IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.PRCFoundationsParsed
Set theory, type theory, and category theory each embed into the Primitive Recognition Calculus once parsed as formal systems with their native distinction mechanisms. The module aggregates those four parse layers and records that all three foundations realize the δ core non-degenerately. Anyone comparing foundational frameworks inside RS cites the aggregate theorems here. The argument is assembly: import the parses and discharge the shared FormalSystem/PRC embedding obligations.
claimEach of set theory, type theory, and category theory, once presented as a formal system with its own distinction relation, admits a Primitive Recognition Calculus embedding that realizes the $\delta$ core; the embeddings are non-degenerate and each foundation owns its distinction mechanism. Full ZFC with infinity likewise realizes $\delta$.
background
Primitive Recognition Calculus (PRC) is the RS layer that treats recognition as a formal calculus of distinctions. A formal system is admitted into PRC when it supplies a distinction mechanism that can be read as an instance of the $\delta$ core (the minimal recognition/distinction structure underlying the forcing chain).
This module sits above four parse modules: set-theory parse, full ZFC parse, type-theory parse, and category-theory parse. Each parse turns a classical foundation into the shared FormalSystem interface by naming how that foundation draws distinctions (membership, typing, morphisms/objects).
The local claim is comparative, not constructive from scratch: once those parses exist, the three named foundations (and ZFC with infinity) all realize $\delta$ inside PRC, without collapsing to a degenerate distinction structure.
proof idea
Definition-and-aggregation module over the four foundation parses. It does not reprove the individual embeddings; it imports PRCSetTheoryParse, PRCFullZFCParse, PRCTypeTheoryParse, and PRCCategoryTheoryParse, then packages the shared consequences: the three foundations realize $\delta$, they are non-degenerate, each owns its distinction mechanism, and set theory with infinity realizes $\delta$. Proof content is thin wrapper/assembly over those upstream parse theorems.
why it matters in Recognition Science
In Recognition Science the forcing chain (T0–T8) and the Recognition Composition Law presuppose a neutral distinction core that is not hostage to one school of foundations. This module records that set theory, type theory, and category theory each embed into that core via PRC, so later physics derivations (J-uniqueness, $\varphi$, eight-tick octave, $D=3$) need not pick a single metamathematical host.
Sibling aggregate facts (three_foundations_realize_delta, non-degeneracy, own-distinction, and ZFC-with-infinity) are the citation surface. No downstream used_by edges are recorded yet; the module is a foundation-side ledger rather than a direct input to a named physics theorem. It closes the "which foundations see $\delta$?" question at the parse layer.
scope and limits
- Does not construct set theory, type theory, or category theory inside Lean from first principles.
- Does not prove logical equivalence or bi-interpretability of the three foundations.
- Does not derive physical constants, J-uniqueness, or forcing steps T5–T8.
- Does not claim every possible formal system realizes $\delta$, only the named parses.
- Does not supply downstream physics theorems; used_by is currently empty.
depends on (4)
-
IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.PRCCategoryTheoryParse -
IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.PRCFullZFCParse -
IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.PRCSetTheoryParse -
IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.PRCTypeTheoryParse