Foundation Primitive Recognition Calculus Prctype Theory Parse Tt System Express

The two-token ledger

In the ledger (a discrete record of events) that Recognition Science uses to model recognition, the smallest possible alphabet has exactly two symbols. The declaration ttSystem_expressive proves that this two-symbol system is expressive: it can represent every distinction the underlying type theory can make. The proof is a theorem in the framework's machine-checked library of formal theorems, meaning it is verified by the computer rather than argued in prose.

The two symbols are the canonical terms of the type Two, which is the two-element boolean type familiar from logic: its terms are false and true. The framework proves two classical facts about this type. First, canonicity: every closed term of Two is either false or true, so the type has exactly two inhabitants. Second, no-confusion: the two terms are distinct, so false is not equal to true. These two facts together give the system its expressive power: it can tell two things apart, and it knows there are exactly two things to tell apart.

The theorem ttSystem_expressive packages these facts into the framework's FormalSystem interface. In that interface, tokens are the two closed terms, the discrimination relation is term inequality, and the expression order is the derivation-length order. The theorem states that this system is expressive, and from it the framework derives a stronger result: the system embeds the entire δ core, the minimal structure that the framework's recognition calculus requires. This embedding is what makes the two-symbol system non-degenerate, meaning it is not a trivial system where everything collapses into one thing.

The theorem ttSystem_expressive does not claim that the two-symbol system is the same as full type theory, nor that it can express every mathematical statement. It claims only that the two-symbol system contains the expressive core that the framework needs to build its recognition calculus. The framework's library proves that type theory's two-element type satisfies canonicity and no-confusion, and that the foundation realizes the δ core. This is a precise, limited claim: the two-symbol system is expressive enough to serve as the foundation for the framework's recognition calculus, not that it is the whole of mathematics.

theorem ttSystem_expressive : ttSystem.Expressive := by
  show (false : Two) ≠ true
  decide

/-- The canonical two-element type `𝟚` of Martin-Löf type theory / CIC, here Lean's
own `Bool`. Its two closed terms are `false` and `true`. -/
abbrev Two := Bool

/-- **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

/-- **MLTT contains the δ core.** -/
theorem ttSystem_embeds_delta : Nonempty (PRCEmbeddingInto ttSystem) :=
  FormalSystemEmbeddingTarget_proved ttSystem ttSystem_expressive

What this page does not claim

The two-symbol system is not the same as full type theory. The theorem does not claim that the two-symbol system can express every mathematical statement. The framework does not claim that the two-symbol system is the whole of mathematics.

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)

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.

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