Foundation Pair Kernel Response Ancestry S21 Forced Responses Physically Realize

The realization equivalence

A recognition event in this framework is a discrete record of a change, and it forces a small fixed catalog of responses: the spatial, temporal, successor, and balance responses. The question is whether those forced responses must have a physical representative, something in the world that carries them. The theorem forcedResponsesPhysicallyRealized_iff_realizeParents answers that question with an equivalence: the forced responses are physically realized if and only if the physical responses realize their parent configurations. In plainer language, the abstractly forced responses get a physical carrier exactly when the physical system's responses trace back to the parent states that the recognition event commits to.

The equivalence is proved in the framework's machine-checked library of formal theorems, with no unproved assumptions and no new axioms. It is a theorem about a specific structure: a posting physical response system, which pairs each recognition event with a set of physical channels, and each channel with a response. The theorem says the two properties are interchangeable. If every forced response has some channel that produces it, then the responses realize their parents, and conversely. This is a tight logical link, not a loose analogy.

The theorem does not say that the physical carrier itself is created or identified by the recognition event. The source semantics, the committed recognition acts, are complete on the response catalog, but they cannot create or pick out points of an independently supplied physical carrier. Full, incomplete, extra, and collapsed physical response systems all read the recognition-forced responses while differing on physical realization. The forcing predicate is total on the five-response catalog: it contributes source-side ancestry but does not shrink the physical obligations. The two remaining physical arrows are exactly separation and realization, and this theorem settles the realization arrow.

What the theorem does not claim is as important as what it proves. It does not claim that every physical system must realize the forced responses; it only says that if a system does, that is equivalent to the responses realizing their parents. It does not claim that physical channels are distinguishable by probes; that is a separate property, proved in a companion theorem. And it does not claim that the physical carrier is derived from the recognition event. The equivalence is a bridge between two descriptions, not a creation story.

theorem forcedResponsesPhysicallyRealized_iff_realizeParents
    {N : ℕ} [NeZero N]
    (system : PostingPhysicalResponseSystem3 N) :
    RecognitionForcedResponsesArePhysicallyRealized3
        system ↔
      PostingResponsesRealizeParents3 system := by
  constructor
  · intro hforced event parent
    exact hforced event
      (parentTagResponse3 parent)
      (every_response_is_recognitionForced
        event (parentTagResponse3 parent))
  · intro hrealize event response _hforced
    obtain ⟨channel, hchannel⟩ :=
      hrealize event
        (responseParentTag3 response)
    refine ⟨channel, ?_⟩
    rw [hchannel,
      parentTagResponse_responseParentTag]

/-- Every Recognition-forced response has one unique parent channel. This is
source realization and unique channel decomposition, not physical carrier
surjectivity. -/
theorem recognitionForcedResponse_unique_parent
    {N : ℕ} [NeZero N]
    {event : RealizedPostingEvent3 N}
    (response : RecognitionParentResponse3)
    (_hforced :
      RecognitionForcedResponse3 event response) :
    ∃! parent :
        PostingConfigurationDegreeCarrier3
          (realizedPostingEventConfiguration3 event),
      @parentTagResponse3 N _ event parent = response := by
  refine
    ⟨responseParentTag3 response,
      parentTagResponse_responseParentTag response,
      ?_⟩
  intro parent hparent
  apply parentTagResponse3_injective
  rw [hparent,
    parentTagResponse_responseParentTag]

/-- 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)

What this page does not claim

The theorem does not claim that the physical carrier is created or identified by the recognition event. The theorem does not claim that physical channels are distinguishable by probes. The theorem does not claim that every physical system must realize the forced responses.

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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Derived articles

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