Encyclopedia Foundation Foundation Pair Kernel Coherence Event Constructor S15 Canonical Recognition Coh

ARTICLE 5 claims 5 theorems

Foundation Pair Kernel Coherence Event Constructor S15 Canonical Recognition Coh

A machine-checked theorem pins down what a single recognition event is, without yet saying how long it lasts in physical time or what its energy costs.

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The committed semantics

In the Recognition Science framework, a ledger is a discrete record of events. The theorem canonicalRecognitionCoherenceEventSemantics_model shows that a specific, concrete way of describing those events satisfies the framework's definition of a coherent event. The description fixes three things. First, an event is one oriented pair drawn from a batch, and each event's relation-set multiplicity is exactly one. Second, appending that event to the ledger advances a native clock by one successor tick. Third, the event's configuration dimension is forced to a canonical value. The theorem proves these three layers hold together as a single coherent model.

The three layers are not idle bookkeeping. The clock is defined so that each tick has spacing tau0, and the theorem recognitionClockSemantics_implies_fundamentalDuration derives that any event read from such a clock lasts exactly one fundamental tick. The configuration dimension feeds an energy-pricing rule: the theorem configurationPricingSemantics_implies_coherenceEnergy shows that when the dimension is forced and energy prices that dimension, the event carries the coherence energy E_coh. Together these imply the S14 claim that one realized primitive posting realizes one coherence event. The theorem primitiveCoherenceEvent_eightTickCycleAction then shows that eight such events produce the product E_coh times the octave duration.

The theorem deliberately stops short of physical time and energy. No committed theorem turns a successor tick into a real duration in seconds, and no theorem prices each event's configuration degrees by phi to the minus one. Those remain separate target-blind predicates. The docstring names explicit countermodels: null energy preserves posting parents but not coherence, cycle-distributed energy gives E_coh over eight events but not per event, and octave duration is not the fundamental duration. The model is a construction, not a derivation of those physical constants.

What the theorem does establish is a clean, inspectable foundation. It separates event identity, occurrence count, clock, configuration count, energy pricing, and action into layers that can be checked independently. The proof is complete in the machine-checked library, with no sorry and no new axiom. The consequence is that the framework's claim about a single coherence event now rests on explicit, verified structure rather than on an unexamined primitive.

THEOREM canonicalRecognitionCoherenceEventSemantics_model · IndisputableMonolith/Foundation/PairKernelCoherenceEventConstructorS15.lean
canonicalRecognitionCoherenceEventSemantics_model · IndisputableMonolith/Foundation/PairKernelCoherenceEventConstructorS15.lean:326 
/-- The canonical assignments satisfy the decomposed package as a MODEL. This
is a non-vacuity theorem, not a derivation from posting dynamics. -/
theorem canonicalRecognitionCoherenceEventSemantics_model
    {N : ℕ} [NeZero N] :
    RecognitionCoherenceEventSemantics3
      nativeRecognitionClock
      (@canonicalPostingConfigurationDimension3 N _)
      (canonicalCoherenceKinematics3 N) := by
  refine
    ⟨nativeRecognitionClock_has_nativeTickSpacing,
      ?_, ?_, ?_⟩
  · intro event
    simp [postingEventEndTick3, postingEventStartTick3,
      ledgerTick, LedgerTime.writeHead, LedgerTime.commit,
      nativeRecognitionClock, canonicalCoherenceKinematics3]
  · intro event
    rfl
  · intro event
    change
      Constants.E_coh =
        Constants.phi ^
          (-(GapDerivation.configDim GapDerivation.D : ℤ))
    exact GapDerivation.Constants_E_coh_eq_configDim
THEOREM canonicalRecognitionCoherenceEventSemantics_model · IndisputableMonolith/Foundation/PairKernelCoherenceEventConstructorS15.lean
canonicalRecognitionCoherenceEventSemantics_model · IndisputableMonolith/Foundation/PairKernelCoherenceEventConstructorS15.lean:326 
/-- The canonical assignments satisfy the decomposed package as a MODEL. This
is a non-vacuity theorem, not a derivation from posting dynamics. -/
theorem canonicalRecognitionCoherenceEventSemantics_model
    {N : ℕ} [NeZero N] :
    RecognitionCoherenceEventSemantics3
      nativeRecognitionClock
      (@canonicalPostingConfigurationDimension3 N _)
      (canonicalCoherenceKinematics3 N) := by
  refine
    ⟨nativeRecognitionClock_has_nativeTickSpacing,
      ?_, ?_, ?_⟩
  · intro event
    simp [postingEventEndTick3, postingEventStartTick3,
      ledgerTick, LedgerTime.writeHead, LedgerTime.commit,
      nativeRecognitionClock, canonicalCoherenceKinematics3]
  · intro event
    rfl
  · intro event
    change
      Constants.E_coh =
        Constants.phi ^
          (-(GapDerivation.configDim GapDerivation.D : ℤ))
    exact GapDerivation.Constants_E_coh_eq_configDim
THEOREM recognitionClockSemantics_implies_fundamentalDuration · IndisputableMonolith/Foundation/PairKernelCoherenceEventConstructorS15.lean
recognitionClockSemantics_implies_fundamentalDuration · IndisputableMonolith/Foundation/PairKernelCoherenceEventConstructorS15.lean:216 
/-- The temporal sub-arrows derive exactly S14's duration half. -/
theorem recognitionClockSemantics_implies_fundamentalDuration
    {N : ℕ} [NeZero N]
    {clock : Tick → ℝ}
    {kinematics : PostingEventKinematics3 N}
    (hspacing :
      RecognitionClockHasNativeTickSpacing clock)
    (hread :
      PostingDurationReadsRecognitionClock3
        clock kinematics) :
    RealizedPostingLastsFundamentalTick3 kinematics := by
  intro event hevent
  let realized : RealizedPostingEvent3 N :=
    ⟨event, hevent⟩
  rw [hread realized,
    postingEventEndTick3_eq_succ realized]
  exact hspacing (postingEventStartTick3 N)
THEOREM configurationPricingSemantics_implies_coherenceEnergy · IndisputableMonolith/Foundation/PairKernelCoherenceEventConstructorS15.lean
configurationPricingSemantics_implies_coherenceEnergy · IndisputableMonolith/Foundation/PairKernelCoherenceEventConstructorS15.lean:273 
/-- The configuration and pricing sub-arrows derive exactly S14's energy
half, using the already-proved global exponent identity. -/
theorem configurationPricingSemantics_implies_coherenceEnergy
    {N : ℕ} [NeZero N]
    {dimension : PostingConfigurationDimension3 N}
    {kinematics : PostingEventKinematics3 N}
    (hdimension :
      PostingEventHasForcedConfigurationDimension3 dimension)
    (hpricing :
      PostingEnergyPricesConfigurationDimension3
        dimension kinematics) :
    RealizedPostingCarriesCoherenceEnergy3 kinematics := by
  intro event hevent
  let realized : RealizedPostingEvent3 N :=
    ⟨event, hevent⟩
  rw [hpricing realized, hdimension realized]
  exact GapDerivation.Constants_E_coh_eq_configDim.symm
THEOREM primitiveCoherenceEvent_eightTickCycleAction · IndisputableMonolith/Foundation/PairKernelCoherenceEventConstructorS15.lean
primitiveCoherenceEvent_eightTickCycleAction · IndisputableMonolith/Foundation/PairKernelCoherenceEventConstructorS15.lean:388 
/-- Once the decomposed physical event semantics close, eight copies of one
primitive event carry the coherence energy integrated over one octave. -/
theorem primitiveCoherenceEvent_eightTickCycleAction
    {N : ℕ} [NeZero N]
    {clock : Tick → ℝ}
    {dimension : PostingConfigurationDimension3 N}
    {kinematics : PostingEventKinematics3 N}
    (hsemantics :
      RecognitionCoherenceEventSemantics3
        clock dimension kinematics)
    (event : RealizedPostingEvent3 N) :
    postingMultiplicityAction3 kinematics 8 event.1 =
      Constants.E_coh * Constants.octave := by
  have hcoherence :=
    recognitionCoherenceEventSemantics_implies_S14_arrow
      hsemantics
  unfold postingMultiplicityAction3
  rw [primitiveCoherenceEvent_action_eq_hbar
    hcoherence event.2,
    Constants.hbar_action_identity]
  change
    (8 : ℝ) * (Constants.E_coh * Constants.tau0) =
      Constants.E_coh * Constants.octave
  rw [show Constants.octave =
      8 * Constants.tau0 by rfl]
  ring

What this page does not claim

The theorem does not turn a successor tick into a physical real duration. The theorem does not price each event's configuration degrees by phi to the minus one. The theorem does not mention hbar, exact-J field units, Green response, or empirical data.

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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/PairKernelCoherenceEventConstructorS15.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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