Foundation Pair Kernel Production Quotient Identification S23 Consumer Hidden Im

The hidden implementation theorem

The declaration hiddenImplementation_consumer is a theorem in the framework's machine-checked library of formal theorems. It concerns a specific constructed example of a production response system, a formal model of a physical system that records and responds to events. The theorem establishes three facts about this example. First, the system's observable states, the states a detector could actually distinguish, do form recognition classes, which are the equivalence classes of events that the system treats as the same. Second, the raw physical carrier, the underlying state space before any quotient is taken, does not itself form recognition classes. Third, for every possible event, the quotient that identifies events with the same observable outcome has exactly five classes.

The point of the theorem is to show that the framework's five-class physical carrier, the structure on which its readout surfaces are compiled, is compatible with a hidden implementation that has more internal states. The example uses six internal states. The theorem proves that such a six-state system can still present exactly five observable classes after the quotient is taken. This is a conditional result: it holds for a specific constructed system, not for all systems. The declaration also compiles several other consumers, which are the framework's interfaces for readout semantics, and these compile unchanged on the resulting five-class carrier.

What the theorem does not claim is equally important. The framework's own documentation states that no committed declaration selects such a production response system. The realization premise, the assumption that any real physical system instantiates this structure, remains a HYPOTHESIS for external production physics. The theorem is about a formal model, not about any specific measured physical system. It does not claim that any real system has six hidden states, nor that the five-class quotient is physically realized. It establishes a consistency result: the framework's five-class observable structure can accommodate a hidden implementation with more states, and the readout surfaces compile on that carrier.

The theorem matters because it addresses a potential objection to the framework's structure. If the framework forces a five-class observable carrier, one might ask whether it can handle systems whose internal state space is larger. This theorem shows that it can, at least for the constructed example. The hidden implementation is quotiented away, and the observable structure remains five classes. The result is a formal consistency check, not an empirical claim about any particular physical system.

/-- A six-point hidden implementation supplies a concrete production
presentation whose physical observational quotient still has five classes. -/
theorem hiddenImplementation_consumer :
    PhysicalObservableStatesAreRecognitionClasses3
        (hiddenImplementationResponseSystem3 3) ∧
      ¬ RawPhysicalCarrierIsRecognitionClasses3
        (hiddenImplementationResponseSystem3 3) ∧
      ∀ event : RealizedPostingEvent3 3,
        Fintype.card
          (ProductionObservableStateQuotient3
            (hiddenImplementationResponseSystem3 3)
            event) = 5 :=
  ⟨hiddenImplementation_observableStatesExhaustive,
    hiddenImplementation_rawCarrier_not_identified,
    hiddenImplementation_observableQuotient_card_eq_five⟩

/-- The constructed S22 quotient system exhausts its own Recognition classes. -/
theorem constructedQuotient_observableStatesExhaustive :
    PhysicalObservableStatesAreRecognitionClasses3
      (responseQuotientPhysicalResponseSystem3 3) :=
  (physicalObservableStates_iff_responsesRealize
    (responseQuotientPhysicalResponseSystem3 3)).2
      responseQuotient_responsesRealize

/-- The active source-side obstruction: committed ancestry admits systems that
split on physical observable exhaustion. -/
def s23_observableExhaustion_independence_compiles :=
  committed_response_ancestry_admits_observableExhaustion_split

/-- A six-point hidden implementation supplies a concrete production
presentation whose physical observational quotient still has five classes. -/
theorem hiddenImplementation_consumer :
    PhysicalObservableStatesAreRecognitionClasses3
        (hiddenImplementationResponseSystem3 3) ∧
      ¬ RawPhysicalCarrierIsRecognitionClasses3
        (hiddenImplementationResponseSystem3 3) ∧
      ∀ event : RealizedPostingEvent3 3,
        Fintype.card
          (ProductionObservableStateQuotient3
            (hiddenImplementationResponseSystem3 3)
            event) = 5 :=
  ⟨hiddenImplementation_observableStatesExhaustive,
    hiddenImplementation_rawCarrier_not_identified,
    hiddenImplementation_observableQuotient_card_eq_five⟩

What this page does not claim

The theorem does not claim that any real physical system has six hidden internal states. The theorem does not claim that the five-class observable quotient is physically realized in any measured system. The theorem does not claim that all production response systems with hidden implementations present exactly five observable classes.

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/PairKernelProductionQuotientIdentificationS23Consumer.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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