Foundation Pair Kernel Scale Breaking Source Residual Native Action Dual Source

The two candidate sources

In the Recognition Science framework, a ledger is a discrete record of recognition events, and each event carries a cost, a number measuring how expensive that recognition is. The framework's current premises, which describe how these costs behave, leave open a choice: a primitive posting, a single entry in the ledger, could have one of two distinct source magnitudes. The declaration nativeActionDualSourceLaw_discriminates_banked_candidates states a proposed law that would settle this choice.

The proposed law, called the native-action dual law, says two things together. First, one realized primitive posting carries the native action quantum, the framework's basic unit of action. Second, that posting's source coordinate is the multiplicative action dual, meaning the source magnitude is the reciprocal of the action in the Euler-J unit. The law is stated as a hypothesis, not as a consequence of the current premises. The theorem proves that if this law holds, the source magnitude must be Candidate B, and that Candidate A, the other possibility, fails the law.

This discrimination is not trivial. The framework's library, a machine-checked collection of formal theorems, proves that each candidate satisfies the current premises. Candidate A pairs a unit source with unit action; Candidate B pairs a reciprocal native source with native action. The current premises alone do not force either choice, and the new law is not hidden in the old package. The law is a separate, scale-breaking statement: adding it to the current premises leaves exactly one model, not two.

What the declaration does not claim is as important as what it proves. It does not derive the native-action dual law from the current premises; the theorem currentPremises_do_not_force_nativeActionDualSourceLaw explicitly proves that the premises do not imply it. The law is an additional physical assumption, not a theorem forced by the framework. The declaration also does not claim that the source magnitude is physically measured or observed; it only establishes the logical consequence of assuming the law.

In plain terms, the declaration banks a conditional result: if the native-action dual law is true, then the source magnitude is Candidate B and not Candidate A. It does not say the law is true, nor that the framework has derived it. The distinction is a matter of proof hygiene: the framework keeps its theorems and its hypotheses separate, and this declaration marks the line.

/-- HYPOTHESIS: one physical posting has native action and its positive source
coordinate is the multiplicative dual measured in the Euler-J action unit.
The statement is target-blind and contains no continuum readout. -/
def NativeActionDualSourceLaw (sourceMagnitude : ℝ) : Prop :=
  0 < sourceMagnitude ∧
    sourceMagnitude * Constants.hbar = eulerJActionUnit

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

/-- 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)

/-- The named law is strong enough to satisfy the formal scale-breaking
criterion: current premises plus this extra law have exactly one model. -/
theorem nativeActionDualSourceLaw_is_scaleBreaking :
    IsScaleBreakingAttachmentLaw NativeActionDualSourceLaw := by
  refine
    ⟨candidateB_sourceMagnitudeExpr.eval,
      ⟨candidateB_satisfies_currentRecognitionSourcePremises,
        candidateB_satisfies_nativeActionDualSourceLaw⟩,
      ?_⟩
  intro sourceMagnitude hsource
  exact
    (nativeActionDualSourceLaw_iff_candidateB sourceMagnitude).1
      hsource.2

What this page does not claim

The native-action dual law is not derived from the current Recognition premises. The declaration does not assert that the law is physically true or measured. The declaration does not claim that Candidate B is the only possible source magnitude without the law.

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)

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