct positive real numbers, both free of any factor of pi, and both satisfy every condition in the current package: positivity, pi-freeness, primitive elementary posting, symmetric pair action, and stationarity at that scale. The theorem currentRecognitionSourcePremises_admit_distinct_candidates proves these two candidates exist and have different magnitudes. A companion theorem, currentRecognitionSourcePremises_do_not_select_unique_scale, proves that the current premises alone cannot single out one of them. This is the source-side scale torsor: the framework's existing laws are compatible with more than one possible source magnitude.

The declaration's content is conditional. It does not itself provide such a law, nor does it name which candidate should be rejected. Instead, it forces the structure of any future solution: a genuine attachment law must add new scale-breaking content that goes beyond the current premises. The theorem currentPremisesCannotForceScaleBreakingAttachment makes this explicit, proving that no law implied by the current package can serve as a scale-breaking attachment. The residual must come from a new physical premise, not from a consequence of what is already established.

What the declaration does not claim is equally important. It does not assert that the framework is inconsistent, nor that no attachment law can exist. It does not claim that either candidate is the correct physical source scale, nor that the two candidates exhaust all possibilities. It does not derive any numerical value for a source scale. The theorem is a no-go result with a precise scope: it maps the boundary of what the current primitive package can achieve, and it identifies the exact form that any successful extension must take. Within the framework, this is a constructive step, not a dead end, because it tells the framework's developers where the next axiom must be placed.

/-- Every genuine attachment law must reject at least one of the two source
scales currently admitted by action, posting, and stationarity. -/
theorem scaleBreakingAttachmentLaw_rejects_current_candidate
    (law : ℝ → Prop)
    (hbreaking : IsScaleBreakingAttachmentLaw law) :
    ¬ law candidateA_sourceMagnitudeExpr.eval ∨
      ¬ law candidateB_sourceMagnitudeExpr.eval := by
  by_cases hA : law candidateA_sourceMagnitudeExpr.eval
  · right
    intro hB
    rcases hbreaking with ⟨selected, _hselected, honly⟩
    have hAselected :
        candidateA_sourceMagnitudeExpr.eval = selected :=
      honly _
        ⟨candidateA_satisfies_currentRecognitionSourcePremises, hA⟩
    have hBselected :
        candidateB_sourceMagnitudeExpr.eval = selected :=
      honly _
        ⟨candidateB_satisfies_currentRecognitionSourcePremises, hB⟩
    exact
      candidates_select_distinct_magnitudes
        (hAselected.trans hBselected.symm)
  · exact Or.inl hA

/-- The present primitive package has two distinct positive pi-free source
models. This is the source-side scale torsor stated on the banked candidates. -/
theorem currentRecognitionSourcePremises_admit_distinct_candidates :
    ∃ sourceScale₁ sourceScale₂ : ℝ,
      sourceScale₁ ≠ sourceScale₂ ∧
        CurrentRecognitionSourcePremises sourceScale₁ ∧
        CurrentRecognitionSourcePremises sourceScale₂ :=
  ⟨candidateA_sourceMagnitudeExpr.eval,
    candidateB_sourceMagnitudeExpr.eval,
    candidates_select_distinct_magnitudes,
    candidateA_satisfies_currentRecognitionSourcePremises,
    candidateB_satisfies_currentRecognitionSourcePremises⟩

/-- Action, posting, positivity, pi-freeness, and stationarity do not attach a
unique source magnitude. -/
theorem currentRecognitionSourcePremises_do_not_select_unique_scale :
    ¬ ∃! sourceScale : ℝ,
      CurrentRecognitionSourcePremises sourceScale := by
  intro hunique
  rcases hunique with ⟨selected, _hselected, honly⟩
  have hA :
      candidateA_sourceMagnitudeExpr.eval = selected :=
    honly _ candidateA_satisfies_currentRecognitionSourcePremises
  have hB :
      candidateB_sourceMagnitudeExpr.eval = selected :=
    honly _ candidateB_satisfies_currentRecognitionSourcePremises
  exact candidates_select_distinct_magnitudes (hA.trans hB.symm)

/-- No law implied by the current primitive package can be a scale-breaking
attachment. The remaining residual must add a new physical premise. -/
theorem currentPremisesCannotForceScaleBreakingAttachment
    (law : ℝ → Prop)
    (hforced :
      ∀ sourceScale : ℝ,
        CurrentRecognitionSourcePremises sourceScale →
          law sourceScale) :
    ¬ IsScaleBreakingAttachmentLaw law := by
  intro hbreaking
  rcases
      scaleBreakingAttachmentLaw_rejects_current_candidate
        law hbreaking with hA | hB
  · exact hA
      (hforced _
        candidateA_satisfies_currentRecognitionSourcePremises)
  · exact hB
      (hforced _
        candidateB_satisfies_currentRecognitionSourcePremises)

What this page does not claim

The declaration does not provide a specific scale-breaking attachment law. The declaration does not assert that either candidate is the correct physical source scale. The declaration does not claim the framework is inconsistent or that no attachment law can exist.

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/PairKernelPhysicalSourceScaleNoGo.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

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