Foundation Primitive Recognition Calculus Prctype Theory Parse Type Theory Reali

The two-element type

Martin-Löf type theory, the foundation underlying modern proof assistants, begins with a two-element type. Its two closed terms are conventionally called false and true. The type has exactly two inhabitants, no more and no fewer. This is the classical fact of canonicity: every closed term of the type is one of the two canonical constructors. A second classical fact, no-confusion, states that the two constructors are distinct. These two properties together give the type its basic logical character: it is a genuine two-valued object, not a collapsed or degenerate one.

Recognition Science (RS) starts from the idea that reality keeps a ledger, a discrete record of events, and that distinguishing between events is the fundamental act. The framework's library, a machine-checked collection of formal theorems, defines a minimal formal system for this act: a set of tokens, a way to compare them, and a way to order expressions. The two-element type fits this interface directly. Its tokens are the two closed terms, its discrimination relation is term inequality, and its expression order is the natural order on derivation lengths. This is not a metaphor; it is a definition in the library.

The theorem type_theory_realizes_delta packages three results into one statement. It asserts canonicity, no-confusion, and the existence of an embedding of the RS core into this type-theoretic system. The embedding is the substantive part: it shows that the minimal structure RS calls the delta core, the bare ability to distinguish two things, is already present in the two-element type. The proof is fully checked by the machine, with no gaps and no extra axioms. The theorem also implies that the system is not degenerate, meaning it does not collapse into a single undifferentiated point.

In Recognition Science, this result is read as a grounding step. It shows that the framework's primitive notion of distinction is not an alien addition to type theory but a faithful parse of what the two-element type already does. The framework models its foundational act of recognition as this act of distinguishing two canonical terms. The theorem does not claim that type theory itself is Recognition Science, nor that the two-element type is the whole of the framework. It claims only that the delta core, the minimal seed, is realized there.

/-- **Canonicity.** Every closed term of `𝟚` is one of the two canonical
constructors. The type has exactly two inhabitants. -/
theorem canonicity (b : Two) : b = false ∨ b = true := by
  cases b
  · exact Or.inl rfl
  · exact Or.inr rfl

/-- **No-confusion / constructor disjointness.** The two canonical terms are
distinct: this is the recursor's verdict, the type theory's own distinction. -/
theorem no_confusion : (false : Two) ≠ true := by decide

theorem ttSystem_not_degenerate : ¬ DistinctionDichotomy.Degenerate ttSystem :=
  DistinctionDichotomy.not_degenerate_of_realizesDelta ttSystem ttSystem_embeds_delta

/-- **The faithful parse, packaged.** Type theory's two-element type satisfies
canonicity (exactly two closed terms) and no-confusion (they are distinct), and the
foundation realizes the δ core. -/
theorem type_theory_realizes_delta :
    (∀ b : Two, b = false ∨ b = true)
      ∧ ((false : Two) ≠ true)
      ∧ Nonempty (PRCEmbeddingInto ttSystem) :=
  ⟨canonicity, no_confusion, ttSystem_embeds_delta⟩

What this page does not claim

The theorem does not claim that Martin-Löf type theory is identical to Recognition Science. The theorem does not claim that the two-element type alone is sufficient to derive the full forcing chain. The theorem does not claim that the embedding is unique.

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

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