# Attempt tree · Does recollement need finite unions of orbits?

Started 2026-10-20T16:35:27Z · 3 rows · 24 steps · final step R-1 · verdict proved

Status: ✓ proved · ✗ refuted · ⊘ blocked · ? open · ~ heuristic · ↺ superseded · ◐ partial. "← a, b": the step cites a and b. [error], [minor], [inherited]: audit grade.

## A1 · Solver · round 1 · 16:36–16:49 · proved

```text
16:44  A1-1   ✓ Setup         Levels, cylinders and level-wise categories
16:44  A1-2   ✓ Lemma         Cylinder iff quasi-compact open complement        ← A1-1
16:44  A1-3   ✓ Proposition   Recollement for quasi-compact open pieces         ← A1-1, A1-2
16:45  A1-4   ✓ Proposition   Quasi-compactness is also necessary               ← A1-1, A1-2, A1-3
16:45  A1-5   ✓ Counterexample The point 0 in a lattice: no recollement          ← A1-2, A1-4
16:45  A1-6   ✓ Computation   Finite-level check of orbits and cylinders        ← A1-5
16:46  A1-7   ✓ Example       All recollements hold, orbit devissage fails      ← A1-3, A1-5
16:46  A1-8   ✓ Audit         Where 'finite' is used in the proofs              ← A1-2, A1-3, A1-5, A1-7
16:46  A1-9   ✓ Summary       Verdict: quasi-compactness, not finiteness        ← A1-3, A1-4, A1-5, A1-7, A1-8
```

## A2 · Auditor · round 2 · 16:49–17:04 · proved

```text
16:57  A2-1   ✓ Check         A1-1 levels: holds; (P) is an assumption          ← A1-1
16:57  A2-2   ✓ Check         A1-2 cylinder iff quasi-compact: holds            ← A1-2
16:57  A2-3   ✓ Check         A1-3 recollement: holds; cite repair              ← A1-3
16:57  A2-4   ✓ Check         A1-4 necessity via the constant sheaf: holds      ← A1-4
16:58  A2-7   ✓ Check         A1-6 finite-level computation reproduced          ← A1-6
16:58  A2-5   ✓ Objection     A1-5 counterexample holds; (e) misleads           ← A1-5
16:58  A2-6   ✓ Lemma         Cylinder iff finite-codim components              ← A1-2, A2-5
16:59  A2-8   ✓ Objection     A1-7 holds; its diagnosis needs A2-9              ← A1-7
16:59  A2-10  ✓ Objection     A1-8 finiteness audit: three omissions            ← A1-8
16:59  A2-11  ✓ Lemma         Symmetric orbits are cylinders (Chen-Yi 36)       ← A1-8, A2-10, A1-2, A1-3
17:00  A2-9   ✓ Lemma         Finitely many orbits in support is thick          ← A2-8, A1-7
17:00  A2-13  ✓ Lemma         Criterion for any closed stable Z in LX           ← A2-12, A1-2, A1-3, A1-4, A2-6
17:00  A2-14  ✓ Verdict       A1 stands, with four repairs                      ← A2-1, A2-2, A2-3, A2-4, A2-5, A2-6, A2-7, A2-8, A2-9, A2-10, A2-11, A2-12, A2-13
17:01  A2-12  ✓ Objection     A1-9 scope: Z fixed inside one Y_N                ← A1-9, A1-3, A1-4
```

## R · root

```text
17:04  R-1    ✓ Summary       Verdict: quasi-compactness, not finiteness        ← A1-2, A1-3, A1-4, A1-5, A1-7, A1-8, A2-3, A2-5, A2-6, A2-9, A2-10, A2-11, A2-13, A2-14
```