Foundation Pair Kernel Event Metric Pricing S16 Metric Pricing Semantics Implies

The pricing semantics theorem

The declaration metricPricingSemantics_implies_primitiveCoherenceEvent is a theorem in the framework's machine-checked library of formal theorems. It states that if a posting's duration is read from the recognition clock interval and its energy is read from the recognition configuration price, then the event satisfies the coherence-event arrow. The theorem is proved with no sorry and no new axiom.

The theorem's content is best understood as a composition result. The framework models a recognition clock as a function from ticks to real numbers that preserves zero and adds a fixed native spacing at each successor. The theorem proves this clock is unique: any clock with those properties equals the native recognition clock. It also proves that eight successive ticks span exactly one octave, a unit of recognition time.

For configuration prices, the theorem shows that the price of a configuration with n degrees is phi⁻ⁿ, where phi is the golden ratio. This follows from the fact that the J-cost log aggregate makes independent degree prices compose multiplicatively. The theorem also proves that the tagged parent-degree carrier has cardinality D + 2, fixing the dimension of the event-local configuration.

What the theorem does not claim is as important as what it proves. It does not prove that physical duration or energy readouts are unconditionally true. Those readouts remain explicit, target-blind statements: posting duration reads the recognition clock interval, and posting energy reads the recognition configuration price. The theorem shows that if those readouts hold, then the coherence-event arrow follows. It does not prove that those readouts hold in the physical world.

The theorem also does not claim that the canonical witnesses are physical selection theorems. The canonical model, which uses the native recognition clock, the canonical channel, and the parent-derived configuration dimension, is a MODEL, not a proof that physical reality must select that model. The theorem is a statement about the framework's internal mathematics, not a statement about the physical world.

The practical consequence is that the framework has a clean, machine-checked path from metric pricing semantics to the coherence-event arrow. This path is the mathematical ancestry that S15's four joins require. The theorem closes the available mathematical ancestry and isolates the two exact readout residuals: duration and energy. Those residuals are the remaining physical statements that the framework does not yet prove.

theorem canonicalMetricPricingSemantics_model
    {N : ℕ} [NeZero N] :
    RecognitionMetricPricingSemantics3
      nativeRecognitionClock
      canonicalChannel
      (@parentDerivedPostingConfigurationDimension3 N _)
      (canonicalCoherenceKinematics3 N) := by
  refine
    ⟨nativeRecognitionClock_is_nativeNormalized,
      ?_, parentDerivedConfiguration_readsParentDegrees, ?_⟩
  · exact
      (canonicalRecognitionCoherenceEventSemantics_model
        (N := N)).2.1
  · intro event
    change
      Constants.E_coh =
        recognitionConfigurationPrice canonicalChannel
          (parentDerivedPostingConfigurationDimension3 event)
    rw [recognitionConfigurationPrice_eq_phi_zpow_neg]
    rw [show
      parentDerivedPostingConfigurationDimension3 event =
        GapDerivation.configDim GapDerivation.D by
          exact
            parentDerivedConfiguration_has_forcedDimension
              event]
    exact GapDerivation.Constants_E_coh_eq_configDim

/-- The Recognition NNO has one real-valued native clock. Zero and successor
preservation force the complete coordinate, not just its first tick. -/
theorem nativeNormalizedRecognitionClock_unique
    {clock : Tick → ℝ}
    (hclock : NativeNormalizedRecognitionClock clock) :
    clock = nativeRecognitionClock := by
  funext tick
  cases tick with
  | mk n =>
      induction n with
      | zero =>
          simpa [tickZero, nativeRecognitionClock] using hclock.1
      | succ n ih =>
          have hs := hclock.2 (⟨n⟩ : Tick)
          change
            clock (⟨n + 1⟩ : Tick) =
              nativeRecognitionClock (⟨n + 1⟩ : Tick)
          change
            clock (⟨n + 1⟩ : Tick) =
              clock (⟨n⟩ : Tick) + Constants.tau0 at hs
          rw [hs, ih]
          unfold nativeRecognitionClock
          push_cast
          ring

/-- Independent clock consequence: eight Recognition successors span one
native octave. -/
theorem nativeNormalizedClock_eight_successors_eq_octave
    {clock : Tick → ℝ}
    (hclock : NativeNormalizedRecognitionClock clock)
    (tick : Tick) :
    clock (recognitionTickAfter 8 tick) - clock tick =
      Constants.octave := by
  rw [nativeNormalizedClock_n_successors hclock]
  rfl

/-- Independent pricing consequence: disjoint degree blocks compose
multiplicatively by the J-cost log aggregate. -/
theorem recognitionConfigurationPrice_composes
    (channel : RecognitionDegreeChannel)
    (m n : ℕ) :
    recognitionConfigurationPrice channel (m + n) =
      recognitionConfigurationPrice channel m *
        recognitionConfigurationPrice channel n :=
  channel.occOf_composes m n

theorem postingConfigurationDegreeCarrier_card
    {N : ℕ} [NeZero N]
    (configuration : PostingRecognitionConfiguration3 N) :
    Fintype.card
        (PostingConfigurationDegreeCarrier3 configuration) =
      GapDerivation.configDim GapDerivation.D := by
  simp [PostingConfigurationDegreeCarrier3,
    GapDerivation.configDim]

What this page does not claim

The theorem does not prove that physical duration or energy readouts are unconditionally true. The theorem does not claim that the canonical witnesses are physical selection theorems. The theorem does not claim that the framework's internal mathematics is a statement about the physical world.

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/PairKernelEventMetricPricingS16.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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