§7 · Root · root: registry, redirects, synthesis · 2026-10-18T04:41:45Z

Theorem 7.4 · Verdict: Sakellaridis's theorem over \(\mathbb F_q(\!(t)\!)\); for \(E_6/F_4\) in both characteristics, (A3a) included

proved audit: minor defect ★ key step

Minor defect (second audit). Re-derived from the parents: over every p-adic F, Theorem 8.0.2 for M_θ follows from (A1)–(A5) over O_F (S4/S4b with Summary 3.12's characteristic-free proofs, (A3a) by Theorem 2.8/Summary 6.9), the normalizations of Audit 4.7/Audit 5.5, Audit 7.3, and (7.5) by the inversion-set argument of Computation 1.7 (factor-2 error on p. 62 and the repair re-derived by hand); over F_q((t)) it follows from the transfer Theorem 4.8/Audit 5.6 with (I_Λ) = Theorem 2.8 at W(F_q); my own scripts confirm that the W-orbit of ρ−ω4 meets ρ_L + a_X^* in two W_X-orbits (so r is not onto; a scan of half-integral points finds no other collision) and that P for the labels of 3ω1^∨ separates them (q = 3, 5, 7). Defects, none changing an E6/F4 claim: (1) in (c), 'the other hypotheses hold over F_q((t)) (Summary 3.12, Summary 5.13)' and the summary's 'every hypothesis in both characteristics' need Audit 5.19 for (A4)(iii)_p, which those steps exclude, and 'So (T1)–(T5) hold' skips the list closing Proposition 4.9 ((T2) from Proposition 4.5/A3-3, (T3) from Audit 4.3, (T5) from Audit 4.7 + Audit 7.3 + Theorem 2.8). (2) (a) cites Theorem 8.0.2 under (A1)–(A5) for all X with no caveat, although the root's own finding with Objection 5.20 is that the printed proof of (7.5) fails and is repaired only for multiplicity-free Θ^+ ⊆ Φ̌_X^+, so for general X (a) (and the 'e.g.' route to (T5)) needs H1. (3) In (d), 'the direct route reaches the same conclusion, with Statement 6.3.1 in characteristic p supplied by (A3a)': for E6/F4 that input is (A3a) over W(F_q) pushed through Theorem 8.0.2 at F'_e and the close-field point counts (Audit 5.19: route B coincides with A4's transfer), so it is not independent confirmation. (4) (b) omits Proposition 4.9's standing data (θ-stable split T containing a maximal θ-split S, split over Λ) and its target ring is built from the p-adic invariants, identified with the characteristic-p ones only for M_θ (Audit 5.6 remark (b)); the Chen–Yi finding is cited to Audit 3.1, which does not contain it (it is in A3's blocked routes and in Audit 4.3).

Proposition VI splits into four audited parts. (a) is the published characteristic-0 theorem. (b) is its characteristic-\(p\) form for symmetric varieties under (T1)–(T5). (c) says that \(E_6/F_4\) satisfies every hypothesis in both characteristics, (A3a) included. (d) is the criterion. With Proposition VII, (SG) fails for \(E_6/F_4\) over \(\mathbb F_q(\!(t)\!)\) and over every \(p\)-adic field.

Proposition VI (\(p\) odd).

(a), characteristic 0. Theorem 8.0.2 under (A1)–(A5), cited. For ramified \(F\) the normalizations are checked (Audit 4.7, Audit 5.5), and [Sa08] holds over every field of characteristic 0 (Audit 7.3).
(b), characteristic \(p\). For a symmetric \(X\) over \(W(\mathbb F_q)\) under (T1)–(T5): \(K\backslash X(\mathbb F_q(\!(t)\!))\simeq\Lambda_S^+\) and \(C_K\simeq\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\) with \(\varphi_0\mapsto1\) (Proposition 4.9, audited Audit 5.7). A second, direct form reruns the analytic half of the proof in characteristic \(p\). It needs (A2), (A5), (A3b), (A3c), Statements 6.3.1 and 7.1.5 in characteristic \(p\), the cone condition and H1–H3 (Objection 5.20), and not (A3a).
(c), \(E_6/F_4\). (A3a) holds over every discrete valuation ring with 2 invertible (Theorem 2.8, audited Summary 6.9). The other hypotheses hold over \(\mathbb F_q(\!(t)\!)\) (Summary 3.12, audited Summary 5.13) and over \(p\)-adic fields (S4, S4b). So (T1)–(T5) hold.
(d), the criterion. For \(E_6/F_4\), over \(\mathbb F_q(\!(t)\!)\) (Theorem 4.8, audited Audit 5.6) and over every \(p\)-adic \(F\): \(\mathcal H_K\varphi_0=C_K\) iff \(r\) is onto. Since \(r\) is not onto (Proposition VII), \(C_K=\mathcal H_K\varphi_0\oplus\mathbb C\,\mathbf 1_{Kx_{3\omega_1^\vee}}\). The direct route reaches the same conclusion, with Statement 6.3.1 in characteristic \(p\) supplied by (A3a) (Audit 5.19, Summary 5.21).

Findings beyond the question. On p. 62 Sakellaridis derives (7.5) from \(|\Theta^+|-w|\Theta^+|=\check\rho-w\check\rho\). That identity is false by a factor 2 when \(\Theta^+=\check\Phi_X^+\) (Computation 1.7, Audit 5.17; checked by the root). (7.5) still holds when \(\Theta^+\subseteq\check\Phi_X^+\) has no repeated coroot, which covers \(E_6/F_4\). Chen–Yi prove their Prop. 17(i), the pages' input D0, by citing results over \(\mathbb C\) (Audit 3.1, Audit 4.3, Audit 5.7).

Open. For a general symmetric \(X\) without the transfer: H1–H3, and Sakellaridis's Theorem 6.3.2 in characteristic \(p\). The audit pass (/solve-maths-audit) has not been run.

Builds on

Theorem 2.8 · Theorem: Axiom 2.2.1 holds for \(\mathcal M_\theta\) over every DVR with \(2\) invertible
Summary 6.9 · Verdict: (A3a) holds for \(\mathcal M_\theta\) over every DVR with \(2\) invertible
Summary 3.12 · (Q3) verdict: (A1)–(A3c), (A5), (A4)(i,ii,iv) hold over \(\mathbb F_q((t))\)
Summary 5.13 · Verdict of audit A5: transfer route A4-2…A4-9 and char-p hypotheses A3-1…A3-10
Theorem 4.8 · Theorem: Theorem 8.0.2 for \(\mathcal M_\theta\) over \(\mathbb F_q((t))\); (SG) fails
Proposition 4.9 · Proposition: what the characteristic-\(p\) theorem needs for symmetric \(X\)
Audit 5.6 · Audit of A4-8: Theorem 8.0.2 for M_θ over F_q((t)); uniqueness; (SG)
Audit 5.7 · Audit of A4-3 (τ closed immersion) and A4-9 (general symmetric X)
Audit 7.3 · Check: Sakellaridis's 2008 paper is stated for every local field of characteristic \(0\)
Objection 5.20 · Audit of A1-9: the characteristic-p Theorem 8.0.2 and its hypotheses
Summary 5.21 · Verdict of the second audit: A1-1 … A1-9 (direct rerun over F_q((t)))

Built on by

Summary S1.6 · Verdict inputs A5-20 and A5-21 re-derived; how R-4(b) and its open items read
Summary 7.5 · Verdict after the audit: unchanged for \(E_6/F_4\); the general direct form completed

Table 1. Page production by session
	tokens	US $
The run, its in-run audits and the pages (from 01:50 UTC)01:50–05:20 UTC	349 M	$144
This conversation: the audit pass (eight reviewers, one salvage agent, re-checks) and the simplified VIb page05:21–07:33 UTC	267 M	$106
Removing the detail switch from the published trees05:34–05:51 UTC	15 M	$7
Total	631 M	$257