Foundation Pair Kernel Response Ancestry S21 Physical Probe Extensional Iff Resp

Probe extensionality

A physical measurement channel is a way of reading a response from an event. Two channels are observationally equivalent under a family of probes if every probe in that family returns the same result on both. The classical question is whether such observational equivalence forces the channels to be identical. The theorem physicalProbeExtensional_iff_responseDistinguishing answers this in a precise setting: a physical system's channels are extensional under the Recognition probe family if and only if the system's responses distinguish channels. Extensional here means that two channels which every probe treats alike must in fact be the same channel. Distinguishing means that for any two distinct channels, there is some response that separates them. The theorem proves these two conditions are equivalent, so one can check either one and get the other for free.

This is a theorem in the framework's machine-checked library of formal theorems. The framework models recognition events as discrete postings in a ledger, a record of committed acts. The theorem applies to a physical response system, which pairs events with carriers and produces responses. The proof relies on earlier results: every response is recognition-forced, meaning it traces back to a committed act, and the five operational responses form a coordinate basis with separating probes. The theorem itself is a biconditional, so it works in both directions. If channels are extensional, then responses distinguish them. If responses distinguish them, then channels are extensional.

What the theorem does not claim is just as important. It does not say that any particular physical system is extensional. It only says that extensionality and distinguishing are equivalent properties. A system could fail both. The library provides a concrete example: a collapsed five-carrier system fails both physical arrows, meaning it is neither extensional nor distinguishing. The theorem also does not identify points of an independently supplied physical carrier. The source response basis is complete, but it cannot create or identify points of a physical carrier that is supplied independently. Full parent, incomplete, extra, and collapsed physical response systems all read Recognition-forced responses while differing on physical realization or extensionality.

The practical consequence is a test. To check whether a physical system's channels are truly distinct under the Recognition probe family, one only needs to check whether the system's responses separate channels. The equivalence makes the second condition easier to verify in many cases, since responses are directly observable while channels are not. This is a working tool for anyone building a physical response system inside the framework: prove one side, get the other.

theorem physicalProbeExtensional_iff_responseDistinguishing
    {N : ℕ} [NeZero N]
    (system : PostingPhysicalResponseSystem3 N) :
    PhysicalChannelsAreProbeExtensional3 system ↔
      PostingResponsesDistinguishChannels3 system := by
  constructor
  · intro hextensional event left right hresponse
    apply hextensional event left right
    intro probe hprobe
    obtain ⟨responseProbe, rfl⟩ := hprobe
    change
      responseCoordinateObservation3 responseProbe
          (system.respond event left) =
        responseCoordinateObservation3 responseProbe
          (system.respond event right)
    rw [hresponse]
  · intro hdistinguish event left right hindistinguishable
    apply hdistinguish event
    apply responseCoordinateProbes_separate
    intro probe hprobe
    obtain ⟨responseProbe, rfl⟩ := hprobe
    exact
      hindistinguishable
        (fun channel =>
          responseCoordinateObservation3 responseProbe
            (system.respond event channel))
        ⟨responseProbe, rfl⟩

/-- Every member of the D+2 response catalog has committed Recognition
ancestry. -/
theorem every_response_is_recognitionForced
    {N : ℕ} [NeZero N]
    (event : RealizedPostingEvent3 N)
    (response : RecognitionParentResponse3) :
    RecognitionForcedResponse3 event response := by
  cases response with
  | spatial axis =>
      exact
        RecognitionForcedResponse3.spatial
          axis (spatialResponseProbe3_exists axis)
  | temporal kind =>
      fin_cases kind
      · exact
          RecognitionForcedResponse3.successor
            (successorResponseProbe3_exists event)
      · exact
          RecognitionForcedResponse3.balance
            (balanceResponseProbe3_exists event)

/-- The same-cardinality collapsed carrier fails both physical arrows even
though every response in the source catalog remains Recognition-forced. -/
theorem collapsedFiveCarrier_fails_both_physical_arrows :
    let system :=
      classifiedPhysicalResponseSystem3
        (misclassifiedFiveChannelCarrier3 3)
    ¬ PhysicalChannelsAreProbeExtensional3 system ∧
      ¬ RecognitionForcedResponsesArePhysicallyRealized3 system := by
  dsimp
  constructor
  · intro hextensional
    have hdistinguish :=
      (physicalProbeExtensional_iff_responseDistinguishing
        (classifiedPhysicalResponseSystem3
          (misclassifiedFiveChannelCarrier3 3))).1
        hextensional
    exact misclassifiedFiveCarrier_not_parentInjective
      ((classifiedResponses_distinguish_iff_parentInjective
        (misclassifiedFiveChannelCarrier3 3)).1
        hdistinguish)
  · intro hrealized
    have hrealize :=
      (forcedResponsesPhysicallyRealized_iff_realizeParents
        (classifiedPhysicalResponseSystem3
          (misclassifiedFiveChannelCarrier3 3))).1
        hrealized
    exact misclassifiedFiveCarrier_not_parentSurjective
      ((classifiedResponses_realize_iff_parentSurjective
        (misclassifiedFiveChannelCarrier3 3)).1
        hrealize)

What this page does not claim

This theorem does not claim that any particular physical system is extensional. This theorem does not claim to identify points of an independently supplied physical carrier.

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/PairKernelResponseAncestryS21.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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