Foundation Pair Kernel Physical Posting Attachment S10 Physical Posting Attachme

The forced posting magnitude

In the Recognition Science framework, a ledger is a discrete record of events, and each event can carry two numbers: an action and a source magnitude. The framework's library, a machine-checked collection of formal theorems, proves a conditional statement: if a realized primitive posting is assigned the native action quantum hbar, and if its source coordinate is the action-dual coordinate, then the source magnitude is uniquely phi^5, the fifth power of the golden ratio, about 11.09. The proof is a theorem in the framework's library, meaning it is derived from prior definitions and theorems with no unproved assumptions.

The two conditions are not derived from the ledger alone. The library proves that a ledger unit normalization does not force the physical attachment: there exists a realized posting pair where the source is action-dual but the attachment condition fails. The identification of a posting with hbar, and the identification of the source coordinate as the action dual, are explicit semantic assumptions, left as predicates over assignment functions. The library does not prove either identification; it proves only that if both hold, the magnitude follows.

The theorem also shows the magnitude is the same for every realized primitive posting, so the assignment is uniform across events. The Green scale, a derived quantity from the posting magnitude, is phi^5 / 2. The native action quantum hbar itself is not a free parameter; the library proves it equals phi^(-5), and the theorem connects this to the posting magnitude through the duality condition.

What the declaration does not claim: it does not prove that any physical posting actually carries hbar, nor that the source coordinate is the action dual. Those remain open identifications. It does not use any measured data, such as hydrogen or Coulomb strength. The numerical consequence is conditional on the two identifications, which are not asserted unconditionally.

/-- The explicit event-to-action and action-to-source arrows uniquely force
the physical source magnitude to the reciprocal native action quantum. -/
theorem physicalPostingAttachment_forces_nativeActionQuantumInv
    {N : ℕ} [NeZero N]
    {postingAction sourceMagnitude : PostingPair3 N → ℝ}
    (hattach :
      PhysicalPostingMagnitudeAttachment3 postingAction sourceMagnitude)
    {event : PostingPair3 N}
    (hevent : event ∈ realizedPrimitivePostingPairs3 N) :
    sourceMagnitude event = nativeActionQuantumInv := by
  rcases hattach with ⟨haction, hdual⟩
  have hpos := (hdual event hevent).1
  have hproduct := (hdual event hevent).2
  rw [haction event hevent, eulerJActionUnit_eq_one,
    ← nativeActionQuantum_eq_constants_hbar] at hproduct
  exact
    (nativeQuantumSourceLaw_iff_inv (sourceMagnitude event)).mp
      ⟨hpos, hproduct⟩

/-- 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

/-- The attached source magnitude is invariant under account/event relabeling:
every realized posting receives the same value. -/
theorem physicalPostingMagnitude_account_relabeling_invariant
    {N : ℕ} [NeZero N]
    {postingAction sourceMagnitude : PostingPair3 N → ℝ}
    (hattach :
      PhysicalPostingMagnitudeAttachment3 postingAction sourceMagnitude)
    {event₁ event₂ : PostingPair3 N}
    (h₁ : event₁ ∈ realizedPrimitivePostingPairs3 N)
    (h₂ : event₂ ∈ realizedPrimitivePostingPairs3 N) :
    sourceMagnitude event₁ = sourceMagnitude event₂ := by
  rw [physicalPostingAttachment_forces_nativeActionQuantumInv
      hattach h₁,
    physicalPostingAttachment_forces_nativeActionQuantumInv
      hattach h₂]

/-- The candidate Green coefficient forced by S9 is `phi^5 / 2`. -/
theorem nativeDualCandidate_greenScale_eq_phi_pow_five_half :
    realGreenScaleFromPostingMagnitude nativeActionQuantumInv =
      Constants.phi ^ (5 : ℕ) / 2 := by
  unfold realGreenScaleFromPostingMagnitude
  rw [nativeActionQuantumInv_eq_constants_phi_pow_five]

What this page does not claim

The library does not prove that any realized posting carries hbar. The library does not prove that the source coordinate is the action dual. The theorem does not use any measured physical data.

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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