Foundation Pair Kernel Production Effect Physicality S26 Consumer S26 S23 Produc

The production consumer

The declaration s26_S23_production_consumer_compiles is a named entry in the framework's machine-checked library of formal theorems. It packages a result about production effects, which are the framework's term for the distinguishable outcomes of a production operation. The core theorem, productionEffect_source_consumer, states that the source effect quotient has exactly five classes, that every operation has a probe that observes it, that the coordinate representation is injective, and that the quotient satisfies a universal property: any invariant map from operations factors uniquely through the quotient.

In plain terms, the theorem says that the five-class structure is not an arbitrary choice but is forced by the underlying definitions. The universal property is the technical way of saying the quotient is the most general way to identify operations that have the same effect. The declaration also includes a theorem, effectRealizedProduction_scaleCovariant_consumer_exists, that establishes physical readouts compile, but only under a specific condition: the exact effect realization arrow must be supplied. Without that arrow, the physical side does not follow.

In Recognition Science, this is part of a chain that connects abstract production effects to observable physical channels. The framework models a ledger of recognition events where cost is forced, and this declaration is a step in showing how the abstract structure can be realized physically. The theorem proves that if a system satisfies the realization condition, then there exists a realized posting event with normalized duration, energy, and action equal to the canonical channel price, and with a physical carrier dimension of five.

The declaration does not claim that physical readouts exist without the realization arrow. It does not claim that the five-class quotient is the only possible structure, only that it is the one forced by the definitions. It also does not claim that the realization arrow itself is derived; it is a supplied condition. The theorem is conditional: given the arrow, the physical readouts follow.

What this changes is the status of the physical side of the framework. The abstract five-class structure is proved, and the physical readouts are proved conditional on a single supplied arrow. That is a precise boundary: the framework knows exactly where its derivation stops and where a definitional choice begins.

/-- The complete source-effect result consumed as one package. -/
theorem productionEffect_source_consumer :
    Fintype.card ProductionOperationEffectClass3 = 5 ∧
      (∀ operation : ProductionSourceOperation3,
        ∃ probe : ProductionOperationEffectProbe3,
          productionOperationEffectObservation3 probe operation = 1) ∧
      Function.Injective
        productionOperationEffectCoordinateRepresentation3 ∧
      (∀ {β : Type*}
          (map : ProductionSourceOperation3 → β),
        ProductionOperationEffectInvariantMap3 map →
          ∃! descended :
              ProductionOperationEffectClass3 → β,
            ∀ operation,
              descended
                  (productionOperationEffectClass3 operation) =
                map operation) :=
  ⟨productionOperationEffectClass_card_eq_five,
    everyProductionOperation_has_nontrivialEffect,
    productionOperationEffectCoordinateRepresentation_injective,
    productionOperationEffectInvariantMap_factors_unique⟩

/-- S17-S20 physical readouts compile once the exact effect-to-channel
realization arrow is supplied. -/
theorem effectRealizedProduction_scaleCovariant_consumer_exists
    (system : PostingPhysicalResponseSystem3.{0} 3)
    (hrealize :
      ProductionEffectsRealizePhysicalChannels3 system) :
    ∃ event : RealizedPostingEvent3 3,
      normalizedPostingDuration3
          (observableProductionScaleCovariantPostingReadoutSemantics3
            system
            ((productionOperationSelectors_iff_observableExhaustion
              system).1
              ((productionEffectsRealizeChannels_iff_operationSelectors
                system).1 hrealize)))
          event.1 = 1 ∧
        normalizedPostingEnergy3
          (observableProductionScaleCovariantPostingReadoutSemantics3
            system
            ((productionOperationSelectors_iff_observableExhaustion
              system).1
              ((productionEffectsRealizeChannels_iff_operationSelectors
                system).1 hrealize)))
          event.1 =
            canonicalPostingEventChannelPrice3 event ∧
        normalizedPostingAction3
          (observableProductionScaleCovariantPostingReadoutSemantics3
            system
            ((productionOperationSelectors_iff_observableExhaustion
              system).1
              ((productionEffectsRealizeChannels_iff_operationSelectors
                system).1 hrealize)))
          event.1 =
            canonicalPostingEventChannelPrice3 event ∧
        physicalPostingCarrierDimension3
          (observableProductionPhysicalChannelCarrier3
            system
            ((productionOperationSelectors_iff_observableExhaustion
              system).1
              ((productionEffectsRealizeChannels_iff_operationSelectors
                system).1 hrealize)))
          event = 5 ∧
        PostingCarrierResponseObservability3
          (observableProductionResponseSystem3
            system
            ((productionOperationSelectors_iff_observableExhaustion
              system).1
              ((productionEffectsRealizeChannels_iff_operationSelectors
                system).1 hrealize))) := by
  exact
    operationSelectedProduction_scaleCovariant_consumer_exists
      system
      ((productionEffectsRealizeChannels_iff_operationSelectors
        system).1 hrealize)

What this page does not claim

Physical readouts exist without the exact effect realization arrow. The realization arrow itself is derived rather than supplied. The five-class quotient is the only possible structure, rather than the one forced by the definitions.

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/PairKernelProductionEffectPhysicalityS26Consumer.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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