Foundation Primitive Recognition Calculus Quotient Endpoint Class Eq Of Same

Sameness classes

The declaration endpointClass_eq_of_same is a formal theorem in the Recognition Science framework's machine-checked library of formal theorems. It states a simple bridging fact: if a recognition judgment declares two endpoints to be the same under a given trace, then those two endpoints belong to the same equivalence class. In plain terms, when the ledger's rule says two things are equivalent, the quotient construction that groups endpoints by that equivalence relation must place them together. This is not a claim about physics; it is a claim about how the framework's own definitions behave.

The theorem works through a standard mathematical construction. The framework defines a setoid, an equivalence relation on endpoints, using the judgment's sameness relation. It then forms the quotient type, whose elements are the equivalence classes themselves. The theorem endpointClass_eq_of_same is the proof that the map sending an endpoint to its class respects the equivalence relation: equivalent endpoints map to the same class. This is the well-definedness condition that makes the quotient usable for further reasoning.

In Recognition Science, this theorem is a load-bearing step. It ensures that when the framework reasons about endpoint classes, it can safely replace one endpoint with another that the judgment deems the same. Without this theorem, any function defined on classes would risk depending on which representative was chosen, and the framework's counting and classification arguments would collapse. The theorem is proved by the quotient construction's soundness principle, which is one of the three standard axioms the framework's kernel relies on.

The theorem does not claim that the sameness relation itself is correct, meaningful, or physically true. It takes the judgment's sameness relation as given and only proves that the quotient respects it. It also does not claim that any particular endpoints are the same; it only states the conditional: if the judgment says they are the same, then they share a class. The theorem is a formal bridge between the judgment relation and the quotient structure, not an assertion about the world.

/-- K4.4. SameT endpoints determine the same quotient class. -/
theorem endpointClass_eq_of_same
    (J : TraceJudgment) (T : Trace) {a b : Endpoint}
    (h : J.same T a b) :
    endpointClassOf J T a = endpointClassOf J T b :=
  Quot.sound h

What this page does not claim

The theorem does not assert that any specific endpoints are the same, only that sameness implies shared class membership. The theorem does not establish that the sameness relation is physically meaningful or correct. The theorem does not prove that the quotient construction is the only way to group endpoints.

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/Quotient.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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