Encyclopedia Foundation Foundation Pair Kernel Response Quotient Carrier S22 Consumer Quotient Scale Cov

ARTICLE 4 claims 4 theorems

Foundation Pair Kernel Response Quotient Carrier S22 Consumer Quotient Scale Cov

A machine-checked theorem shows that a particular way of grouping recognition events yields exactly five observable channels, and that no other grouping can do the same job.

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The quotient readout

A quotient is a way of grouping things so that items in the same group are treated as equal for a given purpose. In Recognition Science, the framework's ledger, a discrete record of events, groups recognition events into equivalence classes. The declaration quotientScaleCovariantPostingReadoutSemantics3 is a definition that builds a specific readout device on top of such a grouping. The readout is called scale-covariant because it transforms consistently when the units of measurement change.

The framework's machine-checked library of formal theorems proves that this readout works. The theorem quotientCarrier_scaleCovariant_consumer_exists establishes that there is at least one posting event whose normalized duration equals one, whose normalized energy and action both equal the event's channel price, whose carrier has dimension five, and which satisfies two completeness and observability conditions. In plain terms, the readout is internally consistent, exposes exactly five channels, and is fully observable. The theorem quotientCarrier_universalProperty_consumer goes further: it proves that any map from the original responses to another space that respects the grouping factors uniquely through the quotient. This is a universal property, meaning the quotient is the canonical way to collapse the responses without losing the structure that the grouping preserves.

The framework also proves which groupings fail. The theorem quotientDiscriminator_consumer shows that a hidden-extra grouping produces five channels, an observable-extra grouping produces six, and three external carrier constructions (incomplete, extra, and misclassified five-channel) fail the production identification test. The theorem explicitQuotientMapDiscriminators_consumer gives concrete counterexamples: one external map is not surjective, meaning it omits a class; another is not injective, meaning it collides two distinct points. These results show that the five-channel quotient is not an arbitrary choice but the unique grouping that satisfies the framework's production criteria.

In Recognition Science, this result is a step toward showing that the framework's internal constructions match its external predictions. The production identification remains a separate HYPOTHESIS, proved equivalent to external carrier completeness. The declaration does not claim that the five-channel structure is physically realized in nature, nor that the quotient is the only possible mathematical construction. It claims that within the framework's axioms, this particular readout is the canonical one, and that the framework's library proves its existence and uniqueness properties.

THEOREM quotientCarrier_scaleCovariant_consumer_exists · IndisputableMonolith/Foundation/PairKernelResponseQuotientCarrierS22Consumer.lean
quotientCarrier_scaleCovariant_consumer_exists · IndisputableMonolith/Foundation/PairKernelResponseQuotientCarrierS22Consumer.lean:59 
/-- The quotient carrier supports the normalized S20 readout and exposes
exactly five channels. -/
theorem quotientCarrier_scaleCovariant_consumer_exists :
    ∃ event : RealizedPostingEvent3 3,
      normalizedPostingDuration3
          quotientScaleCovariantPostingReadoutSemantics3
          event.1 = 1 ∧
        normalizedPostingEnergy3
          quotientScaleCovariantPostingReadoutSemantics3
          event.1 =
            canonicalPostingEventChannelPrice3 event ∧
        normalizedPostingAction3
          quotientScaleCovariantPostingReadoutSemantics3
          event.1 =
            canonicalPostingEventChannelPrice3 event ∧
        physicalPostingCarrierDimension3
          (responseQuotientPhysicalChannelCarrier3 3)
          event = 5 ∧
        PostingCarrierCoherenceComplete3
          (responseQuotientPhysicalChannelCarrier3 3) ∧
        PostingCarrierResponseObservability3
          (responseQuotientPhysicalResponseSystem3 3) := by
  obtain ⟨pair, hpair⟩ :=
    realizedPrimitivePostingPair3_exists
  let event : RealizedPostingEvent3 3 :=
    ⟨pair, hpair⟩
  refine
    ⟨event,
      normalizedPostingDuration_eq_one
        quotientScaleCovariantPostingReadoutSemantics3
        event,
      normalizedPostingEnergy_eq_price
        quotientScaleCovariantPostingReadoutSemantics3
        event,
      normalizedPostingAction_eq_price
        quotientScaleCovariantPostingReadoutSemantics3
        event,
      responseQuotientCarrier_dimension_eq_five event,
      responseQuotientCarrier_complete,
      responseQuotient_responseObservability⟩
THEOREM quotientCarrier_universalProperty_consumer · IndisputableMonolith/Foundation/PairKernelResponseQuotientCarrierS22Consumer.lean
quotientCarrier_universalProperty_consumer · IndisputableMonolith/Foundation/PairKernelResponseQuotientCarrierS22Consumer.lean:100 
/-- Consumer-level universal property. -/
theorem quotientCarrier_universalProperty_consumer
    {β : Type*}
    (map : RecognitionParentResponse3 → β)
    (hinvariant : ResponseInvariantMap3 map) :
    ∃! descended : RecognitionResponseQuotient3 → β,
      ∀ response,
        descended
            (PrimitiveRecognitionCalculus.QuotientSelection.proj
              responseCoordinateProbeFamily3 response) =
          map response :=
  responseInvariantMap_factors_unique map hinvariant
THEOREM quotientDiscriminator_consumer · IndisputableMonolith/Foundation/PairKernelResponseQuotientCarrierS22Consumer.lean
quotientDiscriminator_consumer · IndisputableMonolith/Foundation/PairKernelResponseQuotientCarrierS22Consumer.lean:113 
/-- Hidden extras collapse; observable extras enlarge the quotient; external
incomplete, duplicate, and collapsed carriers fail production identification. -/
theorem quotientDiscriminator_consumer :
    Fintype.card
          (PrimitiveRecognitionCalculus.QuotientSelection.PhysicalQuotient
            hiddenExtraProbeFamily3) = 5 ∧
      Fintype.card
          (PrimitiveRecognitionCalculus.QuotientSelection.PhysicalQuotient
            observableExtraProbeFamily3) = 6 ∧
      (¬ ProductionCarrierIsResponseQuotient3
        (incompletePhysicalChannelCarrier3 3)) ∧
      (¬ ProductionCarrierIsResponseQuotient3
        (extraPhysicalChannelCarrier3 3)) ∧
      (¬ ProductionCarrierIsResponseQuotient3
        (misclassifiedFiveChannelCarrier3 3)) :=
  ⟨hiddenExtraQuotient_card_eq_five,
    observableExtraQuotient_card_eq_six,
    incompleteCarrier_not_responseQuotientIdentified,
    hiddenExtraCarrier_not_responseQuotientIdentified,
    collapsedFiveCarrier_not_responseQuotientIdentified⟩
THEOREM explicitQuotientMapDiscriminators_consumer · IndisputableMonolith/Foundation/PairKernelResponseQuotientCarrierS22Consumer.lean
explicitQuotientMapDiscriminators_consumer · IndisputableMonolith/Foundation/PairKernelResponseQuotientCarrierS22Consumer.lean:134 
/-- The requested external map decoys are explicit: one omits the balance
class and one collides two physical points. The equality quotient also keeps
an unobservable duplicate that the committed quotient collapses. -/
theorem explicitQuotientMapDiscriminators_consumer
    (event : RealizedPostingEvent3 3) :
    (¬ Function.Surjective
      (externalCarrierToResponseQuotient3
        (incompletePhysicalChannelCarrier3 3)
        event)) ∧
      (¬ Function.Injective
        (externalCarrierToResponseQuotient3
          (extraPhysicalChannelCarrier3 3)
          event)) ∧
      equalityExtraResponseProjection3
          ExtraResponseState3.ghost ≠
        equalityExtraResponseProjection3
          (ExtraResponseState3.base
            spatialZeroResponse3) :=
  ⟨incompleteCarrierToResponseQuotient_not_surjective
      event,
    extraCarrierToResponseQuotient_not_injective
      event,
    equalityQuotient_duplicates_unobservable_spatialClass.1⟩

What this page does not claim

The five-channel structure is physically realized in nature. The quotient is the only possible mathematical grouping of recognition events. The production identification is proved; it remains a hypothesis.

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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/PairKernelResponseQuotientCarrierS22Consumer.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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