Foundation Pair Kernel Scale Breaking Source Residual Source Action Duality Alon

What the duality alone permits

In the Recognition Science framework, a ledger, a discrete record of recognition events, assigns to each primitive posting two numbers: an action, the cost of the event, and a source magnitude, the strength of the event's source. A natural symmetry principle, called source-action duality, requires that the source magnitude of a posting be the multiplicative inverse of its action, measured in the framework's native unit. The question is whether this principle alone forces a unique value for the source magnitude, or whether it leaves a choice.

The machine-checked theorem sourceActionDuality_alone_admits_distinct_banked_magnitudes proves that duality alone leaves exactly two possible values. It constructs two distinct models of the framework's premises, each satisfying the duality principle, in which a realized primitive posting carries a different source magnitude: Candidate A pairs a unit source with a unit action, while Candidate B pairs a reciprocal native source with the native action quantum. The theorem shows that both assignments satisfy the current recognition source premises and the duality condition, and that the two magnitudes are distinct. The framework's library, a machine-checked collection of formal theorems, records this as a proved fact, not a hypothesis.

This result is a precise statement of what the framework does not yet know. It shows that the principle of source-action duality, while natural, is not sufficient to determine the physical scale of the source. The choice between the two candidates is a separate scale-breaking statement, one that must be supplied by an additional physical law. The module names that law explicitly as NativeActionDualSourceLaw, a hypothesis stating that one physical posting carries the native action quantum and that its source coordinate is the multiplicative dual. This law, if adopted, selects Candidate B, and the theorem nativeActionDualSourceLaw_discriminates_banked_candidates proves that it rejects Candidate A.

The declaration does not claim that the framework's current premises derive this law. It proves the opposite: currentPremises_do_not_force_nativeActionDualSourceLaw shows that Candidate A satisfies every current premise and fails the proposed law. The theorem also does not claim that the two missing physical statements are dependent; in fact, nativePostingAction_alone_admits_distinct_source_maps shows that fixing the posting action alone still leaves the source map free. The value of the theorem is its precision: it banks the discrimination power of the proposed law, its independence from the current premises, and the exact condition under which the source magnitude becomes Candidate B.

/-- Source-action duality alone admits both banked source magnitudes. Candidate
A pairs unit source with unit action; Candidate B pairs reciprocal native
source with native action. The physical action normalization is therefore a
separate scale-breaking statement. -/
theorem sourceActionDuality_alone_admits_distinct_banked_magnitudes :
    PostingSourceIsActionDual3
        (@unitPostingActionCountermodel3 3 inferInstance)
        (@unitSourceMagnitudeCountermodel3 3 inferInstance) ∧
      PostingSourceIsActionDual3
        (@nativePostingActionCandidate3 3 inferInstance)
        (@nativeDualSourceMagnitudeCandidate3 3 inferInstance) ∧
      ∃ event : PostingPair3 3,
        event ∈ realizedPrimitivePostingPairs3 3 ∧
          unitSourceMagnitudeCountermodel3 event =
            candidateA_sourceMagnitudeExpr.eval ∧
          nativeDualSourceMagnitudeCandidate3 event =
            candidateB_sourceMagnitudeExpr.eval ∧
          unitSourceMagnitudeCountermodel3 event ≠
            nativeDualSourceMagnitudeCandidate3 event := by
  refine
    ⟨unitAssignments_satisfy_sourceActionDuality,
      nativeCandidates_source_is_actionDual, ?_⟩
  obtain ⟨event, hevent, _hcert⟩ :=
    canonicalGeneratorSource_consumer_exists
      (N := 3) (by norm_num) (Equiv.refl (Fin 3))
  have hA :
      unitSourceMagnitudeCountermodel3 event =
        candidateA_sourceMagnitudeExpr.eval := by
    change (1 : ℝ) = candidateA_sourceMagnitudeExpr.eval
    exact candidateA_sourceMagnitude_eq_one.symm
  have hB :
      nativeDualSourceMagnitudeCandidate3 event =
        candidateB_sourceMagnitudeExpr.eval := by
    rfl
  refine ⟨event, hevent, hA, hB, ?_⟩
  intro heq
  exact candidates_select_distinct_magnitudes
    (hA.symm.trans (heq.trans hB))

/-- The new law is not hidden in the old package: Candidate A satisfies every
current premise and fails the native-action dual law. -/
theorem currentPremises_do_not_force_nativeActionDualSourceLaw :
    ¬ (∀ sourceMagnitude : ℝ,
      CurrentRecognitionSourcePremises sourceMagnitude →
        NativeActionDualSourceLaw sourceMagnitude) := by
  intro hforce
  exact nativeActionDualSourceLaw_rejects_candidateA
    (hforce _
      candidateA_satisfies_currentRecognitionSourcePremises)

theorem nativeActionDualSourceLaw_discriminates_banked_candidates :
    NativeActionDualSourceLaw
        candidateB_sourceMagnitudeExpr.eval ∧
      ¬ NativeActionDualSourceLaw
        candidateA_sourceMagnitudeExpr.eval :=
  ⟨candidateB_satisfies_nativeActionDualSourceLaw,
    nativeActionDualSourceLaw_rejects_candidateA⟩

What this page does not claim

The declaration does not claim that the framework's current premises derive the native-action dual law. It does not claim that the two missing physical statements are dependent; fixing the posting action alone still leaves the source map free. It does not claim that the source-action duality principle is physically realized in nature, only that it is a consistent premise within the framework.

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/PairKernelScaleBreakingSourceResidual.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.

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