Foundation Primitive Recognition Calculus Quotient Endpoint Class

Endpoint classes

In the Recognition Science framework, a ledger (a discrete record of events) is built from traces, and each trace has endpoints. The declaration EndpointClass defines what it means for two endpoints to be the same for the purposes of a given judgment. It does this by taking the relation "judgment J says endpoints a and b are the same on trace T" and turning it into a setoid, which is a mathematical structure with a reflexive, symmetric, and transitive relation. The endpoint class is then the quotient of all endpoints by that relation: each class collects together every endpoint that the judgment cannot distinguish.

The construction is deliberately minimal. It uses only the standard quotient construction from type theory, which is the formal way of saying "group things by an equivalence relation." The framework proves that if two endpoints are judged the same, they land in the same class, and it provides a lifting rule: any function on endpoints that respects the judgment's sameness relation can be applied to the classes themselves. These are the only properties the declaration establishes. It does not say which endpoints are actually the same, because that depends entirely on the particular judgment J and trace T supplied to it.

In Recognition Science, this quotient is the bridge between raw recognition events and the coarse-grained objects the framework reasons about. The endpoint class is what lets the framework say "these two occurrences are one thing" without committing to any particular criterion for sameness. That neutrality is the point: the judgment supplies the criterion, and the quotient supplies the machinery. The result is a reusable formal tool that works for any judgment, not a claim about any specific physical situation.

What the declaration does not claim is just as important. It does not assert that any two endpoints are actually the same, only that if a judgment says so, they share a class. It does not define what a judgment is, what a trace is, or what an endpoint is; those come from elsewhere in the framework. And it does not prove that any particular physical system has endpoints or that recognition events occur. The declaration is a piece of pure logical plumbing: it says how to build classes from a sameness relation, nothing more.

/-- K2.11. Endpoint trace-classes under SameT. -/
def EndpointClass (J : TraceJudgment) (T : Trace) : Type :=
  Quot (sameSetoid J T)

/-- 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

/-- K4.4. Quotient recursion for SameT-respecting maps. -/
def endpointClassLift
    (J : TraceJudgment) (T : Trace) {α : Sort _}
    (f : Endpoint → α)
    (hf : ∀ {a b : Endpoint}, J.same T a b → f a = f b) :
    EndpointClass J T → α :=
  Quot.lift f (by
    intro a b h
    exact hf h)

What this page does not claim

The declaration does not assert that any two endpoints are actually the same. It does not define what a judgment, trace, or endpoint is. It does not prove that any physical system has endpoints or that recognition events occur.

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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