Foundation Primitive Recognition Calculus Prctype Theory Parse Tt System Embeds

The two-symbol core

In the Recognition Science framework, the foundational core is built from a primitive act of distinction: a ledger, a discrete record of events, must be able to tell two things apart. The declaration ttSystem_embeds_delta proves that this core is already present in the simplest imaginable system: one with exactly two symbols, conventionally called false and true. The theorem states formally that this two-symbol system can faithfully represent, or embed, the framework's delta core, the minimal structure required for meaningful recognition.

The proof rests on two plain facts about the two-symbol type. First, it satisfies canonicity: every closed term of the type is one of the two constructors, so the type has exactly two inhabitants. Second, it satisfies no-confusion: the two constructors are distinct, so false is not true. These two facts together give the type its expressive power. The theorem then packages these facts into a formal embedding, showing that the two-symbol system realizes the delta core of the framework.

The significance is structural. The framework's library, a machine-checked collection of formal theorems, proves that this minimal two-symbol system is not degenerate: it is expressive enough to contain the framework's foundational core. This means the entire edifice of Recognition Science, with its cost functions, golden ratio, and derived constants, can be built on a foundation that requires only the ability to distinguish two things. The theorem is a formal statement about the framework's own construction, not a claim about the physical world.

What the theorem does not claim is equally important. It does not claim that the two-symbol system is the same as the full type theory of Martin-Löf, nor that it captures all of mathematics. It establishes a specific embedding into a specific formal system, not a general equivalence. The theorem is a precise, narrow result: the delta core of the framework is realizable in a two-symbol system. It is a foundational brick, not a claim about the nature of computation or the limits of logic.

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

/-- **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

What this page does not claim

The theorem does not claim that the two-symbol system is equivalent to the full type theory. It does not claim that the embedding captures all of mathematics. It does not claim that the two-symbol system is the only system that realizes the delta core.

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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