Encyclopedia Foundation Foundation Pair Kernel Response Quotient Carrier S22 External Carrier To Respons

ARTICLE 3 claims 2 theorems 1 hypothesis

Foundation Pair Kernel Response Quotient Carrier S22 External Carrier To Respons

A theorem about when an external physical carrier can account for every possible response class, and the precise boundary of that claim.

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The classifier's surjectivity

The declaration establishes a precise condition under which a mapping from an external physical channel carrier to the response quotient is surjective. In plain terms, it proves that if a carrier is complete in the sense of posting carrier coherence, a property meaning the carrier's classification of posting events is bijective and thus loses no distinctions, then every response class in the quotient is hit by at least one element of the external carrier. This result is a formal statement in the machine-checked library: it is proved with no sorry and no new axioms.

The classical picture is a quotient construction. Two responses are equivalent when every committed Recognition coordinate probe agrees on them. The quotient is the set of equivalence classes under that relation. The result says that a complete external carrier maps onto this quotient: for every equivalence class, there is some external carrier element that maps to it. This is a surjectivity result, and it is paired with a companion result proving that an incomplete carrier is not surjective, so the condition is sharp.

In Recognition Science, the framework models physical channels as carriers that realize posting events. The result is part of a chain showing that the response quotient itself supplies a complete, separating, response-realizing channel carrier. The declaration's name, externalCarrierToResponseQuotient3_surjective_of_classifier, reflects that it is a result about an external carrier, not a construction of one.

The result does not claim that every external carrier is surjective, nor that any particular physical carrier in the world is complete. It does not identify an independently supplied production carrier with the response quotient. That final commuting identification is stated explicitly and proved equivalent to S17 carrier completeness, and it remains a physical hypothesis. The result also does not establish injectivity for a complete carrier; a separate result proves that an extra carrier with a ghost state is not injective, so the mapping can collapse distinct external states into the same response class while still covering every class.

THEOREM productionCarrierIsResponseQuotient_iff_complete · IndisputableMonolith/Foundation/PairKernelResponseQuotientCarrierS22.lean
productionCarrierIsResponseQuotient_iff_complete · IndisputableMonolith/Foundation/PairKernelResponseQuotientCarrierS22.lean:830 
/-- The final production identification is exactly carrier completeness. This
records the remaining physical boundary rather than renaming it. -/
theorem productionCarrierIsResponseQuotient_iff_complete
    {N : ℕ} [NeZero N]
    (physical : PostingPhysicalChannelCarrier3 N) :
    ProductionCarrierIsResponseQuotient3 physical ↔
      PostingCarrierCoherenceComplete3 physical := by
  constructor
  · intro hidentified event
    obtain ⟨identification, hcommutes⟩ :=
      hidentified event
    have hquotient :=
      classifyResponseQuotient3_bijective event
    constructor
    · intro left right hclassify
      apply identification.injective
      apply hquotient.1
      rw [hcommutes left, hcommutes right,
        hclassify]
    · intro parent
      obtain ⟨responseClass, hresponseClass⟩ :=
        hquotient.2 parent
      refine
        ⟨identification.symm responseClass, ?_⟩
      have hcommute :=
        hcommutes
          (identification.symm responseClass)
      rw [identification.apply_symm_apply]
        at hcommute
      exact hcommute.symm.trans hresponseClass
  · intro hcomplete event
    let physicalEquiv :
        physical.Carrier event ≃
          PostingConfigurationDegreeCarrier3
            (realizedPostingEventConfiguration3 event) :=
      Equiv.ofBijective
        (physical.classify event)
        (hcomplete event)
    let quotientEquiv :
        RecognitionResponseQuotient3 ≃
          PostingConfigurationDegreeCarrier3
            (realizedPostingEventConfiguration3 event) :=
      Equiv.ofBijective
        (classifyResponseQuotient3 event)
        (classifyResponseQuotient3_bijective event)
    let identification :=
      physicalEquiv.trans quotientEquiv.symm
    refine ⟨identification, ?_⟩
    intro channel
    change
      quotientEquiv
          (quotientEquiv.symm
            (physical.classify event channel)) =
        physical.classify event channel
    exact quotientEquiv.apply_symm_apply _
THEOREM incompleteCarrierToResponseQuotient_not_surjective · IndisputableMonolith/Foundation/PairKernelResponseQuotientCarrierS22.lean
incompleteCarrierToResponseQuotient_not_surjective · IndisputableMonolith/Foundation/PairKernelResponseQuotientCarrierS22.lean:748 
/-- Explicit omission: the incomplete carrier map misses the balance response
class. This is a response-class failure, not a cardinality argument. -/
theorem incompleteCarrierToResponseQuotient_not_surjective
    (event : RealizedPostingEvent3 3) :
    ¬ Function.Surjective
      (externalCarrierToResponseQuotient3
        (incompletePhysicalChannelCarrier3 3)
        event) := by
  intro hsurjective
  obtain ⟨channel, hchannel⟩ :=
    hsurjective
      (proj responseCoordinateProbeFamily3
        (.temporal 1))
  have hresponse :=
    congrArg responseFromQuotientClass3 hchannel
  cases channel with
  | inl axis =>
      exact RecognitionParentResponse3.noConfusion
        hresponse
  | inr one =>
      have hfin : (0 : Fin 2) = 1 := by
        simpa [externalCarrierToResponseQuotient3,
          incompletePhysicalChannelCarrier3]
          using
            RecognitionParentResponse3.temporal.inj
              hresponse
      have hnat : (0 : ℕ) = 1 :=
        congrArg Fin.val hfin
      omega
HYPOTHESIS productionCarrierIsResponseQuotient_iff_complete · IndisputableMonolith/Foundation/PairKernelResponseQuotientCarrierS22.lean
productionCarrierIsResponseQuotient_iff_complete · IndisputableMonolith/Foundation/PairKernelResponseQuotientCarrierS22.lean:830 
/-- The final production identification is exactly carrier completeness. This
records the remaining physical boundary rather than renaming it. -/
theorem productionCarrierIsResponseQuotient_iff_complete
    {N : ℕ} [NeZero N]
    (physical : PostingPhysicalChannelCarrier3 N) :
    ProductionCarrierIsResponseQuotient3 physical ↔
      PostingCarrierCoherenceComplete3 physical := by
  constructor
  · intro hidentified event
    obtain ⟨identification, hcommutes⟩ :=
      hidentified event
    have hquotient :=
      classifyResponseQuotient3_bijective event
    constructor
    · intro left right hclassify
      apply identification.injective
      apply hquotient.1
      rw [hcommutes left, hcommutes right,
        hclassify]
    · intro parent
      obtain ⟨responseClass, hresponseClass⟩ :=
        hquotient.2 parent
      refine
        ⟨identification.symm responseClass, ?_⟩
      have hcommute :=
        hcommutes
          (identification.symm responseClass)
      rw [identification.apply_symm_apply]
        at hcommute
      exact hcommute.symm.trans hresponseClass
  · intro hcomplete event
    let physicalEquiv :
        physical.Carrier event ≃
          PostingConfigurationDegreeCarrier3
            (realizedPostingEventConfiguration3 event) :=
      Equiv.ofBijective
        (physical.classify event)
        (hcomplete event)
    let quotientEquiv :
        RecognitionResponseQuotient3 ≃
          PostingConfigurationDegreeCarrier3
            (realizedPostingEventConfiguration3 event) :=
      Equiv.ofBijective
        (classifyResponseQuotient3 event)
        (classifyResponseQuotient3_bijective event)
    let identification :=
      physicalEquiv.trans quotientEquiv.symm
    refine ⟨identification, ?_⟩
    intro channel
    change
      quotientEquiv
          (quotientEquiv.symm
            (physical.classify event channel)) =
        physical.classify event channel
    exact quotientEquiv.apply_symm_apply _

What this page does not claim

The result does not claim that any particular physical carrier in the world is complete. The result does not claim injectivity for a complete carrier; an extra carrier with a ghost state is proved not injective. The result does not construct an external carrier; it states a condition on a given one.

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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/PairKernelResponseQuotientCarrierS22.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red:

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