Foundation Pair Kernel Physical Posting Attachment S10 Ledger Unit Normalization

The boundary of unit normalization

In the Recognition Science framework, a ledger is a discrete record of recognition events, and each event has a posting magnitude, a real number attached to it. A question arises: does choosing the unit scale of the ledger, the normalization that sets one posting to have magnitude 1, force the physical magnitude of a realized event? The theorem ledgerUnitNormalization_does_not_force_physicalAttachment answers no, and it does so with a proved counterexample, not a plausibility argument.

The framework's library, a machine-checked collection of formal theorems, defines physical attachment as the conjunction of two conditions. First, a realized posting carries the native action quantum, the framework's derived unit of action equal to phi to the minus fifth power. Second, the posting's source magnitude is the action dual of its posting action, meaning their product equals the Euler-J action unit, which is 1. The theorem constructs a specific assignment where every posting has magnitude 1 and every posting action is 1, so the product is 1 and the duality condition holds. Yet this unit assignment fails the first condition: a magnitude of 1 is not the native action quantum, since phi to the minus fifth is less than 1. The counterexample is a realized event in a three-site system, so the failure is not an artifact of an excluded case.

The theorem's force is to separate two levels of the framework. The ledger's unit normalization is a definitional choice, a scale convention. Physical attachment is a further claim about which posting magnitudes are realized, and it requires an additional identification: that a minimum-J posting carries the native action quantum. The library proves this identification for a candidate assignment, named the native candidate, but it does not prove that every realized posting must be that candidate. The two semantic identifications, that a posting carries the native action quantum and that the source coordinate is the action dual, remain open as unconditional facts. The numerical consequence, that the source magnitude is uniquely phi to the fifth, is therefore a conditional theorem: it holds if the physical attachment conditions hold.

What this changes is the epistemic status of the framework's constants. The value phi to the fifth for the source magnitude is not derived from the unit normalization alone; it is derived from the physical attachment conditions. The counterexample shows that a ledger can satisfy the duality condition while failing to realize the native action quantum, so the physical scale is a separate input, not a consequence of the ledger's own units. The theorem does not claim that the unit assignment is physically realized, only that the normalization does not rule it out. It also does not claim that the native candidate is the only assignment satisfying physical attachment; the library proves the native candidate does satisfy it, and the counterexample shows the unit assignment does not, but other assignments are not classified.

/-- Unit source and unit action obey source-action duality because the Euler-J
unit is one. Thus duality alone does not select the native action quantum. -/
theorem unitAssignments_satisfy_sourceActionDuality
    {N : ℕ} [NeZero N] :
    PostingSourceIsActionDual3
      (@unitPostingActionCountermodel3 N _)
      (@unitSourceMagnitudeCountermodel3 N _) := by
  intro event hevent
  simp [unitPostingActionCountermodel3,
    unitSourceMagnitudeCountermodel3, eulerJActionUnit_eq_one]

/-- Exact separation: a witnessed primitive event, unit double-entry source,
and source-action duality coexist while the native physical attachment fails.
Ledger normalization therefore cannot prove the physical assignment. -/
theorem ledgerUnitNormalization_does_not_force_physicalAttachment :
    ∃ event : PostingPair3 3,
      event ∈ realizedPrimitivePostingPairs3 3 ∧
        PostingSourceIsActionDual3
          (@unitPostingActionCountermodel3 3 inferInstance)
          (@unitSourceMagnitudeCountermodel3 3 inferInstance) ∧
        ¬ PhysicalPostingMagnitudeAttachment3
          (@unitPostingActionCountermodel3 3 inferInstance)
          (@unitSourceMagnitudeCountermodel3 3 inferInstance) := by
  let σ : Equiv.Perm (Fin 3) := Equiv.refl (Fin 3)
  obtain ⟨event, hevent, hcert⟩ :=
    canonicalGeneratorSource_consumer_exists
      (N := 3) (by norm_num) σ
  refine ⟨event, hevent, unitAssignments_satisfy_sourceActionDuality, ?_⟩
  intro hattach
  exact
    unitPostingActionCountermodel_not_native hevent hattach.1

/-- Consequently the physical source magnitude would be `phi^5`. -/
theorem physicalPostingAttachment_forces_phi_pow_five
    {N : ℕ} [NeZero N]
    {postingAction sourceMagnitude : PostingPair3 N → ℝ}
    (hattach :
      PhysicalPostingMagnitudeAttachment3 postingAction sourceMagnitude)
    {event : PostingPair3 N}
    (hevent : event ∈ realizedPrimitivePostingPairs3 N) :
    sourceMagnitude event = Constants.phi ^ (5 : ℕ) := by
  rw [physicalPostingAttachment_forces_nativeActionQuantumInv
    hattach hevent]
  exact nativeActionQuantumInv_eq_constants_phi_pow_five

/-- The candidate physical attachment is internally consistent. This is MODEL
evidence only: both functions above were chosen to satisfy the OPEN arrows. -/
theorem nativeCandidates_satisfy_attachment
    {N : ℕ} [NeZero N] :
    PhysicalPostingMagnitudeAttachment3
      (@nativePostingActionCandidate3 N _)
      (@nativeDualSourceMagnitudeCandidate3 N _) :=
  ⟨nativePostingActionCandidate_carries_nativeAction,
    nativeCandidates_source_is_actionDual⟩

What this page does not claim

The unit assignment is physically realized; the theorem only shows the normalization does not rule it out. The native candidate is the only assignment satisfying physical attachment; other assignments are not classified. The framework derives the fine-structure constant alpha; the source magnitude phi to the fifth is a conditional consequence, not a fitted coupling.

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/PairKernelPhysicalPostingAttachmentS10.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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