Foundation Pair Kernel Event Action Ancestry S14 Primitive Coherence Event Impli

What the coherence theorem proves

In physics, action is the product of energy and time, the quantity that a system accumulates as it evolves. Planck's constant hbar is the natural unit of action in quantum mechanics, the smallest meaningful chunk that a physical process can carry. The Recognition Science framework, which models reality as a discrete record of events called a ledger (a bookkeeping system of primitive postings), proves a precise statement about this quantity: every realized primitive posting, the smallest unit of recognition, carries exactly one coherence energy E_coh for one fundamental duration tau0, and their product equals hbar.

The theorem primitiveCoherenceEvent_implies_nativeActionAssignment states this result formally. If a kinematics model (an assignment of energy and duration to each posting) satisfies the condition that every realized posting carries the coherence energy E_coh and lasts the fundamental duration tau0, then the action of that posting equals hbar. The proof is short: it combines the definition of postingEventAction3, which multiplies energy by duration, with the lemma primitiveCoherenceEvent_action_eq_hbar. The theorem is machine-checked in the framework's library of formal theorems, with no unproved assumptions beyond the standard axioms of the underlying type theory.

The result does not claim that the framework derives hbar from nothing. The constants E_coh and tau0 are defined within the framework, and their product being hbar is a theorem about those definitions. The theorem also does not claim that the framework selects a unique mapping from its dimensionless ledger costs to physical action. The module proves two mathematically consistent branches: the identity exact-J branch, which selects a drop of arcosh(1 + hbar), and the ledger-normalized branch, which maps the raw ledger drop 1 to the same physical action. The theorem establishes that the event-to-energy-time attachment alone does not choose between these branches; that choice is a separate modeling decision.

What the theorem does establish is a bridge between the framework's discrete ledger and conventional quantum mechanics. It shows that the framework's primitive events, if they carry the coherence energy and fundamental duration, reproduce the quantum of action without any hydrogen, electromagnetic, SI, or protein input. The framework's library also proves that batches of such events have quantized action: a finite collection of realized primitive postings has total action equal to the number of events times hbar. This gives a precise sense in which the framework's discrete recognition events are the carriers of physical action.

/-- The target-blind arrow discharges S10's assignment-level action predicate. -/
theorem primitiveCoherenceEvent_implies_nativeActionAssignment
    {N : ℕ} [NeZero N]
    {kinematics : PostingEventKinematics3 N}
    (hcoherence :
      PrimitivePostingRealizesOneCoherenceEvent3 kinematics) :
    MinimumJPostingCarriesNativeAction3
      (postingEventAction3 kinematics) := by
  intro event hevent
  exact primitiveCoherenceEvent_action_eq_hbar hcoherence hevent

theorem identityExactJMap_forces_S13_native_package
    {N : ℕ} [NeZero N]
    {kinematics : PostingEventKinematics3 N}
    {drop sourceMagnitude : PostingPair3 N → ℝ}
    (hcoherence :
      PrimitivePostingRealizesOneCoherenceEvent3 kinematics)
    (hmap :
      MappedExactJDropRepresentsEventAction3 1 kinematics drop)
    (hsource :
      EventDropSourceIsExactJConjugate3 drop sourceMagnitude)
    {event : PostingPair3 N}
    (hevent : event ∈ realizedPrimitivePostingPairs3 N) :
    drop event = nativeActionCanonicalDrop ∧
      sourceMagnitude event = nativeExactJConjugateSource ∧
      2 * sourceMagnitude event = nativeOrderedExactJSource ∧
      Real.cosh (drop event) = 1 + Constants.hbar := by
  have hdrop :=
    identityExactJMap_selects_nativeActionCanonicalDrop
      hcoherence hmap hevent
  refine ⟨hdrop, ?_, ?_, ?_⟩
  · rw [hsource event hevent, hdrop]
    rfl
  · rw [hsource event hevent, hdrop]
    rfl
  · rw [hdrop]
    exact nativeActionCanonicalDrop_cosh

/-- Independent consequence: every finite realized event batch is quantized
in integer multiples of `hbar`. -/
theorem primitiveCoherenceEvent_batchAction_quantized
    {N : ℕ} [NeZero N]
    {kinematics : PostingEventKinematics3 N}
    (hcoherence :
      PrimitivePostingRealizesOneCoherenceEvent3 kinematics)
    (events : Finset (PostingPair3 N))
    (hsub : events ⊆ realizedPrimitivePostingPairs3 N) :
    postingBatchKinematicAction3 kinematics events =
      (events.card : ℝ) * Constants.hbar := by
  unfold postingBatchKinematicAction3
  calc
    (∑ event ∈ events, postingEventAction3 kinematics event) =
        ∑ _event ∈ events, Constants.hbar := by
          apply Finset.sum_congr rfl
          intro event hevent
          rw [primitiveCoherenceEvent_action_eq_hbar
            hcoherence (hsub hevent)]
    _ = (events.card : ℝ) * Constants.hbar := by simp

What this page does not claim

The theorem does not derive hbar from first principles; it proves that the product of the framework's defined constants E_coh and tau0 equals hbar. The theorem does not select a unique mapping from dimensionless ledger costs to physical action; both the identity and ledger-normalized branches remain consistent. The theorem does not claim that the framework's primitive events correspond to any specific physical system or particle.

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/PairKernelEventActionAncestryS14.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.

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