Foundation Pair Kernel Coherence Event Constructor S15 Configuration Pricing Sem

Pricing and coherence

A recognition event, in the framework's discrete ledger of occurrences, is a single primitive posting: one oriented pair of sites in a batch. The theorem configurationPricingSemantics_implies_coherenceEnergy connects two properties of such postings. The first property is that the event's configuration dimension, a natural-number label, is forced to a canonical value. The second is that the energy assigned to the event is priced as the golden ratio raised to the negative of that dimension, so a dimension of three prices the event at phi to the minus three.

The theorem proves that any kinematics satisfying both properties must carry the coherence energy E_coh on every realized primitive posting. It is a theorem in the machine-checked library of formal theorems, meaning its proof is verified by a computer and rests on no unproven assumptions. The result is a bridge: it shows that a pricing rule, once the dimension is fixed, is enough to force the energy value that the framework associates with a single coherence event.

The declaration does not claim that any particular pricing rule is physically realized. It does not assert that the configuration dimension is actually three in nature, nor that the energy of a real event is in fact phi to the minus three. Those are separate, unformalized targets. The theorem also does not mention the reduced Planck constant, field units, Green response functions, or any empirical data. It is a statement about the internal logic of the framework's event construction, not about measurement.

The framework's library also provides countermodels: explicit constructions showing what the missing semantics must exclude. One such countermodel gives an event with zero energy that still preserves posting parents and legal atomic ticks, proving that energy is not forced by the posting structure alone. Another distributes the coherence energy across eight events, showing that the per-event value is not the only way to reach the total. These countermodels mark the boundary of what the theorem establishes.

What the theorem changes is the shape of the remaining work. The framework can now separate the question of what an event is from the question of what it costs. The event identity, its occurrence count, and its clock advance are closed layers. The pricing of configuration degrees by phi to the minus one, and the conversion of a tick into a physical duration, remain open targets. The theorem narrows the search: any candidate semantics that hopes to realize the coherence event must either satisfy the pricing rule or explain why it does not.

/-- 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

/-- 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_eight_events_eq_coherence
    {N : ℕ} [NeZero N]
    (event : PostingPair3 N) :
    8 * cycleDistributedPostingEnergy3 event =
      Constants.E_coh := by
  unfold cycleDistributedPostingEnergy3
  ring

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

This theorem does not prove that the configuration dimension is three in nature. This theorem does not state that any real physical event carries the energy phi to the minus three. This theorem does not connect the coherence energy to the reduced Planck constant or any empirical measurement.

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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