# IC coordinates and function-detected strata

**Verdict:** IC coordinate inversion gives a certified subset at one field and exact pure recovery over all extensions; original-field no-cancellation remains an additional input.

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 \(q\) be odd, \(k=\overline{\mathbb F}_q\), \(\ell\ne\operatorname{char}k\), \(F_n=\mathbb F_{q^n}((t))\), \(K_n=G(\mathbb F_{q^n}[[t]])\), and \(X=G/G^\theta\). Work in the bounded, finite-orbit-support \(L^+G\)-equivariant \(\overline{\mathbb Q}_\ell\)-category of trace.html. Assume its (R), (C), D0–D3 explicitly: rational Frobenius-stable orbit labels \(\nu\in\Lambda_S^+\), finite lower sets and orbit closures, one rational \(K_n\)-orbit per label, a Frobenius-compatible bounded perverse t-structure with simple objects \(I_\nu=\mathrm{IC}_\nu\), \(\operatorname{End} I_\nu=\overline{\mathbb Q}_\ell\), and finite geometric decomposition of \(M_\lambda=V_\lambda*I_0\) into shifted \(I_\nu\). Normalize \(I_\nu|_{\mathcal O_\nu}=\overline{\mathbb Q}_\ell[\delta(\nu)]\) with open-stalk Frobenius 1 and no Tate twist. For every \(\lambda\), decide what the functions \(f^{(n)}_\lambda=\operatorname{tr}_n M_\lambda\), together with the known IC trace functions, determine about \(\Sigma_\lambda=\{\nu:\exists i,\ I_\nu[-i]\text{ is a geometric direct summand of }M_\lambda\}\).

## 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. Fixed-q equality for each canonical Satake convolution requires a verified no-cancellation statement. All-extension absence needs a supplied multiplicity bound for a finite stopping test.

## 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/.