IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.PRCTypeTheoryParse
Parses Martin-Löf / CIC type theory into the Primitive Recognition Calculus by taking the two-element type as Lean's Bool and packaging a minimal type-theoretic system around it. Establishes canonicity, no-confusion, expressiveness, and a non-degenerate embedding of the distinction dichotomy. Downstream foundation-parsing modules import this bridge. The file is mostly short definitional and embedding lemmas, not a long derivation.
claimThe module fixes the two-point type $2=\{\mathsf{false},\mathsf{true}\}$ (Lean $\mathsf{Bool}$) and a type-theoretic system $\mathcal{T}$ that is expressive, expression-reflexive, and non-degenerate, and that embeds and realizes the PRC distinction dichotomy $\Delta$.
background
Primitive Recognition Calculus treats distinction as a primitive: before measures or dynamics, one needs a clean two-way split (the dichotomy $\Delta$) that can be reflected inside a formal language. The upstream module supplies that dichotomy; this file asks whether ordinary type theory already realizes it.
The main object is the canonical two-element type of Martin-Löf type theory / CIC, identified here with Lean's $\mathsf{Bool}$, whose closed terms are $\mathsf{false}$ and $\mathsf{true}$. Around it the module packages a small type-theoretic system: canonicity (every closed term of $2$ is one of the two constructors) and no-confusion (the constructors are distinct and injective in the usual sense).
Those facts let the system be checked for expressiveness, for an embedding of $\Delta$, for expression-reflexivity, and for non-degeneracy, so that type theory is not a vacuous host for the recognition primitives.
proof idea
Definition-heavy module with short supporting lemmas. It aliases the two-point type to Bool, records canonicity and no-confusion for that type, then assembles a type-theoretic system record. Separate lemmas discharge expressiveness, the embedding of the distinction dichotomy, expression-reflexivity, and non-degeneracy; a final statement packages that type theory realizes $\Delta$. No deep tactic scripts: mostly structure assembly and one-line appeals to Bool's standard API.
why it matters in Recognition Science
Sits in the Foundation / Primitive Recognition Calculus layer: it is the type-theory parse of the distinction dichotomy, not a physics derivation. The sole recorded consumer is the foundations-parsed aggregator module, which imports this bridge so later PRC material can cite a concrete formal host for $\Delta$. In the broader Recognition stack this is scaffolding under the forcing chain (T0–T8): before J-cost uniqueness, $\phi$, the eight-tick octave, or $D=3$, the calculus needs a verified two-valued distinction that ordinary type theory already supplies. It does not itself force physical constants; it certifies that the logical substrate is non-empty and non-degenerate.
scope and limits
- Does not derive J-cost uniqueness, $\phi$, or any T5–T8 forcing step.
- Does not claim a full CIC metatheory; only a minimal system around the two-point type.
- Does not prove physical mass, $\alpha$, or continuum limits from type theory.
- Does not replace the upstream distinction-dichotomy module; it only hosts and embeds it.
- Does not assert uniqueness of the type-theoretic realization among all formal systems.