Foundation Pair Kernel Event Metric Pricing S16 Parent Degree Readout Implies Fo

The forced configuration

The declaration parentDegreeReadout_implies_forcedConfiguration proves a structural implication inside the Recognition Science framework. In plain terms: if a posting event's configuration dimension is read out from the count of its parent degrees, then that configuration is forced to have a specific dimension. The proof is direct: it rewrites the readout condition and applies the cardinality theorem for the degree carrier, which states that the carrier has cardinality D + 2, where D is the gap derivation dimension.

The theorem is a statement about the mathematical structure of configurations, not about physics. It says nothing about which configuration is physically realized. The library explicitly keeps the physical readouts as targets: posting duration reads the recognition clock interval, and posting energy reads the recognition configuration price. These are stated as explicit, target-blind statements, not as proved theorems. The theorem also does not claim that the J-cost log aggregate is the only way to price configurations; it only shows that if the dimension is read from parent degrees, the configuration is forced.

The proof uses the fact that the degree carrier is a sum of a finite type of size D and a finite type of size 2, so its cardinality is D + 2. This is a purely combinatorial fact. The theorem then applies to any configuration that satisfies the readout condition. It does not depend on any particular choice of clock or channel. The library also proves that the canonical metric pricing semantics is a model of the recognition metric pricing semantics, but this is a separate statement.

The theorem is part of a larger chain that connects the structure of posting events to the forcing of dimensions. It is a key step in showing that the event-local configuration dimension is forced to be D + 2, which is the dimension of the degree carrier. This is a structural result that does not depend on any physical assumptions about which configuration is realized. The physical readouts remain open targets, and the theorem does not close them.

/-- Reading the event-local parent carrier derives S15's configuration join. -/
theorem parentDegreeReadout_implies_forcedConfiguration
    {N : ℕ} [NeZero N]
    {dimension : PostingConfigurationDimension3 N}
    (hread :
      PostingConfigurationReadsParentDegrees3 dimension) :
    PostingEventHasForcedConfigurationDimension3 dimension := by
  intro event
  rw [hread event,
    postingConfigurationDegreeCarrier_card]

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

The theorem does not prove that any physical configuration is realized. The theorem does not derive the value of the J-cost log aggregate from first principles. The theorem does not state that the recognition clock is the only clock that satisfies the native normalization condition.

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)

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