Foundation Pair Kernel Gap2a Integral Lattice Dual Residual Integral Dual Does N

A limit on what the integer dual can force

In Recognition Science, the foundational structure begins with a rank-one integral posting lattice, a discrete record of events indexed by the integers. This lattice has a canonical integer dual, the set of additive functions from the integers to themselves. Within that dual sits a unique positive primitive character: the identity function, which sends the elementary posting to the integer 1. The framework's machine-checked library of formal theorems proves this character is unique among all additive characters that send the generator to 1.

The question the library then asks is whether this integer-valued uniqueness can be transported into the real numbers to select a unique positive source scale for the pair-kernel equation, the framework's core cost equation. The answer is a proved negative: adjoining the integer dual data to the current recognition source premises still admits every positive pi-free coupling, including two distinct banked candidates. The theorem integralDual_does_not_force_nativeActionDualSourceLaw states formally that the Hom-enriched package does not force the NativeActionDualSourceLaw, which would identify the unique positive character with a multiplicative action dual. The integer character evaluates to the integer 1, not to a real source coupling, so it fixes only the lattice generator and leaves the real scale undetermined.

This negative result is a deliberate audit of a gap in the framework, not a failure of it. The library shows that any law forced uniformly by the defined Hom-enriched package cannot be scale-breaking, because both banked candidates satisfy that package. A rescaling countermodel survives after adjoining the integral dual, and the forbidden smuggle form, setting source := 1/action, is not Hom-forced. The framework therefore requires an additional physical identification law to transport the integer character into a unique real cotangent or Green source, and that law is not supplied by the defined package.

What the declaration does not claim is equally precise. It does not classify every possible integral-dual formalization or real transport; it audits only the defined Hom-enriched package. It does not assert that no such identification law can exist, only that this package does not force one. And it does not involve Planck's constant, Real.pi, Constants.G, a Green sum, or an endpoint target. The theorem is a clean boundary marker: it proves a limit of the current premises, leaving the next step as an open target for a future physical identification law.

/-- Uniqueness: any additive character with `χ 1 = 1` is the identity. -/
theorem primitivePositiveCharacter_unique
    (χ : IntegralDual) (hχ : IsPrimitivePositiveCharacter χ) :
    χ = primitivePositiveCharacter := by
  apply AddMonoidHom.ext_int
  simpa [IsPrimitivePositiveCharacter, primitivePositiveCharacter] using hχ

/-- The defined Hom-enriched package still admits both banked positive pi-free
source models. -/
theorem currentPremisesWithIntegralDual_admit_distinct_candidates :
    ∃ sourceScale₁ sourceScale₂ : ℝ,
      sourceScale₁ ≠ sourceScale₂ ∧
        CurrentPremisesWithIntegralDual sourceScale₁ ∧
        CurrentPremisesWithIntegralDual sourceScale₂ :=
  ⟨candidateA_sourceMagnitudeExpr.eval,
    candidateB_sourceMagnitudeExpr.eval,
    candidates_select_distinct_magnitudes,
    candidateA_satisfies_currentPremisesWithIntegralDual,
    candidateB_satisfies_currentPremisesWithIntegralDual⟩

/-- 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)

/-- The unique positive primitive character evaluates the elementary posting to
the integer `1`, not to a real source coupling. -/
theorem primitivePositiveCharacter_eval_primitivePosting :
    (primitivePositiveCharacter : ℤ → ℤ) primitivePosting = 1 := by
  simp [primitivePositiveCharacter, primitivePosting]

/-- Any law forced uniformly by the defined Hom-enriched package cannot be
scale-breaking: both banked candidates satisfy that package. -/
theorem homForcedLaw_cannot_be_scaleBreaking
    (law : ℝ → Prop)
    (hforced :
      ∀ sourceScale : ℝ,
        CurrentPremisesWithIntegralDual sourceScale →
          law sourceScale) :
    ¬ HomToRealSourceLaw law := by
  intro hbreaking
  rcases hbreaking with ⟨selected, ⟨hselPrem, hselLaw⟩, honly⟩
  have hA := hforced _
    candidateA_satisfies_currentPremisesWithIntegralDual
  have hB := hforced _
    candidateB_satisfies_currentPremisesWithIntegralDual
  have hAselected :
      candidateA_sourceMagnitudeExpr.eval = selected :=
    honly _
      ⟨candidateA_satisfies_currentPremisesWithIntegralDual, hA⟩
  have hBselected :
      candidateB_sourceMagnitudeExpr.eval = selected :=
    honly _
      ⟨candidateB_satisfies_currentPremisesWithIntegralDual, hB⟩
  exact
    candidates_select_distinct_magnitudes
      (hAselected.trans hBselected.symm)

/-- 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

This theorem does not classify every possible integral-dual formalization or real transport. It does not assert that no identification law can exist, only that the defined package does not force one. It does not involve Planck's constant, Real.pi, Constants.G, a Green sum, or an endpoint target.

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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