Encyclopedia Foundation Foundation Pair Kernel Response Quotient Carrier S22 Incomplete Carrier To Respo

ARTICLE 4 claims 4 theorems

Foundation Pair Kernel Response Quotient Carrier S22 Incomplete Carrier To Respo

A physical channel that misses one kind of response cannot reach every state of the response quotient, a fact the framework proves and carefully scopes.

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The incomplete carrier

In the Recognition Science framework, a ledger is a discrete record of events, and a recognition is the act of registering one event in that record. The framework builds a mathematical object called the response quotient: the collection of all observable response states, identified by which coordinate probes agree on them. A separate, independently supplied physical channel carrier may try to realize those states. The theorem named incompleteCarrierToResponseQuotient_not_surjective states that a particular carrier, the incomplete physical channel carrier with three posting events, cannot reach every state in that quotient. Its map from the carrier to the quotient is not surjective: some response states remain unreachable.

The proof is machine-checked in the framework's library of formal theorems. It works by exhibiting a concrete obstruction. The incomplete carrier lacks a probe that detects a certain extra response state, the ghost state. Because that probe is missing, the carrier cannot distinguish the ghost state from a base state, and so the quotient it produces is smaller than the full response quotient. The theorem is conditional on a realized posting event with three postings, and it applies to that specific carrier construction, not to all carriers in general.

What the theorem does not claim is just as important. It does not say the incomplete carrier is physically impossible or that the framework rejects it. It does not say the response quotient itself is incomplete; in fact, the library proves the quotient is complete and separates all response states. The theorem only establishes a negative fact about one carrier: its map is not surjective. The framework also proves a companion result for a different carrier, the extra carrier, whose map is not injective, meaning it conflates distinct states. Together these two theorems show that neither a missing probe nor an extra probe yields a carrier that exactly matches the response quotient.

The framework is careful about what follows from these results. A separate theorem, productionCarrierIsResponseQuotient_iff_complete, states that an independently supplied carrier is identified with the response quotient exactly when it is complete. The incomplete carrier fails that test, so it is not so identified. This is a theorem about the framework's own construction, not a claim about any particular physical system in the world. Whether a real production carrier satisfies the completeness condition remains a physical hypothesis, not a proved fact.

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
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
THEOREM extraCarrierToResponseQuotient_not_injective · IndisputableMonolith/Foundation/PairKernelResponseQuotientCarrierS22.lean
extraCarrierToResponseQuotient_not_injective · IndisputableMonolith/Foundation/PairKernelResponseQuotientCarrierS22.lean:778 
/-- Explicit collision: the extra carrier map sends two distinct external
points to the same successor response class. -/
theorem extraCarrierToResponseQuotient_not_injective
    (event : RealizedPostingEvent3 3) :
    ¬ Function.Injective
      (externalCarrierToResponseQuotient3
        (extraPhysicalChannelCarrier3 3)
        event) := by
  intro hinjective
  have hclasses :
      externalCarrierToResponseQuotient3
          (extraPhysicalChannelCarrier3 3) event
          (Sum.inr (0 : Fin 3)) =
        externalCarrierToResponseQuotient3
          (extraPhysicalChannelCarrier3 3) event
          (Sum.inr (2 : Fin 3)) := by
    rfl
  have hpoints := hinjective hclasses
  have hfin : (0 : Fin 3) = 2 :=
    Sum.inr.inj hpoints
  have hnat : (0 : ℕ) = 2 :=
    congrArg Fin.val hfin
  omega
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 _

What this page does not claim

The incomplete carrier is physically impossible or ruled out by the framework. The response quotient itself is incomplete or fails to separate states. Any real-world production carrier is or is not complete; that remains a physical hypothesis.

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/PairKernelResponseQuotientCarrierS22.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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