Foundation Pair Kernel Gap2a Remaining Physical Equality Residual Candidate A Sa

The unforced scale

In Recognition Science, a ledger, a discrete record of events, must be read at some scale. The framework's machine-checked library of formal theorems banks a fact about one candidate scale, called candidate A: it satisfies every premise in the current post-bridge package. That package includes the carriers, the common-type bridge, and the unique real cotangent condition. The declaration candidateA_satisfies_currentPremisesAfterCommonTypeBridge is a theorem: it proves that candidate A is admitted by all current assumptions.

The same library then proves that the premises admit at least two distinct candidate scales. So the package does not select a unique scale; it leaves the choice open. The residual equality, RemainingPhysicalEquality, is the missing primitive that would pick one. The theorem remainingPhysicalEquality_rejects_candidateA shows candidate A fails that equality, while remainingPhysicalEquality_selects_candidateB shows candidate B satisfies it. The equality is not forced by the package: forcesRemainingPhysicalEqualityFromAfterBridge_unforced proves no law uniformly forced by the premises can be scale-breaking.

What the declaration does not claim matters. It does not prove candidate A is the physical source; it only proves candidate A is a permitted one. It does not reopen the unit-drop bridge, the Noether carrier packaging, or the Gauss covector construction. It does not introduce Planck's constant, Real.pi, Constants.G, or any endpoint target. The module banks the residual as a named gap: the smallest new physical primitive is exactly that equality, not a weaker parent. The proof is axiom-clean, with no sorry and no new axiom.

theorem candidateA_satisfies_currentPremisesAfterCommonTypeBridge :
    CurrentPremisesAfterCommonTypeBridge
      candidateA_sourceMagnitudeExpr.eval :=
  candidateA_satisfies_currentPremisesWithUniqueRealCotangent

theorem currentPremisesAfterCommonTypeBridge_admit_distinct_candidates :
    ∃ sourceScale₁ sourceScale₂ : ℝ,
      sourceScale₁ ≠ sourceScale₂ ∧
        CurrentPremisesAfterCommonTypeBridge sourceScale₁ ∧
        CurrentPremisesAfterCommonTypeBridge sourceScale₂ :=
  currentPremisesWithUniqueRealCotangent_admit_distinct_candidates

theorem remainingPhysicalEquality_rejects_candidateA :
    ¬ RemainingPhysicalEquality candidateA_sourceMagnitudeExpr.eval :=
  identifiesPhysicalSource_rejects_candidateA

/-- Any law forced uniformly by the post-bridge package cannot select a unique
scale among the banked candidates. -/
theorem afterBridgeForcedLaw_cannot_be_scaleBreaking
    (law : ℝ → Prop)
    (hforced :
      ∀ sourceScale : ℝ,
        CurrentPremisesAfterCommonTypeBridge sourceScale →
          law sourceScale) :
    ¬ IsScaleBreakingAttachmentLaw law := by
  intro hbreaking
  rcases hbreaking with ⟨selected, ⟨_hselPrem, _hselLaw⟩, honly⟩
  have hA := hforced _
    candidateA_satisfies_currentPremisesAfterCommonTypeBridge
  have hB := hforced _
    candidateB_satisfies_currentPremisesAfterCommonTypeBridge
  have hAselected :
      candidateA_sourceMagnitudeExpr.eval = selected :=
    honly _
      ⟨candidateA_satisfies_currentRecognitionSourcePremises, hA⟩
  have hBselected :
      candidateB_sourceMagnitudeExpr.eval = selected :=
    honly _
      ⟨candidateB_satisfies_currentRecognitionSourcePremises, hB⟩
  exact
    candidates_select_distinct_magnitudes
      (hAselected.trans hBselected.symm)

What this page does not claim

This does not claim candidate A is the physical source, only that it is permitted by current premises. This does not claim the remaining physical equality is derivable from the current package. This does not claim any new physical constant or endpoint target is introduced.

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/PairKernelGap2aRemainingPhysicalEqualityResidual.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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