Encyclopedia Foundation Foundation Pair Kernel Response Ancestry S21 Collapsed Five Carrier Fails Both P

ARTICLE 4 claims 4 theorems

Foundation Pair Kernel Response Ancestry S21 Collapsed Five Carrier Fails Both P

A machine-checked proof shows that a physical system with only five indistinguishable response channels cannot satisfy either of the two physical requirements the framework demands.

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The collapsed carrier

The declaration collapsedFiveCarrier_fails_both_physical_arrows is a formal theorem in the framework's machine-checked library of formal theorems. It concerns a physical response system, which in this framework means a system that pairs each recognition event with one of five possible response channels. The theorem's name says what it proves: a system whose five channels are collapsed, so that they all read as the same response, fails both physical requirements that the framework's earlier results leave open.

The first requirement is physical realization: every Recognition-forced response must have a physical representative among the channels. The second is probe extensionality: distinct physical channels must be distinguishable by the Recognition probe family, the set of tests the framework uses to tell responses apart. The collapsed carrier, where every channel produces the same response, fails the first because it cannot represent all five distinct responses, and fails the second because its channels are indistinguishable. The theorem thus closes a specific gap: it shows that a trivial physical carrier cannot satisfy the two remaining physical arrows, which are exactly the separation and realization requirements from the earlier S20 stage.

The proof is axiom-clean, with no sorry and no new axiom. It builds on a companion result, collapsedResponseProbes_not_separating, which shows that the collapsed probe family, one that maps every response to zero, cannot separate distinct responses. The theorem does not claim that all physical systems fail; it claims only that this particular collapsed five-channel system fails both arrows. It also does not establish that any non-collapsed system succeeds, nor does it derive the physical realization or extensionality arrows themselves. Those remain open targets in the framework's development.

THEOREM collapsedFiveCarrier_fails_both_physical_arrows · IndisputableMonolith/Foundation/PairKernelResponseAncestryS21.lean
collapsedFiveCarrier_fails_both_physical_arrows · IndisputableMonolith/Foundation/PairKernelResponseAncestryS21.lean:621 
/-- 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)
THEOREM collapsedFiveCarrier_fails_both_physical_arrows · IndisputableMonolith/Foundation/PairKernelResponseAncestryS21.lean
collapsedFiveCarrier_fails_both_physical_arrows · IndisputableMonolith/Foundation/PairKernelResponseAncestryS21.lean:621 
/-- 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)
THEOREM collapsedFiveCarrier_fails_both_physical_arrows · IndisputableMonolith/Foundation/PairKernelResponseAncestryS21.lean
collapsedFiveCarrier_fails_both_physical_arrows · IndisputableMonolith/Foundation/PairKernelResponseAncestryS21.lean:621 
/-- 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)
THEOREM collapsedResponseProbes_not_separating · IndisputableMonolith/Foundation/PairKernelResponseAncestryS21.lean
collapsedResponseProbes_not_separating · IndisputableMonolith/Foundation/PairKernelResponseAncestryS21.lean:446 
theorem collapsedResponseProbes_not_separating :
    ¬ ∀ left right : RecognitionParentResponse3,
      ObsEquiv collapsedResponseProbeFamily3 left right →
        left = right := by
  intro hseparate
  have heq :=
    hseparate
      (.temporal 0) (.temporal 1) (by
        intro probe hprobe
        obtain ⟨unit, rfl⟩ := hprobe
        rfl)
  have hfin : (0 : Fin 2) = 1 :=
    RecognitionParentResponse3.temporal.inj heq
  norm_num at hfin

What this page does not claim

The theorem does not prove that any non-collapsed system succeeds. It does not derive the physical realization or extensionality arrows themselves. It does not claim that all physical systems fail, only this specific collapsed one.

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)

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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