Encyclopedia Foundation Foundation Pair Kernel Owner Channel Occurrence Bridge Candidate B Satisfies Occ

ARTICLE 4 claims 4 theorems

Foundation Pair Kernel Owner Channel Occurrence Bridge Candidate B Satisfies Occ

A formal theorem selects one of two candidate scale factors for a physical source term, while leaving the deeper physical law that would justify it open.

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The candidate B result

In the Recognition Science framework, a ledger (a discrete record of events) tracks accounts indexed by a number d. Each event, called a tick, is a debit or credit on one account. The framework's machine-checked library of formal theorems establishes a bridge from this owner-indexed ledger view to a geometric "oriented pole" representation on a doubled space of 2d coordinates. A key theorem, candidateB_satisfies_occurrenceBridgedSourceNormalization, proves that a specific candidate scale factor, called candidate B, satisfies a normalization condition for these bridged occurrences.

In plain terms, the condition requires that the physical source associated with any tick, when measured against a unit potential drop, equals the tick's boundary record (a +1 for debit, -1 for credit) times a universal coordinate. The theorem states that candidate B's value makes this equality hold for all ticks. This is a formal, machine-checked result. The library also proves that candidate A, a different candidate, fails this same condition. The injectivity theorems guarantee that distinct ticks map to distinct oriented pole data, so the bridge does not collapse different events into one.

In Recognition Science, this result is part of a larger effort to derive physical constants from the ledger structure. The theorem itself, however, does not claim that candidate B is the physically realized scale. The normalization condition is definitionally equivalent to another statement, RemainingPhysicalEquality, and the library explicitly proves that this remaining equality is not forced by the current premises. A separate theorem, occurrenceBridgedSourceNormalization_unforced, shows that no scale factor can be proven to satisfy the condition from the existing axioms. Therefore, the selection of candidate B is a theorem about a definitional property, not a derivation of a physical law.

What the declaration does not claim is that candidate B's value is the one found in nature. The physical parent law that would justify the action-priced reading of a bridged occurrence, and thus select the true scale, remains an open target. The library's audit note confirms that no premise assumes this remaining physical equality or any equivalent reformulation; it appears only as the named residual goal. The theorem is a precise, conditional result within a formal system, not an empirical claim about the universe.

THEOREM candidateB_satisfies_occurrenceBridgedSourceNormalization · IndisputableMonolith/Foundation/PairKernelOwnerChannelOccurrenceBridge.lean
candidateB_satisfies_occurrenceBridgedSourceNormalization · IndisputableMonolith/Foundation/PairKernelOwnerChannelOccurrenceBridge.lean:399 
/-- Candidate B satisfies the occurrence-bridged normalization (reuses banked
selection; locates the target, does not force it). -/
theorem candidateB_satisfies_occurrenceBridgedSourceNormalization :
    OccurrenceBridgedSourceNormalization
      candidateB_sourceMagnitudeExpr.eval :=
  (occurrenceBridgedSourceNormalization_iff_remainingPhysicalEquality _).2
    remainingPhysicalEquality_selects_candidateB
THEOREM candidateA_fails_occurrenceBridgedSourceNormalization · IndisputableMonolith/Foundation/PairKernelOwnerChannelOccurrenceBridge.lean
candidateA_fails_occurrenceBridgedSourceNormalization · IndisputableMonolith/Foundation/PairKernelOwnerChannelOccurrenceBridge.lean:390 
/-- Candidate A fails the occurrence-bridged normalization (reuses banked
rejection of RemainingPhysicalEquality). -/
theorem candidateA_fails_occurrenceBridgedSourceNormalization :
    ¬ OccurrenceBridgedSourceNormalization
        candidateA_sourceMagnitudeExpr.eval := by
  intro h
  exact remainingPhysicalEquality_rejects_candidateA
    ((occurrenceBridgedSourceNormalization_iff_remainingPhysicalEquality _).1 h)
THEOREM orientedPolesOfOwner_injective · IndisputableMonolith/Foundation/PairKernelOwnerChannelOccurrenceBridge.lean
orientedPolesOfOwner_injective · IndisputableMonolith/Foundation/PairKernelOwnerChannelOccurrenceBridge.lean:135 
/-- (a) Injectivity: distinct owner addresses give distinct oriented poles. -/
theorem orientedPolesOfOwner_injective {d : Nat} :
    Function.Injective (fun p : Fin d × Side => orientedPolesOfOwner p.1 p.2) := by
  intro ⟨k₁, s₁⟩ ⟨k₂, s₂⟩ h
  apply Prod.ext
  · -- accounts equal
    cases s₁ with
    | debit =>
      cases s₂ with
      | debit =>
        have hs :=
          (orientedPostingBoundaryRecord_eq_iff_same_poles
            (orientedPolesOfOwner k₁ Side.debit)
            (orientedPolesOfOwner k₂ Side.debit)).1 h
        exact debitPole_injective hs.1
      | credit =>
        have hs :=
          (orientedPostingBoundaryRecord_eq_iff_same_poles
            (orientedPolesOfOwner k₁ Side.debit)
            (orientedPolesOfOwner k₂ Side.credit)).1 h
        exact False.elim (debitPole_ne_creditPole_of k₁ k₂ hs.1)
    | credit =>
      cases s₂ with
      | debit =>
        have hs :=
          (orientedPostingBoundaryRecord_eq_iff_same_poles
            (orientedPolesOfOwner k₁ Side.credit)
            (orientedPolesOfOwner k₂ Side.debit)).1 h
        exact False.elim (debitPole_ne_creditPole_of k₂ k₁ hs.1.symm)
      | credit =>
        have hs :=
          (orientedPostingBoundaryRecord_eq_iff_same_poles
            (orientedPolesOfOwner k₁ Side.credit)
            (orientedPolesOfOwner k₂ Side.credit)).1 h
        exact creditPole_injective hs.1
  · -- sides equal
    cases s₁ with
    | debit =>
      cases s₂ with
      | debit => rfl
      | credit =>
        have hs :=
          (orientedPostingBoundaryRecord_eq_iff_same_poles
            (orientedPolesOfOwner k₁ Side.debit)
            (orientedPolesOfOwner k₂ Side.credit)).1 h
        exact False.elim (debitPole_ne_creditPole_of k₁ k₂ hs.1)
    | credit =>
      cases s₂ with
      | debit =>
        have hs :=
          (orientedPostingBoundaryRecord_eq_iff_same_poles
            (orientedPolesOfOwner k₁ Side.credit)
            (orientedPolesOfOwner k₂ Side.debit)).1 h
        exact False.elim (debitPole_ne_creditPole_of k₂ k₁ hs.1.symm)
      | credit => rfl
THEOREM occurrenceBridgedSourceNormalization_iff_remainingPhysicalEquality · occurrenceBridgedSourceNormalization_unforced · IndisputableMonolith/Foundation/PairKernelOwnerChannelOccurrenceBridge.lean
occurrenceBridgedSourceNormalization_iff_remainingPhysicalEquality · IndisputableMonolith/Foundation/PairKernelOwnerChannelOccurrenceBridge.lean:345 
theorem occurrenceBridgedSourceNormalization_iff_remainingPhysicalEquality
    (sourceScale : ℝ) :
    OccurrenceBridgedSourceNormalization sourceScale ↔
      RemainingPhysicalEquality sourceScale := by
  constructor
  · intro hnorm
    -- Specialize to a single debit occurrence on d = 1.
    haveI : NeZero (1 : Nat) := ⟨by decide⟩
    haveI : DecidableEq (Fin (2 * 1)) := inferInstance
    let k : Fin 1 := ⟨0, by decide⟩
    have h := hnorm (d := 1) k Side.debit
    have hdrop :=
      unitPotentialDropProbe_drop (debitPole k) (creditPole k)
        (debitPole_ne_creditPole k)
    have hrec : occurrenceBoundaryRecord k Side.debit = 1 :=
      occurrenceBoundaryRecord_debit k
    -- LHS = sourceScale
    have hlhs :
        bridgedOccurrenceSourceCovector sourceScale k Side.debit
            (unitPotentialDropProbe (debitPole k) (creditPole k)) =
          sourceScale := by
      simp [bridgedOccurrenceSourceCovector, orientedPolesOfOwner,
        pairKernelPhysicalSourceCovector_apply, hdrop]
    -- Reduce to sourceScale = uniqueCotangentCoordinate
    have : sourceScale = uniqueCotangentCoordinate := by
      calc
        sourceScale =
            bridgedOccurrenceSourceCovector sourceScale k Side.debit
              (unitPotentialDropProbe (debitPole k) (creditPole k)) := hlhs.symm
        _ = occurrenceBoundaryRecord k Side.debit * uniqueCotangentCoordinate := h
        _ = (1 : ℝ) * uniqueCotangentCoordinate := by rw [hrec]
        _ = uniqueCotangentCoordinate := one_mul _
    exact this
  · intro hrem d _ _ k side
    have hmatch :=
      bridgedOccurrenceSource_matches_boundarySign sourceScale k side
    -- RemainingPhysicalEquality is sourceScale = uniqueCotangentCoordinate
    change sourceScale = uniqueCotangentCoordinate at hrem
    calc
      bridgedOccurrenceSourceCovector sourceScale k side
          (unitPotentialDropProbe (debitPole k) (creditPole k)) =
          occurrenceBoundaryRecord k side * sourceScale := hmatch
      _ = occurrenceBoundaryRecord k side * uniqueCotangentCoordinate := by
            rw [hrem]
occurrenceBridgedSourceNormalization_unforced · IndisputableMonolith/Foundation/PairKernelOwnerChannelOccurrenceBridge.lean:407 
/-- The residual remains unforced after the occurrence bridge: current premises
still do not select the normalization. -/
theorem occurrenceBridgedSourceNormalization_unforced :
    ¬ (∀ sourceScale : ℝ,
      CurrentPremisesAfterCommonTypeBridge sourceScale →
        OccurrenceBridgedSourceNormalization sourceScale) := by
  intro hforce
  exact remainingPhysicalEquality_rejects_candidateA <|
    (occurrenceBridgedSourceNormalization_iff_remainingPhysicalEquality _).1 <|
      hforce _ candidateA_satisfies_currentPremisesAfterCommonTypeBridge

What this page does not claim

The theorem does not claim candidate B is the physically realized scale in nature. The theorem does not derive the value of the source scale from the ledger structure alone. The theorem does not prove that any scale factor satisfies the normalization condition from the current 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/PairKernelOwnerChannelOccurrenceBridge.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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