# The lowest-degree convolution stalk

**Verdict:** The constant section injects into the lowest stalk through ordinary restriction, the edge isomorphism and proper base change, in the stated finite-level Betti realization.

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

Let the base field be \(\mathbb C\) and coefficient field be \(\Bbbk=\mathbb C\), kept distinct. Let \(G\) be connected reductive, \(X\) a smooth affine homogeneous \(G\)-variety with basepoint \(x_0\), and suppose \(L^+X=L^+Gx_0\). Put \(\mathcal O=\mathbb C[[t]]\), \(F=\mathbb C((t))\), \(\operatorname{Gr}=LG/L^+G\). Fix a dominant cocharacter \(\lambda\), \(Z=\operatorname{Gr}_{\le\lambda}\), \(U=\operatorname{Gr}_\lambda\), \(j:U\hookrightarrow Z\), \(d=\dim_{\mathbb C}Z=\langle2\rho,\lambda\rangle\), and \(V=\operatorname{IC}_Z=j_{!*}\underline\Bbbk_U[d]\in D^b_c(Z,\Bbbk)\). The symbol \(\mathcal H^q\) denotes ordinary cohomology sheaves. Let \(E=Z\widetilde\times L^+X\), \(p:E\to Z\), \(m:E\to LX\), \([g,x]\mapsto gx\). The page's conventions are \(\operatorname{IC}_0=\underline\Bbbk_{L^+X}\) extended to \(LX\), without a shift, and \(K=V*\operatorname{IC}_0=Rm_!p^*V\), without a shift. These operations are interpreted in the finite-level Betti realization used on the page. For every \(y\in LX(\mathbb C)\), put \(F_y=\{gL^+G\in Z:g^{-1}y\in L^+X\}\), \(r_y:F_y\hookrightarrow Z\). The target is a canonical injection
$$H^0(F_y,\underline\Bbbk)\hookrightarrow H^{-d}(K_y),$$
which is nonzero whenever \(F_y\ne\varnothing\); consequently the set of nonzero stalks of \(K\) is exactly \(m(E)=Z\cdot L^+X\). A complete answer proves this injection, derives the stalk formula through a finite-type proper model, checks shifts and t-structures, and computes an illustrative finite-dimensional example.

## 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/.