Foundation Pair Kernel Recognition Transport Residuals S18 Carrier Complete Iff

The completeness test

A complete physical channel in the Recognition Science framework is one that neither loses distinctions nor misses possibilities. The theorem carrierComplete_iff_injective_and_surjective states this in exact terms: a channel is complete if and only if its classification map is injective and surjective. Injectivity means no two distinct events are ever classified the same way; surjectivity means every possible classification is actually realized by some event. The proof is a single equivalence in the framework's machine-checked library of formal theorems, with no unproved assumptions.

The theorem splits the notion of completeness into two independent halves, each with its own failure mode. The library shows a carrier that adds an extra classification slot fails injectivity: it can classify two different events identically. A carrier that omits a slot fails surjectivity: some classification is never reached. A third carrier fails both. Each half has a separate cardinal bound, and each has a distinct countermodel, so the two conditions are genuinely independent rather than two names for the same constraint.

In Recognition Science, the framework models physical channels as discrete records of events, and this theorem is a structural fact about those records. It does not say which physical channel is the right one, nor does it attach the completeness condition to any specific measured quantity. The theorem governs the abstract classification map; the physical identification of a channel with a concrete system remains a separate modeling choice, not a proved consequence.

The value of the theorem is diagnostic. When a proposed channel fails completeness, the failure now has a precise location: either it conflates distinct events, or it leaves some classification unrealized, or both. The theorem also sharpens the search for a physical channel: any candidate must pass both tests, and the library provides the exact statements to check.

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 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 extraCarrier_not_parentInjective :
    ¬ PostingCarrierParentInjective3
      (extraPhysicalChannelCarrier3 3) := by
  intro hinjective
  obtain ⟨pair, hpair⟩ :=
    realizedPrimitivePostingPair3_exists
  let event : RealizedPostingEvent3 3 := ⟨pair, hpair⟩
  have hbound :=
    carrierInjective_card_le_configDim
      hinjective event
  rw [extraCarrier_dimension_eq_six] at hbound
  norm_num [GapDerivation.configDim,
    GapDerivation.D] at hbound

theorem incompleteCarrier_not_parentSurjective :
    ¬ PostingCarrierParentSurjective3
      (incompletePhysicalChannelCarrier3 3) := by
  intro hsurjective
  obtain ⟨pair, hpair⟩ :=
    realizedPrimitivePostingPair3_exists
  let event : RealizedPostingEvent3 3 := ⟨pair, hpair⟩
  have hbound :=
    configDim_le_card_of_carrierSurjective
      hsurjective event
  rw [incompleteCarrier_dimension_eq_four] at hbound
  norm_num [GapDerivation.configDim,
    GapDerivation.D] at hbound

What this page does not claim

The theorem does not identify which physical channel is the correct one. The theorem does not prove that any real system is complete. The theorem does not attach completeness to any measured quantity such as energy or duration.

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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