Foundation Pair Kernel Coherence Event Constructor S15 Recognition Coherence Eve

The coherence event theorem

In the Recognition Science framework, a ledger is a discrete record of events, and a recognition event is a single atomic posting to that ledger. The theorem recognitionCoherenceEventSemantics_implies_S14_arrow establishes a precise bridge: if a posting event carries the full coherence semantics, then it necessarily realizes exactly one coherence event. This is a formal, machine-checked result in the framework's library of theorems.

The theorem's content decomposes into three concrete claims about the event. First, an event is identified as a subtype of one oriented pair in the S8 event batch, meaning it is a single, well-defined entry. Second, its relation-set multiplicity is exactly one, so it corresponds to a single relation in the underlying structure. Third, appending this event to the ledger advances the recognition clock by exactly one successor tick. These three properties together constitute the realization of one coherence event.

The theorem is conditional: it requires the full coherence event semantics to hold. This includes a native clock with tick spacing equal to the fundamental time constant tau0, a configuration dimension that is forced to the canonical value, and kinematics where the event's duration reads off the clock and its energy equals the coherence energy E_coh. When these conditions are met, the theorem guarantees the S14 arrow, which is the statement that a primitive posting realizes one coherence event.

What the theorem does not claim is equally important. It does not assert that the successor tick corresponds to any physical duration in seconds. It does not price the event's configuration degrees by the golden ratio phi to the minus one. It does not mention the reduced Planck constant hbar, exact J-field units, Green response functions, or any empirical data. These are deliberately kept as separate, target-blind predicates that are not part of this theorem's scope.

The framework also proves several countermodels that show what the missing semantics must exclude. For instance, a null energy function preserves posting parents but fails to carry coherence energy. A cycle-distributed energy assigns E_coh/8 per event, which sums to E_coh over eight events but is not the per-event coherence energy. An octave duration is not the fundamental tick duration. These countermodels sharpen the boundary of what the theorem establishes.

The practical consequence is that the framework has a clean, modular decomposition of the S14 physical claim into independently inspectable layers: event identity, occurrence count, clock behavior, configuration count, energy pricing, and action. Each layer can be checked separately, and the theorem shows how they compose. This is a step toward making the framework's physical claims auditable at the level of formal logic.

/-- The decomposed Recognition semantics discharge S14's exact physical
event arrow. -/
theorem recognitionCoherenceEventSemantics_implies_S14_arrow
    {N : ℕ} [NeZero N]
    {clock : Tick → ℝ}
    {dimension : PostingConfigurationDimension3 N}
    {kinematics : PostingEventKinematics3 N}
    (hsemantics :
      RecognitionCoherenceEventSemantics3
        clock dimension kinematics) :
    PrimitivePostingRealizesOneCoherenceEvent3 kinematics := by
  rcases hsemantics with
    ⟨hspacing, hread, hdimension, hpricing⟩
  exact
    ⟨configurationPricingSemantics_implies_coherenceEnergy
        hdimension hpricing,
      recognitionClockSemantics_implies_fundamentalDuration
        hspacing hread⟩

/-- The event commit is literally one δ successor. This is discrete cadence,
not yet a real-valued physical duration. -/
theorem postingEventEndTick3_eq_succ
    {N : ℕ} [NeZero N]
    (event : RealizedPostingEvent3 N) :
    postingEventEndTick3 event =
      recognitionTimeAlgebra.succ
        (postingEventStartTick3 N) := by
  exact ledger_commit_is_delta_succ
    ([] : List (RealizedPostingEvent3 N)) event

/-- The decomposed Recognition semantics discharge S14's exact physical
event arrow. -/
theorem recognitionCoherenceEventSemantics_implies_S14_arrow
    {N : ℕ} [NeZero N]
    {clock : Tick → ℝ}
    {dimension : PostingConfigurationDimension3 N}
    {kinematics : PostingEventKinematics3 N}
    (hsemantics :
      RecognitionCoherenceEventSemantics3
        clock dimension kinematics) :
    PrimitivePostingRealizesOneCoherenceEvent3 kinematics := by
  rcases hsemantics with
    ⟨hspacing, hread, hdimension, hpricing⟩
  exact
    ⟨configurationPricingSemantics_implies_coherenceEnergy
        hdimension hpricing,
      recognitionClockSemantics_implies_fundamentalDuration
        hspacing hread⟩

What this page does not claim

The theorem does not claim that a posting event's successor tick corresponds to a physical duration in seconds. The theorem does not claim that configuration degrees are priced by phi to the minus one. The theorem does not claim anything about hbar, exact J-field units, Green response, or empirical data.

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)

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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