Foundation Pair Kernel Gap2a Integral Lattice Dual Residual Candidate B Satisfie

What the dual datum does not decide

In the Recognition Science framework, a ledger (a discrete record of posting events) works with a rank-one integer lattice. The question at hand is whether that lattice's canonical integer dual, the set of integer-valued additive functions from the integers to themselves, can pin down a unique positive real source scale for the pair-kernel equation. The declaration candidateB_satisfies_currentPremisesWithIntegralDual answers in the negative: it proves that one of the two banked candidate source magnitudes, candidate B, satisfies the current premises even after adjoining the integral dual datum.

The proof runs through a general theorem: any positive real source scale that is pi-free (not a rational multiple of pi) satisfies the enriched premises. Candidate B's magnitude, expressed as the native action quantum inverse, is positive and pi-free, so it qualifies. The theorem also shows the same for candidate A, and a companion result proves the two candidates select distinct magnitudes. Together these force the conclusion that the integral dual does not break the real source torsor: both candidates remain admissible, so no unique scale emerges.

What the declaration does not claim is just as important. It does not assert that candidate B is the correct source scale, only that it survives the added integrality and primitivity conditions. It does not establish that the integral dual forces a zero-defect dual identification, nor that it forces the native action dual source law. The machine-checked library of formal theorems explicitly records that identifying the unique positive character with a multiplicative action dual would require an additional physical identification law, one that must not be smuggled in as source := 1/action.

The practical upshot: the integral dual datum, despite its canonical uniqueness, does not select a unique real source magnitude. The framework's current premises plus the Hom-enriched package still admit both banked candidates. Any further progress toward a unique source scale must come from a new physical identification law, not from the integral dual alone.

theorem candidateB_satisfies_currentPremisesWithIntegralDual :
    CurrentPremisesWithIntegralDual
      candidateB_sourceMagnitudeExpr.eval := by
  apply currentPremisesWithIntegralDual_of_positive_piFree
  · change 0 < nativeActionQuantumInv
    exact nativeActionQuantumInv_pos
  · exact candidateB_sourceMagnitude_piFree

/-- Integrality, primitivity, and unique positive Hom character do not attach a
unique real source magnitude. -/
theorem currentPremisesWithIntegralDual_do_not_select_unique_scale :
    ¬ ∃! sourceScale : ℝ,
      CurrentPremisesWithIntegralDual sourceScale := by
  intro hunique
  rcases hunique with ⟨selected, _hselected, honly⟩
  have hA :
      candidateA_sourceMagnitudeExpr.eval = selected :=
    honly _ candidateA_satisfies_currentPremisesWithIntegralDual
  have hB :
      candidateB_sourceMagnitudeExpr.eval = selected :=
    honly _ candidateB_satisfies_currentPremisesWithIntegralDual
  exact candidates_select_distinct_magnitudes (hA.trans hB.symm)

/-- Zero-defect dual identification is still not forced by Hom uniqueness. -/
theorem integralDual_does_not_force_zeroDefect :
    ¬ (∀ coupling : ℝ,
      NamedPremises coupling →
        IsPrimitivePositiveCharacter primitivePositiveCharacter →
          ZeroDefectDualIdentification coupling) := by
  intro hforce
  exact zeroDefectDualIdentification_excludes_two
    (hforce 2 (namedPremises_all_couplings 2)
      primitivePositiveCharacter_isPrimitivePositive)

/-- Identifying the unique positive character with the multiplicative action
dual is an additional premise, not a theorem of the defined Hom-enriched
package. That package does not force `NativeActionDualSourceLaw`. -/
theorem integralDual_does_not_force_nativeActionDualSourceLaw :
    ¬ (∀ sourceScale : ℝ,
      CurrentPremisesWithIntegralDual sourceScale →
        PairKernelScaleBreakingSourceResidual.NativeActionDualSourceLaw
          sourceScale) := by
  intro hforce
  exact
    PairKernelScaleBreakingSourceResidual.nativeActionDualSourceLaw_rejects_candidateA
      (hforce _
        candidateA_satisfies_currentPremisesWithIntegralDual)

What this page does not claim

Candidate B is the correct source scale. The integral dual datum classifies every possible integral-dual formalization or real transport. The Hom-enriched package forces any specific HomToRealSourceLaw.

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/PairKernelGap2aIntegralLatticeDualResidual.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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