Foundation Pair Kernel Physical Readout Selection S17 Misclassified Five Carrier

A miscounted carrier

A physical channel carrier is, in plain terms, a way of assigning to each event a finite set of possible readout values. The framework's ledger, a discrete record of events, needs such carriers to connect its formal events to something a physicist could measure. The carrier in question, misclassifiedFiveChannelCarrier3, is built by taking the standard configuration degrees and then deliberately mislabeling every readout: its classifying function sends every value to the same wrong slot. The theorem misclassifiedFiveCarrier_dimension_eq_five proves that, despite this deliberate mislabeling, the carrier still has exactly five possible readout values for every event. The number five is not an accident; it is the same dimension that any complete, correctly classified carrier must have.

The proof itself is short in the machine-checked library of formal theorems: it reduces the carrier's size to a finite sum and computes it. The interesting content is in what the theorem does not say. A separate theorem, misclassifiedFiveCarrier_not_complete, proves that this same carrier fails the completeness condition: its classifier is not bijective, meaning two distinct readout values can be sent to the same classification. So the dimension-five result is a necessary condition, not a sufficient one. Having the right number of readout values does not make a carrier physically meaningful, any more than having five fingers makes a hand a tool.

The framework's library also proves the positive counterpart. For any carrier that is complete, meaning its classifier is a bijection onto the parent degrees, the dimension is forced to be five, and all such complete carriers are equivalent up to relabeling. The misclassified carrier sits outside that equivalence class. Its dimension matches, but its classification does not. The distinction matters for the framework's physical readout program: the dimension is a structural invariant, while completeness is a semantic requirement about what the readout values actually mean.

In Recognition Science, the declaration is a boundary marker. It shows that dimension alone cannot select a physical readout. The framework models this by separating the four readout interfaces and by proving that the canonical instances are only MODEL witnesses, not forced choices. The misclassified carrier is a counterexample to any claim that dimension five by itself picks out the physical channel. What the theorem establishes is narrow and exact: a miscounted carrier still has dimension five. What it does not establish is that such a carrier is valid, complete, or physically selected.

theorem misclassifiedFiveCarrier_dimension_eq_five
    {N : ℕ} [NeZero N]
    (event : RealizedPostingEvent3 N) :
    physicalPostingCarrierDimension3
      (misclassifiedFiveChannelCarrier3 N) event = 5 := by
  rw [show
    physicalPostingCarrierDimension3
      (misclassifiedFiveChannelCarrier3 N) event =
        GapDerivation.configDim GapDerivation.D by
      exact
        postingConfigurationDegreeCarrier_card
          (realizedPostingEventConfiguration3 event)]
  exact GapDerivation.configDim_at_D3

theorem misclassifiedFiveCarrier_not_complete :
    ¬ PostingCarrierCoherenceComplete3
      (misclassifiedFiveChannelCarrier3 3) := by
  intro hcomplete
  obtain ⟨pair, hpair⟩ :=
    realizedPrimitivePostingPair3_exists
  let event : RealizedPostingEvent3 3 := ⟨pair, hpair⟩
  have hinjective := (hcomplete event).1
  have heq :
      (Sum.inr (0 : Fin 2) :
        PostingConfigurationDegreeCarrier3
          (realizedPostingEventConfiguration3 event)) =
      Sum.inr (1 : Fin 2) :=
    hinjective rfl
  have hfin : (0 : Fin 2) = 1 :=
    Sum.inr.inj heq
  norm_num at hfin

/-- At committed spatial dimension three, every complete physical event carrier
has exactly five channels. -/
theorem completeCarrier_dimension_eq_five
    {N : ℕ} [NeZero N]
    {physical : PostingPhysicalChannelCarrier3 N}
    (hcomplete : PostingCarrierCoherenceComplete3 physical)
    (event : RealizedPostingEvent3 N) :
    physicalPostingCarrierDimension3 physical event = 5 := by
  rw [completeCarrier_dimension_eq_configDim hcomplete event,
    GapDerivation.configDim_at_D3]

/-- Independent carrier consequence: any two complete physical carriers are
equivalent for each event, so completeness is invariant under channel
relabeling. -/
theorem completeCarriers_equivalent
    {N : ℕ} [NeZero N]
    {left right : PostingPhysicalChannelCarrier3 N}
    (hleft : PostingCarrierCoherenceComplete3 left)
    (hright : PostingCarrierCoherenceComplete3 right)
    (event : RealizedPostingEvent3 N) :
    Nonempty (left.Carrier event ≃ right.Carrier event) := by
  let leftEquiv :
      left.Carrier event ≃
        PostingConfigurationDegreeCarrier3
          (realizedPostingEventConfiguration3 event) :=
    Equiv.ofBijective (left.classify event) (hleft event)
  let rightEquiv :
      right.Carrier event ≃
        PostingConfigurationDegreeCarrier3
          (realizedPostingEventConfiguration3 event) :=
    Equiv.ofBijective (right.classify event) (hright event)
  exact ⟨leftEquiv.trans rightEquiv.symm⟩

What this page does not claim

The misclassified carrier is a valid physical readout. The dimension-five result alone identifies the physical channel. The framework currently selects the canonical readout instances as forced physical choices.

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/PairKernelPhysicalReadoutSelectionS17.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: