Foundation Pair Kernel Gap2a Real Cotangent Normalization Residual Unique Real C

The unforced identification

In the Recognition Science framework, a discrete ledger of recognition events carries a primitive character, a kind of basic counting function. The framework's native action quantum prices an embedding of that integer lattice into the real numbers. A theorem in the machine-checked library proves there is exactly one real-linear functional that extends this character along that embedding. Its value at 1 is the reciprocal of the action quantum, a number the framework writes as nativeActionQuantumInv.

The open question is whether this unique cotangent, as the framework calls the extended functional, is the same as the physical pair-kernel source scale. The declaration uniqueRealCotangent_does_not_force_nativeActionDualSourceLaw proves it is not. Even after adjoining the uniqueness of the real-linear extension to the current recognition source premises, the library exhibits two distinct positive real numbers, candidateA and candidateB, that both satisfy all those premises. One of them is the reciprocal of the action quantum; the other is not. The theorem shows that no forced law can identify the physical source with the unique cotangent for every scale, because doing so would reject candidateA, which the premises admit.

The library also proves that the exact-J variational conjugate, a different real number arising from the framework's cost function, is not equal to the unique cotangent coordinate. So first variation of the cost does not supply the identification either. What would close the gap is a separate, independently physical premise, named UniqueCotangentNormalizationByActionUnit, that selects the cotangent as the source without inserting the identification as a definition.

What the declaration does not claim is that the identification is impossible. It only proves the current premises do not force it. The gap remains open, a target for a future physical law rather than a demonstrated contradiction. The framework's library is honest about this boundary: the uniqueness of the mathematical extension is theorem-backed, but the physical selection is not.

/-- Uniqueness: the character-extension condition forces the unique
real-linear functional already named above. -/
theorem uniqueRealCharacterExtension
    (φ : ℝ →ₗ[ℝ] ℝ)
    (hφ : ExtendsPrimitiveCharacterAlongAction φ) :
    φ = canonicalRealCharacterExtension := by
  have hprim : φ nativeActionQuantum = 1 := by
    have h := hφ primitivePosting
    -- h : φ (actionLatticeEmbed 1) = 1
    simpa [actionLatticeEmbed, primitivePosting, primitiveCharacterReal] using h
  have hone : φ 1 = (nativeActionQuantum)⁻¹ := by
    have hsmul :
        φ (nativeActionQuantum • (1 : ℝ)) =
          nativeActionQuantum • φ 1 := map_smul φ nativeActionQuantum 1
    have hrewritten :
        φ nativeActionQuantum = nativeActionQuantum * φ 1 := by
      simpa [smul_eq_mul, mul_comm] using hsmul
    have hprod : nativeActionQuantum * φ 1 = 1 := by
      rw [← hrewritten, hprim]
    exact eq_inv_of_mul_eq_one_right hprod
  rw [realLinear_eq_smul_id φ, hone]
  rfl

/-- Bridge 1 / native-action dual remains unforced after adjoining unique real
cotangent mathematics. -/
theorem uniqueRealCotangent_does_not_force_nativeActionDualSourceLaw :
    ¬ (∀ sourceScale : ℝ,
      CurrentPremisesWithUniqueRealCotangent sourceScale →
        NativeActionDualSourceLaw sourceScale) := by
  intro hforce
  exact nativeActionDualSourceLaw_rejects_candidateA
    (hforce _
      candidateA_satisfies_currentPremisesWithUniqueRealCotangent)

/-- The scalar-extension uniqueness package still admits both banked positive
pi-free source models. -/
theorem currentPremisesWithUniqueRealCotangent_admit_distinct_candidates :
    ∃ sourceScale₁ sourceScale₂ : ℝ,
      sourceScale₁ ≠ sourceScale₂ ∧
        CurrentPremisesWithUniqueRealCotangent sourceScale₁ ∧
        CurrentPremisesWithUniqueRealCotangent sourceScale₂ :=
  ⟨candidateA_sourceMagnitudeExpr.eval,
    candidateB_sourceMagnitudeExpr.eval,
    candidates_select_distinct_magnitudes,
    candidateA_satisfies_currentPremisesWithUniqueRealCotangent,
    candidateB_satisfies_currentPremisesWithUniqueRealCotangent⟩

/-- Exact-J variational conjugate is not the unique character-extension
cotangent coordinate, so first-variation of the exact one-edge action does not
supply the physical identification. -/
theorem exactJConjugate_ne_uniqueCotangentCoordinate :
    nativeExactJConjugateSource ≠ uniqueCotangentCoordinate := by
  rw [uniqueCotangentCoordinate_eq_nativeActionQuantumInv]
  exact nativeExactJConjugateSource_ne_nativeActionQuantumInv

What this page does not claim

The identification of the unique cotangent with the physical source is impossible. The two candidate scales are physically realized; they only demonstrate logical consistency with the premises. The exact-J conjugate is a candidate for the physical source scale; the theorem only proves it differs from the unique cotangent.

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/PairKernelGap2aRealCotangentNormalizationResidual.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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