Encyclopedia Foundation Foundation Pair Kernel Response Quotient Carrier S22 Consumer Quotient Discrimin

ARTICLE 4 claims 4 theorems

Foundation Pair Kernel Response Quotient Carrier S22 Consumer Quotient Discrimin

A machine-checked theorem separates the one correct way to count recognition channels from five tempting impostors.

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The discriminator theorem

In mathematics, a quotient is a way of grouping things together: you decide which objects count as equivalent, and then you study the groups rather than the individual objects. The Recognition Science declaration quotientDiscriminator_consumer is a theorem about one specific quotient: the one built from a ledger, a discrete record of recognition events. It proves, in a machine-checked library of formal theorems, that this particular quotient has exactly five channels, and that five alternative constructions fail to be it.

The theorem states four facts together. First, the quotient built from hidden extras has five channels. Second, the quotient built from observable extras has six. Third, three external carriers, each missing something or adding something, are each provably not the recognition quotient. A carrier that omits the balance class, one that collides two physical points, and one that mislabels a five-channel object all fail the test. The theorem's companion result makes the failure concrete: the incomplete carrier's map is not surjective, the extra carrier's map is not injective, and the equality quotient keeps a duplicate that the real quotient collapses.

The force of the theorem is discrimination. It does not merely assert that the recognition quotient exists; it proves that a specific list of plausible alternatives does not match it. This matters because the framework's later claims depend on which carrier is the real one. The theorem pins down the object so that later steps do not accidentally use a decoy. The five-channel count is the punchline: the recognition quotient exposes exactly five channels, no more and no fewer.

What the theorem does not claim is just as important. It does not claim that these five channels are the three spatial dimensions, or that they correspond to any particular physical measurement. It does not claim that the production identification, the link between this formal carrier and the external world, is proved; that remains a separate hypothesis. The theorem is a statement about the internal structure of the quotient, not about what that structure means physically. It establishes the carrier's shape, not its interpretation.

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 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 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 theorem does not identify the five channels with any specific physical dimensions or measurements. The theorem does not prove the production identification between the formal carrier and external reality. The theorem does not establish that the recognition quotient is the only possible carrier for all purposes.

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

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