Foundation Pair Kernel Coherence Event Constructor S15

The event constructor

In Recognition Science, a recognition event is a discrete record of a single act of comparison: one object is checked against another, and the outcome is a number. The foundation pair kernel coherence event constructor s15 is a machine-checked library of formal theorems that pins down what such an event is, and what it means for one to be recorded. It is not a physical theory; it is a piece of bookkeeping infrastructure. The library takes a primitive posting, which is a single oriented pair of objects from an earlier batch, and proves that this posting can be viewed as an event with a definite identity, a definite position in a sequence, and a definite effect on a clock.

The classical background here is the theory of discrete dynamical systems and the algebra of sequences. A tick is one step in a discrete time sequence, and a clock is a function that assigns a real number to each tick. The library defines a native recognition clock that assigns the constant tau0 to each tick, so that the clock advances by tau0 with every step. It then proves that appending one event to the ledger, which is the discrete record of all events, advances the tick index by exactly one. This is the core result: one event, one tick. The proof is a direct computation, and the library states it as a theorem that the tick span of any realized posting event is 1.

The library also separates concerns that a physicist might want to keep apart. It defines a configuration dimension for each event, which is a natural number, and it defines an energy function that prices each event's configuration degrees. It proves that if the energy of an event equals phi raised to the negative of its configuration dimension, then the event carries the coherence energy E_coh. This is a conditional statement: the pricing rule is a definitional choice, and the conclusion follows from it. The library does not prove that the pricing rule is physically true. It keeps that as a separate, unformalized target.

What the library does establish, with machine-checked proofs, is a set of negative results that sharpen the semantics. It shows that a null energy function, which assigns zero to every event, preserves the posting parents and the tick span but fails to carry the coherence energy. It shows that a cycle-distributed energy function, which assigns E_coh divided by 8 to every event, sums to E_coh over eight events but fails to be the per-event coherence energy. And it shows that an octave duration, which assigns the constant octave to every event, is not the fundamental tick duration. These countermodels are the library's way of saying what the missing physical semantics must exclude.

In plain language, this library is a precision tool. It takes the vague idea of a recognition event and gives it a formal skeleton: an event is a subtype of one oriented pair, its relation-set multiplicity is one, and recording it advances the clock by one successor. It then proves the basic facts about this skeleton. The payoff is that later libraries can build on this skeleton without re-arguing what an event is. The library does not claim to know what an event physically is, only what it formally is, and it is explicit about the gap.

/-- The canonical native-unit clock. This is a MODEL coordinate realization,
not a theorem that a physical system reads this clock. -/
def nativeRecognitionClock (tick : Tick) : ℝ :=
  (tick.index : ℝ) * Constants.tau0

theorem postingEventTickSpan3_eq_one
    {N : ℕ} [NeZero N]
    (event : RealizedPostingEvent3 N) :
    postingEventTickSpan3 event = 1 := by
  simp [postingEventTickSpan3, postingEventEndTick3,
    postingEventStartTick3, ledgerTick, LedgerTime.writeHead,
    LedgerTime.commit]

/-- 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

/-- Null-event countermodel: all posting, multiplicity, cadence, and
conservation parents remain available while zero energy fails the coherence
price. -/
theorem nullEnergy_preserves_postingParents_but_not_coherence :
    ∃ event : RealizedPostingEvent3 3,
      postingPairMultiplicity3 3 event.1.1 event.1.2 = 1 ∧
        (∃ A B : LedgerPostingAdjacency.LedgerState 3,
          PostingStep A B ∧ LegalAtomicTick A B) ∧
        postingEventTickSpan3 event = 1 ∧
        ¬ RealizedPostingEnergyEqualsCoherence3
          (@nullPostingEnergy3 3 _) := by
  obtain ⟨pair, hpair⟩ :=
    realizedPrimitivePostingPair3_exists
  let event : RealizedPostingEvent3 3 :=
    ⟨pair, hpair⟩
  obtain ⟨A, B, hpost, hlegal, _⟩ :=
    realizedPostingEvent_has_ledger_tick event
  refine
    ⟨event,
      realizedPostingEvent_pairMultiplicity_one event,
      ⟨A, B, hpost, hlegal⟩,
      postingEventTickSpan3_eq_one event, ?_⟩
  intro henergy
  have hzero := henergy event.1 event.2
  change 0 = Constants.E_coh at hzero
  exact (ne_of_lt Constants.E_coh_pos) hzero

theorem cycleDistributedEnergy_not_per_event :
    ¬ RealizedPostingEnergyEqualsCoherence3
      (@cycleDistributedPostingEnergy3 3 _) := by
  intro henergy
  obtain ⟨event, hevent⟩ :=
    realizedPrimitivePostingPair3_exists
  have h := henergy event hevent
  unfold cycleDistributedPostingEnergy3 at h
  have hpos := Constants.E_coh_pos
  nlinarith

What this page does not claim

The library does not prove that any physical event actually carries the coherence energy E_coh. The library does not derive the value of the fundamental tick tau0 or the octave constant. The library does not establish that the recognition clock measures physical time.

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

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