Encyclopedia Foundation Foundation Pair Kernel Event Action Ancestry S14

ARTICLE 5 claims 4 theorems 1 model

Foundation Pair Kernel Event Action Ancestry S14

A formal bridge connects the framework's discrete accounting steps to physical energy and time, and proves the conversion cannot pick its own units.

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The event action bridge

In physics, action is the product of an energy and a duration, the quantity a system carries as it evolves. Recognition Science keeps a different book: a discrete record of events, called a ledger, where each posting step has a forced cost. The module named foundation pair kernel event action ancestry s14 asks how one book translates into the other. Its answer, in plain terms, is that a single realized event in the ledger carries exactly one unit of coherence energy for one fundamental duration, and that the translation from ledger cost to physical action is not free to choose its own scale.

The starting point is a proved theorem. Every realized primitive posting pair, the smallest complete event in the ledger, has an underlying posting step that is legal, minimal in cost, costs exactly one unit in the ledger's L1 accounting, and conserves the double-entry balance. The same theorem shows this step carries no physical duration or energy by itself. The module then adds a model: a realized posting carries one coherence energy E_coh for one fundamental duration tau0. This is a definitional choice, not a derivation. The product of those two, E_coh times tau0, is the physical action of the event. The framework proves this product equals hbar, its native unit of action, which is phi^-5 in its own units.

Here the module makes its sharpest point. The physical action and the dimensionless ledger cost are separate objects, so attaching them requires an explicit conversion map. The module proves two mathematically consistent branches exist. In the first, the identity map on the exact-J cost selects a drop of arcosh(1 + hbar), the framework's native action scale. In the second, a ledger-derived scale, hbar divided by the exact-J cost of one unit, maps the raw ledger drop of 1 to the same physical action. The two branches are genuinely different: the identity drop is not equal to 1, and the coherent event model does not make the identity ledger map valid. The module proves both branches are internally consistent, so event-to-energy-time attachment alone does not select the field unit map.

The two branches then diverge in their consequences. The identity exact-J branch forces the S13 native source, tangent, and Green package, with source 2*sinh(1), tangent curvature cosh(1), and Green scale sinh(1)/cosh(1). The ledger-normalized branch instead has source 2*sinh(1), tangent curvature cosh(1), and Green scale sinh(1)/cosh(1). The module also proves that doubling the energy or stretching the duration of a coherent event breaks the coherence condition, so the one-energy-one-duration assignment is not arbitrary. The batch action of any set of realized events is quantized: it equals the number of events times hbar. No hydrogen, electromagnetic, SI, or protein input is used anywhere in the module, and the proofs contain no unproved assumptions beyond the ambient type theory.

THEOREM realizedPrimitivePostingPair_has_legalAtomicTick_and_cost · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean
realizedPrimitivePostingPair_has_legalAtomicTick_and_cost · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean:59 
/-- 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
MODEL PrimitivePostingRealizesOneCoherenceEvent3 · primitiveCoherenceEvent_action_eq_forced_configExponent · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean
PrimitivePostingRealizesOneCoherenceEvent3 · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean:120 
/-- Exact OPEN Recognition-event semantic arrow. It says that one realized
primitive posting is one coherence-energy event lasting one fundamental tick.
It is target-blind: `hbar`, exact-J cost, and field drop do not occur. -/
def PrimitivePostingRealizesOneCoherenceEvent3
    {N : ℕ} [NeZero N]
    (kinematics : PostingEventKinematics3 N) : Prop :=
  RealizedPostingCarriesCoherenceEnergy3 kinematics ∧
    RealizedPostingLastsFundamentalTick3 kinematics
primitiveCoherenceEvent_action_eq_forced_configExponent · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean:155 
/-- 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
THEOREM identityExactJMap_forces_S13_native_package · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean
identityExactJMap_forces_S13_native_package · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean:333 
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
THEOREM ledgerScaleExactJMap_forces_dropOne_package · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean
ledgerScaleExactJMap_forces_dropOne_package · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean:373 
theorem ledgerScaleExactJMap_forces_dropOne_package
    {N : ℕ} [NeZero N]
    {scale : ℝ}
    {kinematics : PostingEventKinematics3 N}
    {drop sourceMagnitude : PostingPair3 N → ℝ}
    (hcoherence :
      PrimitivePostingRealizesOneCoherenceEvent3 kinematics)
    (hscale : PrimitivePostingActionScaleLaw scale)
    (hmap :
      MappedExactJDropRepresentsEventAction3 scale kinematics drop)
    (hsource :
      EventDropSourceIsExactJConjugate3 drop sourceMagnitude)
    {event : PostingPair3 N}
    (hevent : event ∈ realizedPrimitivePostingPairs3 N) :
    drop event = 1 ∧
      sourceMagnitude event = exactJVariationalSource 1 ∧
      2 * sourceMagnitude event =
        ledgerNormalizedOrderedExactJSource ∧
      Real.cosh (drop event) =
        ledgerNormalizedExactJTangentCurvature := by
  have hdrop :=
    ledgerScaleExactJMap_selects_drop_one
      hcoherence hscale hmap hevent
  refine ⟨hdrop, ?_, ?_, ?_⟩
  · rw [hsource event hevent, hdrop]
  · rw [hsource event hevent, hdrop]
    rfl
  · rw [hdrop]
    rfl
THEOREM primitiveCoherenceEvent_batchAction_quantized · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean
primitiveCoherenceEvent_batchAction_quantized · IndisputableMonolith/Foundation/PairKernelEventActionAncestryS14.lean:453 
/-- 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 module does not prove which unit map is physically correct; both branches are proved consistent. The module does not derive the fine-structure constant or any electromagnetic quantity. The coherence energy and fundamental duration are model choices, not derived constants.

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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)

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