Foundation Pair Kernel Gap2a Integral Lattice Dual Residual Current Premises Wit

The integral dual and the missing scale

The declaration currentPremisesWithIntegralDual_do_not_select_unique_scale is a formal theorem in the Recognition Science framework's machine-checked library of formal theorems. It concerns the ledger, a discrete record of recognition events, and the attempt to pin down a unique real-valued source scale for a pair-kernel equation. The theorem establishes a negative result: the current recognition source premises, even when enriched with the canonical integer dual of the rank-one posting lattice, do not force a unique positive real source scale.

To understand the result, consider the rank-one integer posting lattice, which is simply the integers ℤ. Its canonical integer dual is the set of additive characters from ℤ to ℤ, denoted Hom(ℤ, ℤ). The unique positive primitive character in this dual is the identity map, which sends the generator 1 to 1. The framework proves that any additive character with this property is the identity. The question is whether this uniqueness datum, combined with the existing recognition source premises, selects a unique real source scale.

The theorem answers in the negative. It shows that there exist at least two distinct positive, pi-free real source scales, labeled candidate A and candidate B, that both satisfy the enriched premises. The proof demonstrates that if a unique scale were selected, the two candidates would have to be equal, contradicting the fact that they are distinct. This is a formal, machine-checked argument with no unproved assumptions. The result is stated as a theorem in the Lean file PairKernelGap2aIntegralLatticeDualResidual.lean.

The theorem also establishes that the integral dual does not force a zero-defect dual identification, and that any law uniformly forced by the defined Hom-enriched package cannot be scale-breaking. In plain terms, the integer-valued character fixes only the lattice generator, not a real cotangent or Green source. Transporting the integer character into a unique real source requires an additional physical identification law, which is not supplied by the defined package.

What the declaration does not claim is equally important. It does not claim that no such additional law exists, only that it is not provided by the Hom-enriched package. It does not classify every possible integral-dual formalization or real transport. It explicitly forbids smuggling the identification as source := 1/action. The theorem is a precise boundary on what the current premises can and cannot force, leaving the selection of a unique scale as an open problem requiring further physical input.

/-- 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)

/-- 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⟩

/-- 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

/-- 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)

What this page does not claim

The theorem does not claim that no additional law can select a unique scale, only that the Hom-enriched package does not. The theorem does not classify every possible integral-dual formalization or real transport. The theorem does not claim that the identification `source := 1/action` is valid; it explicitly forbids it.

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)

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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