Foundation Pair Kernel Event Metric Pricing S16

The metric pricing module

In Recognition Science, the ledger, a discrete record of events, assigns each posting a time and a price. The module under discussion, foundation pair kernel event metric pricing s16, closes a gap in that accounting. It proves that any clock which starts at zero and advances by a fixed step on each tick is the native clock, whose value at tick index n is n times a base unit tau0. Eight successive ticks span exactly one octave. The proof is a theorem in the framework's machine-checked library of formal theorems.

The module also pins down prices. Each posting has a parent configuration, a record of its spatial location, temporal successor, and balance facts. The number of possible parent configurations is D plus 2, where D is the spatial dimension. A self-similar recognition channel has survival probability phi to the minus one, and the price of a configuration with n degrees is phi to the minus n. This is because the log of the J-cost aggregate makes independent degree prices compose multiplicatively, so the price of n degrees is the product of n single-degree prices. The theorem recognitionConfigurationPrice_composes states this multiplication law exactly.

The module does not stop at definitions. It also isolates exactly what remains unproved. Two physical statements, that posting duration reads the recognition clock interval and that posting energy reads the recognition configuration price, are explicit and target-blind. They are not theorems. The module shows that if these two readouts held, they would imply the four joins of the previous module, S15, and therefore the coherence-event arrow of S14. But the canonical witnesses for these readouts are models, not physical selection theorems. The module also proves a negative result: the ledger cost energy does not read the recognition price for the canonical channel, so the energy readout is not automatic.

The practical consequence is a clean separation. The mathematical ancestry for clock and price is closed, with exact theorems. The physical readouts for duration and energy are isolated as the only remaining assumptions. This tells a reader exactly where the framework's physical claims begin and where its mathematical certainty ends. It is a precise map of what is proved and what is postulated.

/-- 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]

theorem ledgerCostEnergy_does_not_read_recognitionPrice :
    ¬ PostingEnergyReadsRecognitionConfigurationPrice3
      canonicalChannel
      (@parentDerivedPostingConfigurationDimension3 3 _)
      (ledgerCostEventKinematics3 3) := by
  intro hread
  obtain ⟨pair, hpair⟩ :=
    realizedPrimitivePostingPair3_exists
  let event : RealizedPostingEvent3 3 :=
    ⟨pair, hpair⟩
  have h := hread event
  change
    Cost.Jlog 1 =
      recognitionConfigurationPrice canonicalChannel
        (parentDerivedPostingConfigurationDimension3 event) at h
  rw [recognitionConfigurationPrice_eq_phi_zpow_neg] at h
  rw [show
      parentDerivedPostingConfigurationDimension3 event =
        GapDerivation.configDim GapDerivation.D by
          exact
            parentDerivedConfiguration_has_forcedDimension
              event] at h
  exact ledgerJlogUnitCost_ne_constants_hbar
    (h.trans GapDerivation.hbar_exponent_eq_configDim.symm)

What this page does not claim

This module does not prove that posting duration or energy physically reads the clock or price. It does not derive the value of the base time unit tau0 or the golden ratio phi from first principles. It does not claim that the canonical model is the only possible model for the readouts.

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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