research(cantor-diagonalization-oq-04-oq-01): S2 — trivial setoid + refinement-descent (build pending)

[rjwalters]

Summary

Session 2 rounding-out of the Lawvere FPT for Setoids in proofs/Proofs/CantorDiagonalizationOQ04OQ01.lean. S1 (PR #16393) established the failure side (Bool/discrete-setoid impossibility, ℕ-mod-parity impossibility, Cantor diagonal); S2 establishes the success side and the refinement structure that organizes both.

File: 166 → 221 lines, 8 → 12 theorems, 3 → 5 definitions, 0 sorries unchanged, 0 axioms unchanged. Status remains verified / original.

Part VIII — Trivial setoid (success side)

def trivialSetoid (Y : Type*) : Setoid Y where
  r := fun _ _ => True
  iseqv := ⟨fun _ => trivial, fun _ => trivial, fun _ _ => trivial⟩

theorem trivial_setoid_codes_endomorphisms (Y : Type*) [Inhabited Y] :
    CodesEndomorphismsSetoid Y (trivialSetoid Y) where
  encode := fun _ => default
  decode := fun _ => id
  retract := fun _ _ => trivial

theorem bool_trivial_setoid_codes_endomorphisms :
    CodesEndomorphismsSetoid Bool (trivialSetoid Bool) :=
  trivial_setoid_codes_endomorphisms Bool

The pair bool_trivial_setoid_codes_endomorphisms ↔ cannot_code_endomorphisms_bool_setoid (S1) shows directly that coding-feasibility is genuinely setoid-dependent, not a fixed property of the underlying type.

Part IX — Refinement-descent

def IsRefinement {Y : Type*} (s t : Setoid Y) : Prop :=
  ∀ a b : Y, s.r a b → t.r a b

theorem coding_descends_to_coarser {Y : Type*} {s t : Setoid Y}
    (hst : IsRefinement s t) (c : CodesEndomorphismsSetoid Y s) :
    CodesEndomorphismsSetoid Y t where
  encode := c.encode
  decode := c.decode
  retract := fun g y => hst _ _ (c.retract g y)

theorem refines_trivial {Y : Type*} (s : Setoid Y) : IsRefinement s (trivialSetoid Y) :=
  fun _ _ _ => trivial

coding_descends_to_coarser shows coding-existence is monotone-downward in the refinement lattice on Setoid Y: finer admits coding ⇒ coarser does too. refines_trivial places the trivial setoid at the canonical top of that lattice. Together with the discrete setoid (canonical bottom, discreteSetoid from S1), they bracket the full coding spectrum.

Why this is real progress (and the limit thereof)

• Establishes the success side: S1 had three failure witnesses (Bool/discrete, ℕ/parity, Cantor). The trivial-setoid result is the canonical success witness — every inhabited type codes vacuously under the one-class setoid.
• Sets up OQ-04-OQ-01-OQ-02 (characterization): the refinement-descent lemma is the structural backbone for a future characterization of admissible setoids — they form a downward-closed subset of the refinement lattice on Setoid Y. The boundary is open (cardinality? algebraic? Lawvere-style?).
• No axiom/sorry delta: this is a structural rounding-out, not Phase-3 work. Status remains verified / original.
• Honesty: the four new theorems are all 1-2 line proofs. The trivial-setoid case is genuinely vacuous (the retract holds because True is always provable). The refinement-descent lemma is essentially function composition. This is solid structural setup, not a deep result.

Build status: pending

The worktree's proofs/.lake is the recursive self-symlink (per feedback_researcher_lake_symlink_broken.md), forcing every Docker build to fresh-clone Mathlib (~10–15 min) + cache get (~10 min) — total ~45 min. Following the established convention from PRs #16936 (cantor-OQ-01-OQ-01-OQ-02-OQ-01 S5) and #17008 (ehrhart-cube-proven-oq-02 S4), this PR is opened as draft.

The new theorems use only basic Mathlib primitives already exercised in this file:

• Setoid constructor with Equivalence proof from anonymous tuple
• default (from Inhabited), id, trivial
• Anonymous constructor of CodesEndomorphismsSetoid records using fields

Risk of build failure is low; review of the proof bodies should be straightforward by inspection.

Files Modified

• proofs/Proofs/CantorDiagonalizationOQ04OQ01.lean (+55 lines: 2 defs + 4 theorems + 2 section dividers + module-header Key Results list extended)
• src/data/proofs/cantor-diagonalization-oq-04-oq-01/meta.json (lineCount 166→221, theoremCount 8→12, definitionCount 3→5 in both meta and leanFile blocks; +4 originalContributions; +2 sections for Parts VIII/IX; conclusion summary updated)
• src/data/research/problems/cantor-diagonalization-oq-04-oq-01.json (phase COMPLETED→ACT, status completed→active, iteration 1→2; +6 builtItems, +4 insights, +3 nextSteps; lastUpdate; leanFiles entry)
• research/problems/cantor-diagonalization-oq-04-oq-01/knowledge.md (Session 2 narrative)

Test plan

• Re-run ./proofs/scripts/docker-build.sh Proofs.CantorDiagonalizationOQ04OQ01 from a worktree with warm Mathlib cache
• Confirm trivialSetoid, trivial_setoid_codes_endomorphisms, bool_trivial_setoid_codes_endomorphisms, IsRefinement, coding_descends_to_coarser, refines_trivial type-check
• pnpm build for gallery integration

Related

• Session 1 (PR research(cantor-diagonalization-oq-04-oq-01): Lawvere FPT for Setoids #16393): initial Lawvere FPT for Setoids — 8 theorems, 3 definitions, 0 sorries/axioms
• Parent: cantor-diagonalization-oq-04 (Type-level retraction version of Lawvere FPT)
• Sibling: cantor-diagonalization-oq-03 (surjection version, candidate for parallel setoid generalization)
• Open follow-up: cantor-diagonalization-oq-04-oq-01-oq-02 (characterization of admissible setoids — S2 lays the structural backbone)

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