Foundation Pair Kernel Gap2a Remaining Physical Equality Residual Candidate B Sa

The candidate that survives

In the Recognition Science framework, a ledger is a discrete record of recognition events, and the framework's library is a machine-checked collection of formal theorems. The declaration candidateB_satisfies_currentPremisesAfterCommonTypeBridge is a theorem stating that one specific candidate value for a physical source magnitude, called candidate B, satisfies all the premises that remain after a common-type bridge has been established. The bridge is a formal step that connects two carriers, and this theorem confirms candidate B still fits within that post-bridge premise package.

The theorem is part of a larger audit of what the framework's current assumptions can and cannot force. The library proves that the current premises admit at least two distinct candidate source scales, meaning the assumptions alone do not select a unique scale. It also proves that a property called RemainingPhysicalEquality, which would identify the physical source with a unique cotangent coordinate, is unforced by those premises. Candidate B is the one that satisfies this equality property, while candidate A is rejected by it. The theorem remainingPhysicalEquality_selects_candidateB confirms this selection, and remainingPhysicalEquality_rejects_candidateA confirms the rejection.

What this declaration does not claim is important. It does not prove that candidate B is the correct physical source magnitude, only that it is consistent with the current premises. It does not establish that RemainingPhysicalEquality must hold, since the library explicitly proves this property is unforced. It also does not claim that the common-type bridge itself is reopened, or that any new axiom or sorry is introduced. The theorem is a clean result within the existing framework, with no new axioms and no unresolved proofs.

The practical consequence is that the framework has narrowed its search to a single candidate that fits all current constraints, while formally documenting that the final selection still requires an additional primitive. This is a precise statement of where the framework stands: one candidate survives the current assumptions, but the framework does not yet force that candidate as the unique answer.

theorem candidateB_satisfies_currentPremisesAfterCommonTypeBridge :
    CurrentPremisesAfterCommonTypeBridge
      candidateB_sourceMagnitudeExpr.eval :=
  candidateB_satisfies_currentPremisesWithUniqueRealCotangent

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

/-- Core residual: adjoining unique cotangent mathematics and the common-type
bridge comparison machinery does not force RemainingPhysicalEquality. -/
theorem remainingPhysicalEquality_unforced_after_commonTypeBridge :
    ¬ (∀ sourceScale : ℝ,
      CurrentPremisesAfterCommonTypeBridge sourceScale →
        RemainingPhysicalEquality sourceScale) := by
  intro hforce
  exact remainingPhysicalEquality_rejects_candidateA
    (hforce _ candidateA_satisfies_currentPremisesAfterCommonTypeBridge)

theorem remainingPhysicalEquality_selects_candidateB :
    RemainingPhysicalEquality candidateB_sourceMagnitudeExpr.eval :=
  identifiesPhysicalSource_selects_candidateB

theorem remainingPhysicalEquality_rejects_candidateA :
    ¬ RemainingPhysicalEquality candidateA_sourceMagnitudeExpr.eval :=
  identifiesPhysicalSource_rejects_candidateA

What this page does not claim

This theorem does not prove candidate B is the correct physical source magnitude. This theorem does not establish that RemainingPhysicalEquality must hold, only that it is unforced. This theorem does not reopen the common-type bridge or introduce new axioms.

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)

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