Encyclopedia Foundation Foundation Pair Kernel Response Ancestry S21 Committed Response Ancestry Admits

ARTICLE 5 claims 5 theorems

Foundation Pair Kernel Response Ancestry S21 Committed Response Ancestry Admits

A machine-checked library of formal theorems shows that a minimal set of response types can be traced back to committed acts of recognition, while leaving the physical carrier that realizes them entirely unforced.

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The source-side ancestry

The declaration committed_response_ancestry_admits_realization_split is a theorem in the framework's machine-checked library of formal theorems. It establishes a precise boundary: the five operational response types, spatial, temporal, successor, balance, and coordinate, are all forced by the recognition ledger, a discrete record of events. The theorem proves that every one of these responses has a unique parent configuration, and that the full catalog of responses is complete on the source side.

This source-side completeness does not extend to the physical world. The theorem proves that the recognition-forced responses can be read by any physical channel, but it does not prove that a physical system must realize them. The declaration leaves open the two physical arrows: that every forced response has a physical representative, and that physical channels are extensional under the recognition probe family. A collapsed system that fails both arrows is explicitly shown to exist, so the source semantics do not shrink the physical obligations.

The theorem also clarifies the meaning of the three spatial responses. They are the complete family of admitted minimum-cost counterfactual probe contexts, not three simultaneous updates by the same event. One event has one active spatial-axis witness, a distinction the library proves is theorem-visible.

What the declaration does not claim is equally sharp. It does not claim that the recognition ledger can create or identify points of an independently supplied physical carrier. It does not claim that the physical realization is forced. It does not claim that the physical channels are extensional. The theorem is a boundary statement: it proves what the source side forces, and it names exactly what it leaves open.

THEOREM recognitionForcedResponse_unique_parent · IndisputableMonolith/Foundation/PairKernelResponseAncestryS21.lean
recognitionForcedResponse_unique_parent · IndisputableMonolith/Foundation/PairKernelResponseAncestryS21.lean:323 
/-- 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]
THEOREM every_physicalChannel_reads_recognitionForcedResponse · IndisputableMonolith/Foundation/PairKernelResponseAncestryS21.lean
every_physicalChannel_reads_recognitionForcedResponse · IndisputableMonolith/Foundation/PairKernelResponseAncestryS21.lean:473 
theorem every_physicalChannel_reads_recognitionForcedResponse
    {N : ℕ} [NeZero N]
    (system : PostingPhysicalResponseSystem3 N) :
    PhysicalChannelsReadRecognitionForcedResponses3 system := by
  intro event channel
  exact every_response_is_recognitionForced
    event (system.respond event channel)
THEOREM forcedResponsesPhysicallyRealized_iff_realizeParents · physicalProbeExtensional_iff_responseDistinguishing · IndisputableMonolith/Foundation/PairKernelResponseAncestryS21.lean
forcedResponsesPhysicallyRealized_iff_realizeParents · IndisputableMonolith/Foundation/PairKernelResponseAncestryS21.lean:493 
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]
physicalProbeExtensional_iff_responseDistinguishing · IndisputableMonolith/Foundation/PairKernelResponseAncestryS21.lean:537 
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⟩
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 realizedEvent_has_unique_activeSpatialAxis · IndisputableMonolith/Foundation/PairKernelResponseAncestryS21.lean
realizedEvent_has_unique_activeSpatialAxis · IndisputableMonolith/Foundation/PairKernelResponseAncestryS21.lean:152 
/-- The event's chosen ledger witness has exactly one active spatial axis.
This does not identify all three counterfactual spatial basis probes with the
same event update. -/
theorem realizedEvent_has_unique_activeSpatialAxis
    {N : ℕ} [NeZero N]
    (event : RealizedPostingEvent3 N) :
    ∃ before after :
        LedgerPostingAdjacency.LedgerState 3,
      JMinimalPostingStep before after ∧
        ∃! axis : Fin GapDerivation.D,
          FramedPostingAxis
            (Equiv.refl (Fin 3))
            before after axis := by
  obtain
    ⟨before, after, _hposting, _hlegal,
      hminimal, _hl1, _hjlog, _hbalance⟩ :=
    realizedPostingEvent_has_ledger_tick event
  exact
    ⟨before, after, hminimal,
      jMinimalPostingStep_unique_framed_axis
        (Equiv.refl (Fin 3)) hminimal⟩

What this page does not claim

The theorem does not prove that any physical system must realize the recognition-forced responses. The theorem does not prove that physical channels are extensional under the recognition probe family. The theorem does not prove that the recognition ledger can create or identify points of a 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)

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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