§6 · Root · root: registry, redirects, synthesis · 2026-10-17T22:43:24Z
Remark 6.2 · Registry after the split agent and the first audit: audit the split candidate and the mechanism
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The non-split counterexample survived its audit. The split agent made Sakellaridis's Example 8.0.1 concrete: functions fail over every local field, sheaves generate given D1–D2 for it. A fresh auditor takes the split candidate; the first auditor continues with the mechanism.
| family | agents | state |
|---|---|---|
| Frobenius twisting | A2, A1, A4 | settled: the counterexample holds (Summary 4.12; six one-line repairs) |
| fusion and multiplicity two | A3 | candidate: (SG) fails over every \(F\) (Theorem 3.5); \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\) given D1–D2 for \(X\) (Corollary 3.9) |
| trace formalism | A1 | unaudited |
| inside (R) and (C), symmetric | A1, A2, A3 | blocked at the uncovered labels |
Decision. Two audits in parallel. A fresh auditor (A5) gets the eleven steps of the split candidate verbatim: the Mackey argument in positive characteristic and the orbit, fibre and multiplicity computations on \(LX\) are the points to check. The first auditor (A4) continues with the mechanism (Theorems A and T, the criteria, close fields, the reduction under (R) and (C)); its context on the setting is the asset there. No new search agent is launched: inside (R) and (C) all three search agents stopped at the same theorem-strength obstruction, and no new mechanism has appeared.
Builds on
- Summary 4.12 · Verdict of the audit: the unitary twist is a valid counterexample; the repairs are minor
- Theorem 3.5 · (SG) fails for Sakellaridis's \(X\) over \(\mathbb F_q(\!(t)\!)\), every \(q\)
- Corollary 3.9 · Given D1–D2 for \(X\): \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\), \(m^{(n)}=D_ncD_n'\)
- Summary 3.13 · Verdict: \(\mathrm{SL}_2^3\backslash\mathrm{Sp}_4^2\) fails (SG); sheaves given D1–D2
- Theorem 1.2 · Theorem A: decategorification is base change of one \(R\)-matrix
- Theorem 1.3 · Theorem T: trace of Frobenius on \(\mathcal C_k\) is \(C_K\); the missing retract classes
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