Foundation Pair Kernel Executable Effect Physical Existence S27 Executable Effec

Observational equivalence

Observational equivalence is a standard idea in physics and computer science: two systems are equivalent when every possible experiment or probe returns the same result for both. In Recognition Science, the framework models physical reality as a ledger, a discrete record of posting events, and it treats the observable states of a system as equivalence classes of that ledger's internal states. Two internal states belong to the same class when no probe can distinguish them.

The framework's machine-checked library of formal theorems proves a central result about this setup: the property of being a realized executable effect is invariant under observational equivalence. In plain language, if an effect is physically realized by one internal state, and a second internal state is observationally equivalent to the first, then the effect is also realized by the second. The theorem is anchored by the declaration executableEffect_realization_invariant_under_observationalEquivalence, and it follows from the injectivity of the response map, which the library proves in productionOperationEffectClass_response_injective.

This invariance has a sharp consequence for the framework's canonical construction. The library builds a system where each effect is realized by its own equivalence class, and it proves that this realization is unique: no two distinct classes can realize the same effect in the same event. The theorem duplicateCanonicalRealization_is_impossible states this directly, and it is what licenses the framework to select operations, transport event responses, and exhaust observable states on the strength of the quotient alone.

What the theorem does not claim is just as important. It does not say that a separately supplied raw physical ontology, one chosen independently of the framework's own construction, is identified with that quotient. The library keeps that realization claim separate and weaker: it proves that such an external ontology can carry every effect, but it does not prove that the external carrier is the same as the framework's canonical one. The theorem externalRawOntology_realization_remains_independent states this independence explicitly, and the framework flags the gap as an open target rather than a proved identity.

theorem executableEffectPhysicalSystem_identifiedWithResponseQuotient :
    ProductionResponseSystemIsResponseQuotient3
      executableEffectPhysicalResponseSystem3 :=
  (productionSystemIdentified_iff_complete
    executableEffectPhysicalResponseSystem3).2
      executableEffectPhysicalCarrier_complete

/-- Raw duplicate realization is absent in the canonical effect-generated
system, independently of its cardinality. -/
theorem duplicateCanonicalRealization_is_impossible
    (target : RealizedPostingEvent3 3)
    (effect left right : ProductionOperationEffectClass3)
    (hleft :
      PhysicalChannelCarriesProductionEffect3
        executableEffectPhysicalResponseSystem3 target left effect)
    (hright :
      PhysicalChannelCarriesProductionEffect3
        executableEffectPhysicalResponseSystem3 target right effect) :
    left = right := by
  exact
    (executableEffect_channelCarries_iff_eq target left effect).1 hleft |>.trans
      ((executableEffect_channelCarries_iff_eq target right effect).1 hright).symm

/-- The exact remaining external-ontology boundary. Committed effects and
their canonical physical observational realization coexist with external
systems that do and do not realize them. Thus S27 does not promote a
separately supplied raw ontology. -/
theorem externalRawOntology_realization_remains_independent :
    CommittedProductionOperations3 ∧
      ∃ left right : PostingPhysicalResponseSystem3.{0} 3,
        ProductionEffectsRealizePhysicalChannels3 left ∧
          ¬ ProductionEffectsRealizePhysicalChannels3 right :=
  committedEffects_admit_physicalRealization_split

What this page does not claim

The theorem does not identify any separately supplied raw physical carrier with the framework's canonical observational quotient. The theorem does not prove that every physical system's observable states are exhausted by the executable effects; that is a separate theorem with its own conditions.

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/PairKernelExecutableEffectPhysicalExistenceS27.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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