Foundation Pair Kernel Recognition Transport Residuals S18 Misclassified Five Ca

Carrier completeness

A carrier is a rule that assigns each recognized event to a parent channel. In the framework's library, a carrier is complete only if its assignment is both injective and surjective: no two distinct events collapse into one parent, and no parent goes without an event. The theorem misclassifiedFiveCarrier_not_parentSurjective proves that a specific five-channel carrier fails the surjective half of that test. The failure is not a numerical accident; it is a machine-checked consequence of the carrier's definition.

The proof is one of three companion results. Two other carriers fail the injective half, and the same five-channel carrier also fails injectivity. Together the three theorems show that carrier completeness splits into two independent conditions, each with its own counterexample. The framework's library records these as formal theorems, not as empirical observations.

What the theorem does not claim is just as important. It does not say that the five-channel carrier is the wrong choice for physics, nor that any other carrier is the right one. It establishes only a negative fact: this carrier, under the framework's definition of surjectivity, is not complete. The theorem leaves open which carrier, if any, the physical world selects.

theorem carrierComplete_iff_injective_and_surjective
    {N : ℕ} [NeZero N]
    (physical : PostingPhysicalChannelCarrier3 N) :
    PostingCarrierCoherenceComplete3 physical ↔
      PostingCarrierParentInjective3 physical ∧
        PostingCarrierParentSurjective3 physical := by
  constructor
  · intro hcomplete
    exact
      ⟨fun event => (hcomplete event).1,
        fun event => (hcomplete event).2⟩
  · rintro ⟨hinjective, hsurjective⟩ event
    exact
      ⟨hinjective event, hsurjective event⟩

theorem misclassifiedFiveCarrier_not_parentSurjective :
    ¬ PostingCarrierParentSurjective3
      (misclassifiedFiveChannelCarrier3 3) := by
  intro hsurjective
  obtain ⟨pair, hpair⟩ :=
    realizedPrimitivePostingPair3_exists
  let event : RealizedPostingEvent3 3 := ⟨pair, hpair⟩
  obtain ⟨channel, hchannel⟩ :=
    hsurjective event
      (Sum.inr (1 : Fin 2))
  change
    Sum.inr (0 : Fin 2) =
      Sum.inr (1 : Fin 2) at hchannel
  have hfin : (0 : Fin 2) = 1 :=
    Sum.inr.inj hchannel
  norm_num at hfin

What this page does not claim

The five-channel carrier is the physically correct one. Any other specific carrier is complete. The theorem identifies which parent channels are missing events.

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/PairKernelRecognitionTransportResidualsS18.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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