Foundation Pair Kernel Physical Posting Attachment S10

The posting magnitude bridge

In Recognition Science, a ledger is a discrete record of recognition events, and a posting is a single entry in that record. The module Foundation Pair Kernel Physical Posting Attachment S10 asks a precise question: when a posting has the minimum possible cost, what magnitude must its source have? The answer, proved in the framework's machine-checked library of formal theorems, is that the source magnitude is forced to be phi^5, where phi is the golden ratio, approximately 1.618. This is not a fitted value; it follows from the definitions and the forcing chain that produces the framework's constants.

The proof rests on two independently derived normalizations. The first is the native action quantum, the smallest possible action in the ledger's units, which the framework derives as hbar = phi^(-5). The second is the Euler-J action unit, which the framework derives as exactly 1. The module then defines a physical posting magnitude attachment as a pair of assignments: one for the posting's action and one for its source magnitude, satisfying two conditions. First, a minimum-cost posting must carry the native action quantum. Second, the source magnitude and the posting action must be duals, meaning their product equals the Euler-J action unit.

The central theorem, physicalPostingAttachment_forces_phi_pow_five, states that if such an attachment exists for a realized primitive posting, then the source magnitude is uniquely phi^5. The proof is short: the attachment forces the source magnitude to equal the inverse of the native action quantum, which is phi^5. A companion theorem shows the Green scale, a measure of how a source influences its surroundings, is then phi^5 / 2. The module also proves the attachment is invariant under relabeling of accounts, meaning the magnitude does not depend on which specific posting is chosen.

The module is careful about what it does not prove. The two semantic identifications, that a minimum-cost posting actually carries the native action quantum and that the pair-kernel source coordinate is its action dual, are left as explicit predicates over assignment functions. The library currently proves neither. Thus the numerical consequence is a conditional theorem: if the identifications hold, the magnitude is phi^5. The identifications themselves remain open targets. The module also shows that a unit assignment, where posting action and source magnitude are both 1, satisfies the duality condition but fails the native action condition, so it is not a physical attachment.

What this establishes in plain language is a bridge between the abstract ledger and physical magnitudes. If the framework's minimal posting carries its native action, then the source magnitude is not free; it is forced to be the golden ratio raised to the fifth power. This is a concrete, checkable consequence of the framework's axioms, and it provides a target for future work: proving the identifications would turn the conditional theorem into an unconditional one.

/-- 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 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⟩

/-- The reciprocal native action quantum is exactly `phi^5`. -/
theorem nativeActionQuantumInv_eq_constants_phi_pow_five :
    nativeActionQuantumInv = Constants.phi ^ (5 : ℕ) := by
  simpa only using nativeActionQuantumInv_eq_phi_pow_five

/-- 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]

/-- 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]

What this page does not claim

The module does not prove that any physical posting actually carries the native action quantum. The module does not derive the fine-structure constant or any other measured coupling. The module does not claim the identifications are true; they are explicit open targets.

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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