Foundation Pair Kernel Gap2a Noether Symplectic Cotangent Residual Candidate B S

A candidate that survives

In the Recognition Science framework, the physical scale of a source is not yet uniquely pinned down. The framework's library of machine-checked theorems currently banks two candidate values, called Candidate A and Candidate B, for the magnitude of a real source. The declaration candidateB_satisfies_currentPremisesWithNoetherSymplectic proves, in the framework's formal system, that Candidate B meets every condition in the current recognition source premises, and also meets an additional certificate built from Noether's theorem and symplectic geometry.

The additional certificate is a package of already-existing results. It includes the statement that a certain cost functional is a conserved charge along a one-parameter rescaling flow, and that this charge is independent of the source scale. The certificate also records that the recognition cost is a symplectic trace, and that space and time translation symmetries imply conserved charges. None of these conjuncts mentions a pair-kernel source coupling, and none depends on the source scale. The certificate is homogeneous in the source scale, meaning it does not prefer one magnitude over another.

What the declaration does not claim is the more important part. The theorem does not say that Candidate B is the unique scale, or that the framework has selected it as the physical value. In fact, the same library proves the opposite: the enriched premises still admit both Candidate A and Candidate B as distinct, positive, pi-free source models. A separate theorem states that no law forced uniformly by the Noether/symplectic package alone can break the scale ambiguity. The certificate is a necessary condition that both candidates pass, not a sufficient one that picks a winner.

The library also records what is missing. The exact missing parent result would be a momentum-map identification equating the conserved charge of an elementary posting unit orbit with the real Gauss source covector, without assuming a product relation and without Green aggregation. That identification remains open. The dual product law, which would physically reject Candidate A, is not a corollary of the Noether/symplectic package; it is used only as a decoy discriminator to show the package's limits. So the honest summary is: Candidate B is consistent with everything currently proved, but the framework has not yet proved that it is the scale.

theorem candidateB_satisfies_currentPremisesWithNoetherSymplectic :
    CurrentPremisesWithNoetherSymplectic
      candidateB_sourceMagnitudeExpr.eval := by
  apply currentPremisesWithNoetherSymplectic_of_positive_piFree
  · change 0 < nativeActionQuantumInv
    exact nativeActionQuantumInv_pos
  · exact candidateB_sourceMagnitude_piFree

/-- The defined Noether/symplectic enrichment still admits both banked
positive pi-free source models. -/
theorem currentPremisesWithNoetherSymplectic_admit_distinct_candidates :
    ∃ sourceScale₁ sourceScale₂ : ℝ,
      sourceScale₁ ≠ sourceScale₂ ∧
        CurrentPremisesWithNoetherSymplectic sourceScale₁ ∧
        CurrentPremisesWithNoetherSymplectic sourceScale₂ :=
  ⟨candidateA_sourceMagnitudeExpr.eval,
    candidateB_sourceMagnitudeExpr.eval,
    candidates_select_distinct_magnitudes,
    candidateA_satisfies_currentPremisesWithNoetherSymplectic,
    candidateB_satisfies_currentPremisesWithNoetherSymplectic⟩

/-- Any law forced uniformly by the defined Noether/symplectic package cannot
be scale-breaking: both banked candidates satisfy that package. -/
theorem noetherForcedLaw_cannot_be_scaleBreaking
    (law : ℝ → Prop)
    (hforced :
      ∀ sourceScale : ℝ,
        CurrentPremisesWithNoetherSymplectic sourceScale →
          law sourceScale) :
    ¬ NoetherForcedSourceLaw law := by
  intro hbreaking
  rcases hbreaking with ⟨selected, ⟨_hselPrem, _hselLaw⟩, honly⟩
  have hA := hforced _
    candidateA_satisfies_currentPremisesWithNoetherSymplectic
  have hB := hforced _
    candidateB_satisfies_currentPremisesWithNoetherSymplectic
  have hAselected :
      candidateA_sourceMagnitudeExpr.eval = selected :=
    honly _
      ⟨candidateA_satisfies_currentPremisesWithNoetherSymplectic, hA⟩
  have hBselected :
      candidateB_sourceMagnitudeExpr.eval = selected :=
    honly _
      ⟨candidateB_satisfies_currentPremisesWithNoetherSymplectic, hB⟩
  exact
    candidates_select_distinct_magnitudes
      (hAselected.trans hBselected.symm)

What this page does not claim

Candidate B is the unique or physically selected source scale. The Noether/symplectic certificate forces the native-action dual product law. The framework has derived the fine-structure constant or any other specific physical coupling from this certificate.

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/PairKernelGap2aNoetherSymplecticCotangentResidual.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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