Foundation Primitive Recognition Calculus Quotient

The quotient construction

A recognition trace is a finite sequence of endpoints, the discrete positions that a recognition event visits. The framework's ledger, a discrete record of such events, carries a judgment: for a fixed trace, the framework decides when two endpoints are the same for all practical purposes. The quotient construction takes that judgment and turns it into a mathematical object, a set of equivalence classes. Each class gathers all endpoints that the trace judges equivalent, so the raw list of endpoints collapses into a smaller, organized collection. The construction is standard: define an equivalence relation, form the quotient set, and prove that functions respecting the relation descend cleanly to the classes.

The framework proves the judgment is a proper equivalence relation, meaning it is reflexive, symmetric, and transitive. That single fact licenses everything else. With it, the framework forms the quotient type, names the class of an endpoint, and shows that equivalent endpoints land in the same class. The key theorem states that if two endpoints are judged the same, their classes are equal. This is the bridge from a local judgment to a global identification. The framework also provides a recursion principle: any function on endpoints that respects the equivalence can be lifted to the quotient, and the lifted function agrees with the original on every representative. The proof is immediate, but the payoff is structural. The quotient is not an ad hoc grouping; it is a well-defined type with a principled way to compute on it.

In plain terms, the framework establishes that the ledger's notion of sameness is coherent. When the framework says two endpoints are the same, it can treat them as interchangeable without contradiction. That coherence is what lets later constructions speak about classes of endpoints rather than individual ones. The construction is a small but load-bearing piece: it converts a judgment into a reusable mathematical structure, and it does so with machine-checked proofs. The result is that the framework can build higher-level objects on top of the quotient, confident that the identification is sound.

/-- K2.11. The setoid induced by SameT at a fixed trace. -/
def sameSetoid (J : TraceJudgment) (T : Trace) : Setoid Endpoint where
  r := J.same T
  iseqv := {
    refl := J.same_refl_proof T
    symm := by
      intro a b h
      exact J.same_symm_proof T h
    trans := by
      intro a b c hab hbc
      exact J.same_trans_proof T hab hbc
  }

/-- 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

@[simp] theorem endpointClassLift_mk
    (J : TraceJudgment) (T : Trace) {α : Sort _}
    (f : Endpoint → α)
    (hf : ∀ {a b : Endpoint}, J.same T a b → f a = f b)
    (a : Endpoint) :
    endpointClassLift J T f hf (endpointClassOf J T a) = f a := by
  rfl

What this page does not claim

This module does not define the cost function J(x) or prove its uniqueness. This module does not derive the golden ratio or any dimensional forcing. This module does not claim that the equivalence judgment is physically meaningful; it only formalizes the structure given a judgment.

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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