Foundation Pair Kernel Production Effect Physicality S26 Consumer Effect Realize

The realized effect theorem

In the Recognition Science framework, a ledger is a discrete record of events, and a recognition is the act of recording one. The theorem named effectRealizedProduction_scaleCovariant_consumer_exists states a conditional existence result: if a physical response system realizes production effects as physical channels, then there exists a realized posting event whose normalized duration is 1, whose normalized energy equals the canonical channel price, whose normalized action equals that same price, whose physical carrier dimension is 5, and whose response system is observable.

The statement is a theorem in the framework's machine-checked library of formal theorems, meaning it is proved from the framework's axioms with no unverified assumptions. Its proof relies on three prior equivalences: production effects realizing channels is equivalent to operation selectors, operation selectors are equivalent to observable exhaustion, and these together yield the existence of the event. The theorem does not say that such a physical system exists; it says that if one does, then the event with those properties follows.

The theorem is part of a chain labeled S26, which compiles earlier results S13 through S25 into a single package. The chain includes a separate theorem, productionEffect_source_consumer, proving that the source-effect quotient has exactly five classes, that every operation has a probe observing it, that the coordinate representation is injective, and that a universal property holds for maps from operations. These are structural facts about the abstract effect quotient, independent of physical realization.

What the theorem does not claim is as important as what it proves. It does not claim that the physical realization arrow exists; that is a premise, not a conclusion. It does not claim that the realized event is unique, only that one exists. It does not claim that the carrier dimension 5 corresponds to any particular physical space; the number is a property of the formal construction. It does not claim that the canonical price is observable or measurable in any experimental sense; the equality is a formal identity within the framework.

The practical consequence is that within the framework, the step from abstract effects to physical readouts is not a leap of faith. Given the realization premise, the readout properties are forced by the earlier equivalences. The theorem thereby closes a gap in the framework's derivation: the physical side is not an additional assumption but a consequence of the effect structure, conditional on the realization arrow.

/-- 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)

/-- 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⟩

What this page does not claim

The theorem does not claim that the physical realization arrow exists; it is a premise. The theorem does not claim the realized event is unique, only that one exists. The theorem does not claim the carrier dimension 5 corresponds to any particular physical space. The theorem does not claim the canonical price is experimentally measurable; the equality is formal.

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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