Foundation Primitive Recognition Calculus Prcinevitability Instances Type Theory

The type-theoretic witness

In type theory, the canonical two-element type, written as the disjoint union of two unit types, has exactly two closed terms. These terms are distinct by construction. The declaration typeTheorySystem_embeds_delta establishes that this type-theoretic foundation, built from those two distinct terms, realizes the delta core, the minimal structure of the framework's primitive recognition calculus. The delta core is the part of the calculus that requires two distinguishable primitives, a discrete record of events called a ledger that records which primitive occurred, and a monotone notion of extending that record.

The proof is generic. A single theorem, two_distinct_realizes_delta, shows that any type with two distinct elements can be turned into a formal system that embeds the delta core. The type-theoretic system is one instance: its two terms are the left and right injections of the unit type. The same theorem covers three other foundations. The two-valued logic carrier distinguishes true from false, Peano arithmetic distinguishes 0 from 1, and set theory distinguishes the empty set from the singleton set. The machine-checked library of formal theorems proves all four embeddings in one combined statement.

The declaration does not claim that the type-theoretic foundation is a full faithful parse of the entire primitive recognition calculus. The library's own documentation says that corpus task remains open. It claims only that the primitive distinction, the ability to tell two primitives apart, appears in type theory, and that this minimal distinction is enough to realize the delta core. The result is a witness, not a reduction. It shows the distinction is not an artifact of one foundation's notation, but it does not show that type theory, or any other foundation, is the unique or privileged carrier of the calculus.

theorem typeTheorySystem_embeds_delta :
    Nonempty (PRCEmbeddingInto typeTheorySystem) :=
  two_distinct_realizes_delta (Sum.inl () : Unit ⊕ Unit) (Sum.inr ()) (by decide)

/-- **Item 4 (generic).** Any foundation exposing two distinguishable primitives
realizes the δ core. -/
theorem two_distinct_realizes_delta {α : Type} (a₀ a₁ : α) (hne : a₀ ≠ a₁) :
    Nonempty (PRCEmbeddingInto (ofTwoDistinct a₀ a₁ hne)) :=
  FormalSystemEmbeddingTarget_proved _ (ofTwoDistinct_expressive a₀ a₁ hne)

/-- **Item 4, widened.** Four structurally different foundations, the logical
two-valued carrier, the arithmetic `0 ≠ 1`, the set-theoretic `∅ ≠ {∅}`, and the
type-theoretic `𝟚`, each realize the δ core. The primitive distinction is not an
artifact of one foundation's notation; it appears wherever two primitives can be
told apart. -/
theorem named_foundations_embed_delta :
    Nonempty (PRCEmbeddingInto boolLogicSystem)
      ∧ Nonempty (PRCEmbeddingInto peanoSystem)
      ∧ Nonempty (PRCEmbeddingInto setFoundationSystem)
      ∧ Nonempty (PRCEmbeddingInto typeTheorySystem) :=
  ⟨boolLogicSystem_embeds_delta, peanoSystem_embeds_delta,
    setFoundationSystem_embeds_delta, typeTheorySystem_embeds_delta⟩

What this page does not claim

The type-theoretic foundation is a full faithful parse of the entire primitive recognition calculus. Type theory, or any other foundation, is the unique or privileged carrier of the delta core. The declaration proves that the primitive distinction is necessary for any formal system.

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/PRCInevitabilityInstances.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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