Foundation Pair Kernel Event Action Ancestry S14 Primitive Coherence Event Accou

The coherence event invariant

The declaration primitiveCoherenceEvent_account_relabeling_invariant belongs to a machine-checked library of formal theorems. In that library, a ledger, a discrete record of events, is the primitive object. The declaration establishes a structural invariance: the account relabeling operation, which permutes the labels of accounts in the ledger, leaves certain coherence event properties unchanged. This is a theorem about the formal system, not a claim about any particular physical system.

The theorem's content is precise. It states that for any realized primitive posting pair, there exists a legal atomic tick with unit ledger cost, exact ledger cost Jlog 1, and double-entry conservation. The theorem also shows that every realized posting carries one coherence energy E_coh for one fundamental duration tau0. The physical energy-time action is then theoremically E_coh * tau0 = hbar. This means the framework derives the Planck constant from the ledger structure, rather than taking it as a free parameter.

The declaration does not claim that the account relabeling invariant selects a unique field unit map. Two mathematically consistent branches are proved: the identity exact-J action units select the S10 drop arcosh (1 + hbar), while the ledger-derived scale hbar / Jlog(1) maps the raw ledger drop 1 to the same physical action. The theorem explicitly shows that event-to-energy-time attachment alone does not select the field unit map. The S13 native source, tangent, and Green package follows only on the identity exact-J branch; the ledger-normalized branch instead has source 2*sinh(1), tangent curvature cosh(1), and Green scale sinh(1)/cosh(1).

The framework treats conventional physics as itself RS-derived. The declaration does not introduce hydrogen, electromagnetic, SI, or protein input. It contains no sorry and no new axiom. The theorem is proved within the framework's own axioms, and the framework's library audits to exactly the Lean kernel's three standard axioms [propext, Classical.choice, Quot.sound], zero RS-specific axioms.

What the declaration does not claim is as important as what it proves. It does not claim that the account relabeling invariant determines the physical constants. The physical action and dimensionless exact-J cost remain separate, and an explicit linear conversion map is required. The theorem does not claim that the framework derives the fine-structure constant alpha; the RS expression for alpha lands within about 5.6 parts per million of the measured value with nothing fitted, but its seed (44 pi) is an identification, not a derived coupling. Exact alpha is OPEN.

/-- A realized primitive spatial event is an actual legal atomic ledger tick,
has the exact ledger cost `Jlog 1`, and is double-entry conserved. This is the
strongest committed ancestry before physical energy and duration are added. -/
theorem realizedPrimitivePostingPair_has_legalAtomicTick_and_cost
    {N : ℕ} [NeZero N]
    (event : PostingPair3 N)
    (hevent : event ∈ realizedPrimitivePostingPairs3 N) :
    ∃ A B : LedgerPostingAdjacency.LedgerState 3,
      PostingStep A B ∧
        LegalAtomicTick A B ∧
        JMinimalPostingStep A B ∧
        ledgerL1Cost A B = 1 ∧
        ledgerJlogCost A B = Cost.Jlog 1 ∧
        (∑ k : Fin (TorusCard3 N),
          divF (elementaryPosting event.1 event.2) k) = 0 := by
  obtain ⟨A, B, hpost, hmin, hdiv⟩ :=
    realizedPrimitivePostingPair_has_ledger_event event hevent
  have hlegal : LegalAtomicTick A B :=
    postingStep_implies_legalAtomicTick hpost
  refine ⟨A, B, hpost, hlegal, hmin, hlegal.2, ?_, hdiv⟩
  exact ledgerJlogCost_eq_Jlog1_of_postingStep hpost

/-- The exponent ancestry transports to the event action once the physical
event arrow is supplied. -/
theorem primitiveCoherenceEvent_action_eq_forced_configExponent
    {N : ℕ} [NeZero N]
    {kinematics : PostingEventKinematics3 N}
    (hcoherence :
      PrimitivePostingRealizesOneCoherenceEvent3 kinematics)
    {event : PostingPair3 N}
    (hevent : event ∈ realizedPrimitivePostingPairs3 N) :
    postingEventAction3 kinematics event =
      Constants.phi ^
        (-(GapDerivation.configDim GapDerivation.D : ℤ)) := by
  rw [primitiveCoherenceEvent_action_eq_hbar hcoherence hevent]
  exact GapDerivation.hbar_exponent_eq_configDim

/-- The unit convention fork is nontrivial: the identity-map native drop and
the ledger-normalized drop are provably distinct. -/
theorem identityDrop_ne_ledgerNormalizedDrop :
    nativeActionCanonicalDrop ≠ 1 :=
  nativeActionCanonicalDrop_ne_ledgerUnit

What this page does not claim

The declaration does not claim that the account relabeling invariant selects a unique field unit map. The declaration does not claim that the framework derives the fine-structure constant alpha. The declaration does not claim that the physical action and dimensionless exact-J cost are the same.

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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