§6 · Root · root: registry, redirects, synthesis · 2026-10-17T21:39:18Z
Remark 6.1 · Registry after the first two returns: a non-split counterexample found twice; launch its audit
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Two agents reached the same non-split counterexample by different routes, and the trace agent proved why the dictionary cannot give the converse. The counterexample goes to an independent auditor; the split agent continues.
| family | agents | state |
|---|---|---|
| trace formalism: one matrix over \(R=\mathbb Z[\overline{\mathbb Q}_\ell^\times]\), twisted Hochschild homology | A1 | proofs returned, unaudited |
| Frobenius twisting: non-split forms of the group case | A2, and A1 in Counterexample 1.5 | counterexample, unaudited |
| fusion and multiplicity two (split) | A3 | running |
| inside (R) and (C) | A1 Proposition 1.8, A2 Remark 2.12 | reduced to the labels without dominant lifts |
Independent agreement. The norm torus \(\operatorname{Res}_{\mathbb F_{q^2}/\mathbb F_q}\mathbb G_m\to\mathbb G_m\) was found by the trace agent (Counterexample 1.5, by the torus criterion \(\ker(X_*(H)_\sigma\to X_*(T)_\sigma)\ne0\)) and by the non-split agent (Counterexample 2.3), which extended it to the unitary twist of every split group (Theorem 2.6).
Decision. The counterexample decides task (1) of Proposition IX once it survives audit, so an auditor (A4) gets the eleven steps it rests on, verbatim, with no opinion attached. The split agent (A3) continues: under (R) all twisting effects vanish (Proposition 1.8, Remark 2.12), and only fusion or cancellation at \(n=1\) can produce a split counterexample.
Builds on
- Counterexample 1.5 · Counterexample outside (R): the norm torus quotient, and the torus criterion
- Summary 1.12 · Verdict on part (2), with consequences for (1) and (3)
- Theorem 2.6 · Theorem: the unitary twist of \((H\times H)/H\) fails (SG) over \(\mathbb F_q(\!(t)\!)\)
- Summary 2.13 · Verdict: sheaves do not imply (SG) when \(T\) is non-split; nothing new under (R)
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