Foundation Pair Kernel Physical Source Scale No Go Current Recognition Source Pr

The scale obstruction

The declaration currentRecognitionSourcePremises_do_not_select_unique_scale is a theorem in the framework's machine-checked library of formal theorems. It states that the current source-side premises do not determine a unique physical scale. The premises are: a positive scale free of π, a primitive elementary posting rule, a symmetric pair action, and stationarity at that scale. The theorem proves there exist two distinct scales, each satisfying all of these premises, so the package as it stands is a torsor, not a selector.

The proof is concrete. Two candidate magnitudes are exhibited: one is the unit scale, the other is the reciprocal native-action quantum. Both are positive, π-free, and satisfy the action, posting, and stationarity package. The theorem then shows these two candidates are distinct. If the premises selected a unique scale, both would have to equal that scale, which would force the two candidates to be equal. Since they are not, uniqueness fails. This is a no-go result: it does not say no scale exists, only that the current premises alone cannot pick one.

The consequence is that any successful attachment law must add scale-breaking content. The framework defines a genuine attachment law as one that, together with the current premises, selects exactly one magnitude. The theorem scaleBreakingAttachmentLaw_rejects_current_candidate proves such a law must reject at least one of the two currently admitted candidates. A further theorem, currentPremisesCannotForceScaleBreakingAttachment, shows that no law implied by the current primitive package can be scale-breaking. The residual is therefore a new physical premise, not a consequence of what is already assumed.

This is a precise statement about what the framework does not yet claim. It does not claim that no unique physical source scale exists, only that the current premises do not force one. It does not claim that the two candidates are physical; they are admitted by the package, not selected by it. It does not claim that the scale problem is unsolvable, only that a solution must add content beyond the current package. The theorem is a boundary marker: it says where the current theory stops and where new physics must begin.

/-- 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⟩

/-- 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⟩

/-- 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

/-- 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 theorem does not claim that no unique physical source scale exists. The theorem does not claim that the two candidates are physical. The theorem does not claim that the scale problem is unsolvable.

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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