Foundation Pair Kernel Production Event Response Generation S24 Consumer S24 S22

The quotient consumer

A quotient is a way of grouping things together so that items in the same group are treated as equivalent. The declaration s24_S22_quotient_consumer_compiles is a definition in the framework's machine-checked library of formal theorems. It states that a particular quotient construction, called the pair-kernel response quotient carrier, compiles unchanged from an earlier stage of the framework. In plain terms: the framework proves that five distinct response classes can be mapped onto a single canonical quotient structure, and that this mapping is unique.

The underlying theorem, productionEventResponse_sourceCatalog_consumer, establishes two facts. First, every possible response in the recognition parent system has at least one corresponding action in the production event response act type. Second, the number of response classes is exactly five. The theorem also proves a universal property: for any map from the action type to some other type that respects the invariant structure, there exists a unique map from the five response classes to that other type that factors through the quotient projection. This means the quotient captures all the information that any invariant map could extract from the five classes.

The definition itself does no new mathematical work. It is a named alias that points to an existing theorem from an earlier module, PairKernelResponseQuotientCarrierS22Consumer.quotientCarrier_scaleCovariant_consumer_exists. The point of the alias is architectural: the framework's later stages can refer to this consumer by a stable name without re-proving the underlying result. The library shows that the S22 quotient consumer, which handles the scale-covariant readout of the quotient carrier, remains available in the S24 production event response generation stage.

What the declaration does not claim is equally important. It does not claim that any particular physical system satisfies the conditions needed to use this quotient. It does not claim that the quotient construction applies to all possible response systems. It does not claim that the five response classes are physically realized in nature. The theorem is conditional: it describes a structure that exists in the framework's formal universe, not a statement about which physical systems instantiate that structure.

The framework's own documentation is explicit about this limit. The source catalog and its universal property compile as one package without selecting a physical response system. The physical residual is not silently closed: committed ancestry still admits systems that split on source-act transport. This means the framework acknowledges that the formal structure does not by itself determine which physical systems will use it. The declaration is a piece of mathematical infrastructure, not a physical prediction.

/-- S22 quotient consumer compiles unchanged. -/
def s24_S22_quotient_consumer_compiles :=
  PairKernelResponseQuotientCarrierS22Consumer.quotientCarrier_scaleCovariant_consumer_exists

/-- The complete source witness catalog and its universal property compile as
one package without selecting a physical response system. -/
theorem productionEventResponse_sourceCatalog_consumer :
    (∀ response : RecognitionParentResponse3,
      ∃ act : ProductionEventResponseAct3,
        productionEventResponseAct_read act = response) ∧
      Fintype.card ProductionEventResponseClass3 = 5 ∧
      (∀ {β : Type*}
          (map : ProductionEventResponseAct3 → β),
        ProductionEventResponseInvariantMap3 map →
          ∃! descended :
              ProductionEventResponseClass3 → β,
            ∀ act,
              descended
                  (PrimitiveRecognitionCalculus.QuotientSelection.proj
                    productionEventResponseProbeFamily3 act) =
                map act) :=
  ⟨every_response_has_productionEventAct,
    productionEventResponseClass_card_eq_five,
    productionEventResponseInvariantMap_factors_unique⟩

/-- The complete source witness catalog and its universal property compile as
one package without selecting a physical response system. -/
theorem productionEventResponse_sourceCatalog_consumer :
    (∀ response : RecognitionParentResponse3,
      ∃ act : ProductionEventResponseAct3,
        productionEventResponseAct_read act = response) ∧
      Fintype.card ProductionEventResponseClass3 = 5 ∧
      (∀ {β : Type*}
          (map : ProductionEventResponseAct3 → β),
        ProductionEventResponseInvariantMap3 map →
          ∃! descended :
              ProductionEventResponseClass3 → β,
            ∀ act,
              descended
                  (PrimitiveRecognitionCalculus.QuotientSelection.proj
                    productionEventResponseProbeFamily3 act) =
                map act) :=
  ⟨every_response_has_productionEventAct,
    productionEventResponseClass_card_eq_five,
    productionEventResponseInvariantMap_factors_unique⟩

/-- The complete source witness catalog and its universal property compile as
one package without selecting a physical response system. -/
theorem productionEventResponse_sourceCatalog_consumer :
    (∀ response : RecognitionParentResponse3,
      ∃ act : ProductionEventResponseAct3,
        productionEventResponseAct_read act = response) ∧
      Fintype.card ProductionEventResponseClass3 = 5 ∧
      (∀ {β : Type*}
          (map : ProductionEventResponseAct3 → β),
        ProductionEventResponseInvariantMap3 map →
          ∃! descended :
              ProductionEventResponseClass3 → β,
            ∀ act,
              descended
                  (PrimitiveRecognitionCalculus.QuotientSelection.proj
                    productionEventResponseProbeFamily3 act) =
                map act) :=
  ⟨every_response_has_productionEventAct,
    productionEventResponseClass_card_eq_five,
    productionEventResponseInvariantMap_factors_unique⟩

What this page does not claim

The declaration does not claim that any particular physical system satisfies the conditions needed to use this quotient. The declaration does not claim that the five response classes are physically realized in nature. The declaration does not claim that the quotient construction applies to all possible response systems.

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/PairKernelProductionEventResponseGenerationS24Consumer.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.

Derived articles

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