# Exact extension by zero

**Verdict:** Open extension is ordinary-sheaf exact and stable exact; the thick-span argument is valid under H2, while perverse t-exactness can fail.

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

Fix a field E of characteristic zero, complex algebraic spaces with analytic constructible E-sheaves, and an open immersion j:U→Z with closed complement i:D→Z. On the target page E=k=ℂ, Z is a closed finite union of L⁺G-orbits in LX, and C_W=Shv_cons(L⁺G\\W) is the small idempotent-complete stable category of bounded equivariant constructible complexes, for each stratum or union W. H2 assumes that the indicated operations restrict to these categories. For every such j, decide whether j_!:C_U→C_Z preserves finite cofibers and retracts; distinguish this from ordinary sheaf-heart exactness and from middle-perverse t-exactness. Completion means proving the preservation needed for T2, explaining the page's induction, and either proving universal perverse t-exactness or giving a certified counterexample.

## 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/.