Foundation Pair Kernel Executable Effect Physical Existence S27

What S27 establishes

The module recognition (a discrete record of events) proves a clean separation of three existence claims that earlier work had conflated. First, every operation effect has a class in the observational quotient. Second, every such class has an explicit operation representative and therefore a realized posting event. Third, a separately supplied raw physical ontology can carry every effect, but this realization is independent of the committed source signature. The module proves all three claims with no sorry and no new axiom.

The central construction is the executableEffectPhysicalResponseSystem3. This system uses the effect quotient itself as its event-local carrier, and each effect is realized by its own quotient class. The module proves that this system satisfies four key properties: it selects operation channels, transports every event response act, exhausts all physical observable states, and is identified with the response quotient. These four properties together mean that the executable effects are not abstract symbols but have concrete physical instances within the framework's own semantics.

A notable theorem is executableEffectPhysicalCarrier_dimension_eq_five, which proves that the physical posting carrier dimension for this system is exactly 5. This is a specific, non-trivial numerical result that emerges from the construction. The module also contains a family of theorems about the impossibility of silent effects and the distinguishability of distinct effect classes, which together establish that the observational structure is rich enough to tell every effect apart.

The module also studies what happens when you try to add hidden state. A theorem named hiddenState_collapses_observationally proves that a hidden "ghost" state is observationally indistinguishable from the base state. Another theorem, observableExtra_is_detected_and_enlarges, shows that if you instead add an observable extra state, it is detected and enlarges the quotient from five to six classes. This contrast sharpens the boundary between what is physically real and what is merely a bookkeeping artifact.

Finally, the module proves that a separately supplied raw physical ontology can realize all effects, but that this realization remains independent of the committed source signature. The theorem externalRawOntology_realization_remains_independent shows that there exist two different physical response systems, one that realizes all effects and one that does not. This establishes that the framework's claims about executable effects do not depend on a specific choice of raw physical ontology.

/-- The weakest principle follows from the committed Recognition
`PhysicalQuotient`: use the operation's quotient projection, then use S26
effect extensionality for uniqueness. -/
theorem witnessedProductionEffects_determine_physicalObservationClasses :
    WitnessedProductionEffectsDeterminePhysicalObservationClasses3 := by
  intro operation _hoperation
  refine
    ⟨productionOperationEffectClass3 operation, ?_, ?_⟩
  · intro probe
    rfl
  · intro other hother
    apply productionOperationEffectCoordinateRepresentation_injective
    funext probe
    simpa [productionOperationEffectCoordinateRepresentation3] using
      hother probe

theorem executableEffectPhysicalCarrier_dimension_eq_five
    (event : RealizedPostingEvent3 3) :
    physicalPostingCarrierDimension3
        executableEffectPhysicalChannelCarrier3 event = 5 := by
  unfold physicalPostingCarrierDimension3
    executableEffectPhysicalChannelCarrier3
  exact productionOperationEffectClass_card_eq_five

theorem executableEffects_select_operationChannels :
    ProductionOperationsSelectPhysicalChannels3
      executableEffectPhysicalResponseSystem3 :=
  (productionEffectsRealizeChannels_iff_operationSelectors
    executableEffectPhysicalResponseSystem3).1
      executableEffects_have_physicalInstances

theorem executableEffects_transport_eventResponseActs :
    ProductionTransportsEveryEventResponseAct3
      executableEffectPhysicalResponseSystem3 :=
  (productionEffectsRealizeChannels_iff_eventActTransport
    executableEffectPhysicalResponseSystem3).1
      executableEffects_have_physicalInstances

theorem executableEffects_exhaust_physicalObservableStates :
    PhysicalObservableStatesAreRecognitionClasses3
      executableEffectPhysicalResponseSystem3 :=
  (productionEffectsRealize_iff_observableExhaustion
    executableEffectPhysicalResponseSystem3).1
      ((productionEffectsRealizeChannels_iff_observableClasses
        executableEffectPhysicalResponseSystem3).1
          executableEffects_have_physicalInstances)

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

/-- An unobservable hidden state collapses in the Recognition physical
quotient. -/
theorem hiddenState_collapses_observationally :
    proj hiddenExtraProbeFamily3 ExtraResponseState3.ghost =
      proj hiddenExtraProbeFamily3
        (ExtraResponseState3.base spatialZeroResponse3) :=
  hiddenExtra_ghost_collapses

/-- 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

This module does not claim that the five-dimensional carrier is the three spatial dimensions plus time. This module does not claim that the raw physical ontology is uniquely determined or that it is identified with the executable effect system. This module does not claim to derive any specific physical constants or forces.

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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