Foundation Pair Kernel Response Quotient Carrier S22 Consumer S22 Nonlinear Gaus

The compiled consumer

A consumer, in this framework, is a formal recipe that turns a physical response system into a concrete observable readout. The declaration s22_nonlinearGauss_tangentGreen_consumer_compiles establishes that a particular consumer, one built from a nonlinear Gauss law and a tangent Green function, still compiles unchanged after the underlying carrier is replaced by a quotient carrier. A quotient carrier is a reduced representation of the response system that identifies equivalent states, preserving only what the recognition ledger can distinguish. The declaration is a definition that points to an existing theorem, canonicalExactJTangentConsumer_exists, so its content is the existence of that consumer, not a new proof.

The surrounding file proves more than the compile claim. A theorem shows that on the quotient carrier there exists an event whose normalized posting duration equals 1, whose normalized posting energy and action both equal the canonical channel price, whose physical carrier dimension is exactly 5, and for which the carrier is both coherence-complete and response-observable. A second theorem establishes a universal property: any invariant map from the parent response system factors uniquely through the quotient. A third theorem discriminates between carriers: hidden extras collapse to a 5-channel quotient, observable extras enlarge it to 6, and incomplete, duplicate, or collapsed external carriers fail production identification.

The declaration itself does not claim that the nonlinear Gauss consumer is new, that it was re-proved, or that it is physically validated. It only records that the consumer compiles unchanged over the new carrier result. The docstring states that the S13 nonlinear Gauss law, tangent Hessian, and Green consumer compile unchanged; the S20 unit quotient and prediction-ready exact-J ratios also compile unchanged. The production identification remains a separate hypothesis, proved equivalent to external carrier completeness, not a theorem about the physical world.

What this means for a reader is that the framework's library, a machine-checked collection of formal theorems, has verified that a specific consumer construction remains valid when the carrier is quotiented. The result is a formal consistency check, not an empirical prediction. It does not say that the nonlinear Gauss law is true in nature, nor that the quotient carrier is the correct physical model. Those questions remain open.

/-- The S13 nonlinear Gauss law, tangent Hessian, and Green consumer compile
unchanged over the new carrier result. -/
def s22_nonlinearGauss_tangentGreen_consumer_compiles :=
  PairKernelExactJNonlinearGaussS13Consumer.canonicalExactJTangentConsumer_exists

/-- 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⟩

/-- 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

/-- 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⟩

What this page does not claim

The declaration does not prove that the nonlinear Gauss law is physically true. The declaration does not claim that the quotient carrier is the correct physical model. The declaration does not establish that the consumer is new or that it was re-proved.

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)

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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