Foundation Pair Kernel Physical Posting Attachment S10 Native Posting Action Can

The native action candidate

In the Recognition Science framework, a ledger is a discrete record of events, and each event is a posting that moves a quantity between two accounts. The framework's core theorem forces a particular cost function, and from that cost function a chain of results derives the framework's native constants, including its unit of action, written hbar and equal to the golden ratio raised to the negative fifth power. This section concerns a specific declaration, nativePostingActionCandidate_carries_nativeAction, which states that a particular candidate assignment of action to postings indeed makes every realized posting carry that native action quantum.

In plain terms, the theorem says: if you assign to every posting the constant value hbar as its action, then for every posting that the framework realizes as primitive, that assignment is correct. The declaration is a theorem in the machine-checked library of formal theorems, proved by unfolding definitions and simplifying, with no additional axioms. It is a conditional statement: it holds for the candidate assignment, not for any assignment. The theorem does not say that the framework's physics forces this assignment; it says that if you choose it, then the property holds.

The significance is that this candidate assignment is the one that, together with a dual source magnitude, satisfies the full attachment condition. The library proves that the candidate source magnitude, the reciprocal of the action quantum, is phi to the fifth power, and that this pair satisfies the source-action duality. The theorem nativeCandidates_satisfy_attachment combines these to show the candidate pair meets the full physical posting magnitude attachment condition. This is a construction, not a derivation: the library explicitly does not prove that a posting must carry hbar, nor that the source coordinate is the action dual. Those identifications remain open.

The theorem also implies a uniqueness property: for any assignment that satisfies the attachment condition, the source magnitude on any realized posting is forced to be phi to the fifth power. This is the content of physicalPostingAttachment_forces_phi_pow_five. The framework's library shows that if the attachment condition holds, the magnitude is uniquely determined, regardless of which posting you look at. This is a strong consequence, but it is conditional on the attachment condition being satisfied, which itself is not derived.

In Recognition Science, this result is a step in a larger program of deriving physical constants from the ledger structure. The native action quantum hbar is not a free parameter; it is forced by the cost function chain. The theorem here shows how that quantum would attach to postings if the identification is made. It does not claim that the identification is forced, nor that the framework has derived the fine-structure constant or any measured charge. The honesty of the library is explicit: it leaves the semantic identifications as open predicates, not as hidden assumptions.

For a reader, the takeaway is that the framework has a precise, machine-checked statement about a candidate assignment, and a conditional theorem about what follows if that assignment is realized. The open questions are clear: whether a posting actually carries the native action, and whether the pair-kernel source coordinate is its action dual. These are targets for future derivation, not settled facts.

theorem nativePostingActionCandidate_carries_nativeAction
    {N : ℕ} [NeZero N] :
    MinimumJPostingCarriesNativeAction3
      (@nativePostingActionCandidate3 N _) := by
  intro event hevent
  rfl

/-- 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 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⟩

/-- 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

What this page does not claim

The declaration does not prove that any posting must carry the native action quantum. The declaration does not prove that the source coordinate is the action dual. The declaration does not derive the fine-structure constant or any measured charge.

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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