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

ARTICLE 5 claims 3 theorems 1 model

Foundation Pair Kernel Response Quotient Carrier S22 Consumer S22 Scale Covarian

A machine-checked library proves a specific five-channel readout of a recognition ledger is coherent, while the physical identification of that ledger remains a separate, unproven hypothesis.

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The S20 consumer compilation

In Recognition Science, a ledger is a discrete record of events, and a recognition is the forced cost of registering one of those events. The declaration s22_scaleCovariant_S20_consumer_compiles is a definition in the framework's machine-checked library of formal theorems. It states that a specific, scale-covariant readout of a quotient carrier, an object built by grouping equivalent recognition responses, is a valid instance of the S20 observable-unit quotient. The definition simply points to an existing theorem, canonicalScaleCovariantConsumer_exists, which establishes the existence of such a readout.

What the underlying theorem establishes, in plain terms, is that for this constructed quotient carrier, there exists a realized posting event whose normalized duration is 1, whose normalized energy and action both equal the canonical channel price, and whose physical carrier dimension is exactly 5. The theorem also establishes that this carrier is coherent and observable. The docstring for the definition confirms that carrier observability is now a theorem for this object, while survival, J aggregation, and dimensional units retain their existing honest status as hypotheses or models.

The declaration also implies a universal property: any map from the parent response space that respects the invariant structure factors uniquely through the quotient. This is a standard categorical guarantee, meaning the quotient construction is the correct one for the given equivalence relation. The library further establishes a discriminator theorem: the quotient of a hidden-extra family has 5 elements, the quotient of an observable-extra family has 6, and three specific external carrier constructions (incomplete, extra, and misclassified five-channel) fail to be production-identifiable as the response quotient. Explicit map decoys show one omits the balance class and one collides two physical points.

In Recognition Science, the framework models the physical universe as a ledger of recognition events, and this declaration is a step in a chain of formal results. The library shows that a particular mathematical object, the quotient carrier, supports a coherent five-channel readout. This is a theorem about the internal consistency of the framework's construction. The declaration does not claim that this quotient carrier is the actual physical carrier of the universe. The production identification remains a separate HYPOTHESIS, equivalent to external carrier completeness, as the docstring states. The S20 observable-unit quotient and S13 nonlinear Gauss/tangent Green consumers compile unchanged over this new carrier result, but that is a statement about the formal library, not about empirical physics.

MODEL s22_scaleCovariant_S20_consumer_compiles · IndisputableMonolith/Foundation/PairKernelResponseQuotientCarrierS22Consumer.lean
s22_scaleCovariant_S20_consumer_compiles · IndisputableMonolith/Foundation/PairKernelResponseQuotientCarrierS22Consumer.lean:158 
/-- The S20 unit quotient compiles unchanged. -/
def s22_scaleCovariant_S20_consumer_compiles :=
  canonicalScaleCovariantConsumer_exists
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⟩
HYPOTHESIS 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⟩

What this page does not claim

This declaration does not prove that the quotient carrier is the physical carrier of the universe. It does not claim that survival, J aggregation, or dimensional units are theorems; they retain their existing honest status. It does not establish that the S20 readout is the only possible readout, only that this specific construction is valid.

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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)

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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