Foundation Pair Kernel Gap2a Integral Lattice Dual Residual Candidate A Satisfie

A precise non-result

The declaration candidateA_satisfies_currentPremisesWithIntegralDual is a machine-checked theorem in the Recognition Science framework's library of formal theorems. It establishes that a specific candidate value for the recognition source scale, called candidate A, satisfies the current recognition source premises even after adjoining a canonical integer dual structure. In plain terms, the theorem proves that this candidate is compatible with all the framework's current axioms about how recognition events are recorded and scaled.

The integer dual structure in question is the set of additive characters from the integers to themselves, with the identity map as the unique positive primitive character. The framework's library proves this character is unique and evaluates the elementary posting to the integer 1. The theorem shows that candidate A, whose source magnitude evaluates to 1, satisfies the premises with this dual structure attached, and that it is positive and free of any factor of pi.

The main content is what the theorem does not establish. The library also proves that candidate B, a different positive pi-free source model, satisfies the same premises with the same integral dual structure. The two candidates have distinct magnitudes, so the premises with the integral dual do not select a unique source scale. The theorem does not prove that candidate A is forced or that it is the only possible source; it proves only that it is admitted by the current premises.

This is a deliberate non-result in the framework's development. The library shows that the integral dual, primitivity, and uniqueness of the positive character do not attach a unique real source magnitude. A rescaling countermodel exists: the coupling value 2 satisfies the named premises and the primitive positive character, yet it is not 1. The framework therefore requires an additional physical identification law to transport the integer character into a unique real source, and the theorem explicitly forbids smuggling that law in as a definition.

Within the framework, this theorem is a checkpoint on the path to deriving physical constants. It confirms that the current premises are not yet strong enough to force a unique scale, and it names the gap that a future law must close. The consequence is that the framework's derivation of specific constants remains open at this stage, and candidate A is a viable but not unique candidate.

theorem candidateA_satisfies_currentPremisesWithIntegralDual :
    CurrentPremisesWithIntegralDual
      candidateA_sourceMagnitudeExpr.eval := by
  apply currentPremisesWithIntegralDual_of_positive_piFree
  · rw [candidateA_sourceMagnitude_eq_one]
    norm_num
  · exact candidateA_sourceMagnitude_piFree

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)

/-- Rescaling countermodel survives after adjoining the integral dual. -/
theorem integralDual_rescaling_countermodel :
    ∃ coupling : ℝ,
      coupling ≠ 1 ∧
        NamedPremises coupling ∧
          IsPrimitivePositiveCharacter primitivePositiveCharacter :=
  ⟨2, by norm_num, namedPremises_all_couplings 2,
    primitivePositiveCharacter_isPrimitivePositive⟩

What this page does not claim

Candidate A is not proven to be the unique source scale; the theorem only shows it is admitted by the premises. The integral dual structure does not force any scale-breaking law, including the forbidden source := 1/action form. The theorem does not classify every possible integral-dual formalization or real transport.

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.

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