Encyclopedia Foundation Foundation Pair Kernel Owner Channel Occurrence Bridge Candidate A Fails Occurre

ARTICLE 5 claims 5 theorems

Foundation Pair Kernel Owner Channel Occurrence Bridge Candidate A Fails Occurre

A machine-checked proof eliminates one proposed way to set the scale of physical sources, leaving a specific open question about which law governs them.

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The rejected candidate

The declaration candidateA_fails_occurrenceBridgedSourceNormalization is a theorem in the framework's machine-checked library of formal theorems. It proves that a specific proposed assignment for the scale of physical sources, called candidate A, is inconsistent with a required normalization condition. The condition, occurrence bridged source normalization, demands that for every possible event in the ledger, a discrete record of events, the source's effect on a standard probe exactly matches the event's boundary record, a signed number representing the event's orientation.

The proof works by showing that candidate A would force a separate equality, RemainingPhysicalEquality, to hold. That equality is already known to be unprovable from the current premises, and in fact candidate A contradicts it. The theorem therefore rules out candidate A as a possible physical law, without needing any new axioms or unproved assumptions. The library also proves that a different candidate, candidate B, does satisfy the normalization condition, so the rejection is not a dead end but a selection step.

What the theorem does not claim is that the normalization condition itself is forced. The condition is a definitional choice, a proposed reading of what a bridged occurrence means physically. The library proves that this condition is equivalent to RemainingPhysicalEquality, and that this equality is not derivable from the current premises. Closing the gap requires a new physical parent law, to be adjudicated, not a theorem available from today's package. The theorem also does not claim that the ledger's occurrence identity depends on any scalar quantity like heat or flux; the library explicitly proves that two distinct events can share the same boundary record value, so identity is carried by the oriented poles, the source and sink positions, alone.

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 occurrenceBridgedSourceNormalization_iff_remainingPhysicalEquality · 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]
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 occurrenceBridgedSourceNormalization_unforced · IndisputableMonolith/Foundation/PairKernelOwnerChannelOccurrenceBridge.lean
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
THEOREM heat_flux_scalar_join_not_occurrence_identity · IndisputableMonolith/Foundation/PairKernelOwnerChannelOccurrenceBridge.lean
heat_flux_scalar_join_not_occurrence_identity · IndisputableMonolith/Foundation/PairKernelOwnerChannelOccurrenceBridge.lean:445 
/-- (iii) Heat=flux scalar join is absent from the occurrence identity.
Equal unit boundary fluxes do not identify oriented poles (inherited
scalar-coincidence decoy); the bridge uses poles, not heat. -/
theorem heat_flux_scalar_join_not_occurrence_identity {d : Nat}
    (hd : 2 ≤ d) :
    ∃ k₁ k₂ : Fin d,
      occurrenceBoundaryRecord k₁ Side.debit =
          occurrenceBoundaryRecord k₂ Side.debit ∧
        orientedPolesOfOwner k₁ Side.debit ≠
          orientedPolesOfOwner k₂ Side.debit := by
  have hpos : 0 < d := lt_of_lt_of_le (by decide : (0 : Nat) < 2) hd
  have h1 : 1 < d := lt_of_lt_of_le (by decide : (1 : Nat) < 2) hd
  let k₁ : Fin d := ⟨0, hpos⟩
  let k₂ : Fin d := ⟨1, h1⟩
  refine ⟨k₁, k₂, ?_, ?_⟩
  · rw [occurrenceBoundaryRecord_debit, occurrenceBoundaryRecord_debit]
  · intro h
    have hs :=
      (orientedPostingBoundaryRecord_eq_iff_same_poles _ _).1 h
    have hk := debitPole_injective hs.1
    exact (by decide : (0 : Nat) ≠ 1) (congrArg Fin.val hk)

What this page does not claim

The theorem does not prove that the occurrence bridged source normalization condition is forced; it only proves candidate A fails it. The theorem does not claim that the ledger's occurrence identity depends on any scalar quantity like heat or flux. The theorem does not establish which physical law, if any, will close the gap and make the normalization derivable.

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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)

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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