# Finite activity and orbit-standard generation

**Verdict:** With the required recollement and orbitwise generation, standards thick-generate precisely the finite-active objects; infinite orbit closures are allowed only with the enlarged hypotheses.

21 October 2026; 2 agents (solver, fresh auditor); 14.0 minutes from brief preparation through synthesis, including queue time. Validation: in-run candidate audit only.

## Problem and setting

Work over \(\mathbb C\), with a characteristic-zero coefficient field \(E\). Let \(H\) act on a stratified space \(Y=\bigsqcup_{\lambda\in\Lambda}O_\lambda\). Every orbit is locally closed, every orbit closure is a union of orbits, and \(O_\nu\subset\overline{O_\lambda}\) defines a partial order. There is no finiteness assumption on \(\Lambda\) or its lower sets. For every invariant locally closed subspace \(W\) needed below, let \(\mathcal D(W)\) be a small idempotent-complete stable category of bounded equivariant constructible complexes. Assume:

(R) Restriction and extension by zero along the inclusions used below are exact, preserve these categories, satisfy composition and open–closed recollement; closed extension is fully faithful. Orbit restrictions are jointly conservative on each such \(W\), and extension by zero has zero restriction outside its image.

(O) \(\mathcal D(O_\lambda)=\operatorname{Thick}(\underline E_{O_\lambda})\). A sufficient hypothesis is a bounded t-structure with heart \(\operatorname{Vect}^{\rm fd}_E\), generated by the constant sheaf.

In particular every \(\Delta_\lambda=a_{\lambda!}\underline E_{O_\lambda}\) belongs to \(\mathcal D(Y)\), where \(a_\lambda:O_\lambda\hookrightarrow Y\). Define the active orbit set \(S(M)=\{\lambda:a_\lambda^*M\ne0\}\) and \(\mathcal D_{\rm fin}(Y)=\{M:|S(M)|<\infty\}\). The quantified target is
$$\forall M\in\mathcal D(Y),\qquad M\in\operatorname{Thick}\{\Delta_\lambda\}_{\lambda\in\Lambda}\quad\Longleftrightarrow\quad |S(M)|<\infty.$$
Completion means proving both implications, a genuine infinite-orbit example, and the distinction from transfinite/localizing generation and placid pieces. The topological closure of the nonzero-stalk locus is not the active set.

## Completion criterion

Give a proof or certified counterexample at the stated level, specify any additional assumptions, and keep stronger claims with exact unmet obligations. Finite numerical checks or analogies do not complete the question.

## Checklist

- Functor domains, codomains, shifts, and ordinary/perverse meanings.
- Finite thick operations versus arbitrary coproducts.
- Hypotheses of every source and preservation on the intended category.
- Geometric versus rational labels; examples at their actual level.

## Portfolio

Table 1. Independent roles.

| agent | round | role | return |
|---|---:|---|---|
| A1 | 1 | Candidate proof | [A1.json](returns/A1.json) |
| A2 | 2 | Fresh adversarial check | [A2.json](returns/A2.json) |

## Registry

The direct candidate and its independent audit support the stated answer. No mathematical gap in the stated conditional answer. Applicability to the original geometry retains the explicit hypotheses; this is an in-run candidate audit, not a separate audit of the root synthesis.

## References

[Verified literature](references.md), with corrections in [audit-summary.md](audit-summary.md). The original source assertions and locators remain in the immutable returns.

## Files

[Verdict](verdict.md) · [attempt tree](tree.md) · [tree data](tree.json) · [candidate proof](candidate-proof.md) · [presentation plan](presentation.md) · briefs/ · returns/ · scripts/.