§1 · Agent 1 · Direct rerun · 2026-10-18T04:06:04Z

Theorem 1.9 · Theorem 8.0.2 over \(\mathbb F_q((t))\), \(p\) odd, symmetric \(X\): hypotheses

proved audit: error ★ key step

Error. The E6/F4 corollary holds: I re-derived it from Proposition 1.5, Proposition 1.6, Computation 1.7 (beta built directly from Phi_X^+, leading coefficients checked), Proposition 1.2 with sigma(lambda)=-w0 lambda, X(O)=Kx0 (Lang for F4 plus Hensel) and Proposition VII; it is conditional on (A2), (A3b'), (A3c) and (S6.3.1)_p. The general symmetric theorem fails as written, in five places. (1) 'With (iii)_p and (iv)_p, Sakellaridis's proof of Theorem 7.2.1 gives B_w=beta/(beta o w)': that step also uses the uniqueness half of Lemma 7.1.4 (one colour per type-G root, a pair for type T), which is proved only by inspecting the characteristic-0 classification. Because D_V counts colours with multiplicity, two colours of valuation gamma-check satisfy 6.3.1 and 7.1.5 yet square the factor. So a characteristic-0 lemma is used in characteristic p, although the step claims both characteristic-0 places are kept as explicit hypotheses. (2) (7.5) is justified only by Computation 1.7's argument, which covers multiplicity-free Theta+ in Phi_X^+ only. (3) Lemma 1.4 and Proposition 1.5 are applied with F-rational adapted data that are neither assumed nor proved. (4) Non-vanishing of the leading coefficients (hypothesis (c) of Proposition 1.2, including c!=0, which is obtained 'by Lemma 4.2.3' although that lemma uses (4.5)) is not shown for general Theta. (5) 'No type N' is presupposed by (iii)_p. Objection 5.20's H1-H3 repair (1)-(3) and (5): H1 with T=cone(Theta+) also forces the colours of each simple root to be exactly those of 6.3.1. They do not repair (4), and Objection 5.20's 'c!=0 follows from Omega_chi!=0' is a non sequitur for general X. conclusion_holds refers to the E6/F4 corollary, the part used downstream; the general statement is plausible but unproven. The conclusion still holds by another argument.
Used later by: 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))), Theorem 7.4 · Verdict: Sakellaridis's theorem over \(\mathbb F_q(\!(t)\!)\); for \(E_6/F_4\) in both characteristics, (A3a) included.
Corrected by: Objection 5.20 · Audit of A1-9: the characteristic-p Theorem 8.0.2 and its hypotheses, Lemma S1.1 · (7.5) for every \(\Theta\): H1 is equivalent to one colour set per simple root, Lemma S1.3 · Positivity of \(\hat S\) forces \(c\neq0\) and \(a_J/c>0\) over every local field, Theorem S1.5 · Theorem 8.0.2 over \(\mathbb F_q((t))\) for symmetric \(X\): the complete hypothesis list.

States the characteristic-p Theorem 8.0.2 for symmetric varieties with p odd, under (A2), (A5), Cartan and Iwahori axioms with antidominant labels, and Statements 6.3.1/7.1.5 read in characteristic p; Axiom 2.2.1 is not used. For E6/F4 the only analytic hypothesis left open is the char-p value of Statement 6.3.1.

Theorem 1.9.

Theorem (Theorem 8.0.2 over \(\mathbb F_q((t))\)). Let \(F=\mathbb F_q((t))\), \(p\) odd, \(G\) as in (A1), \(\theta\) an involution of \(G\) over \(O\), and \(X=G/G^\theta\). Let \(W_X'=N_W(\mathfrak a)\cap[W/W_L]\) act on \(\Lambda_X\), with closed antidominant chamber \(C^-\), \(\Lambda_X^-=\Lambda_X\cap C^-\), and \(A_X^+\) the points of \(A_X(F)\) with label in \(\Lambda_X^-\). Assume:

(A2) and (A5) (\(X\) affine homogeneous; \(X̊(F)=x_0B(F)\));
(A3b′) \(X(F)=\bigsqcup_{\lambda\in\Lambda_X^-}x_\lambda K\) with \(x_\lambda\in A_X^+\) (Axiom 2.4.1 with antidominant labels);
(A3c) \(xJ\subseteq xB(O)\) for \(x\in A_X^+\) (Axiom 2.4.2);
(A4)(iii)\(_p\) Statement 6.3.1 for the cocycle \(B_w\) of Proposition 1.6(d), for a set of reflections \(w_\gamma\) (\(\gamma\in\Delta_X\)) generating \(W_X'\);
(A4)(iv)\(_p\) Statement 7.1.5 for \(\Theta\), and (E) \(\operatorname{cone}(\Theta^+)\cap C^-=\{0\}\).

Then there is a canonical isomorphism \(C_c^\infty(X(F))^K\cong\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X'}\). Under it \(f\in\mathcal H_K\) acts as multiplication by \(r(\mathcal Sf)\) and \(\mathbf 1_{x_0K}\mapsto1\). Hence \((\mathrm{SG})\iff r\) onto.

Axiom 2.2.1 ((A3a)), "no type \(N\)" and "\(L_\gamma\) classical" ((A4)(i),(ii)) are not used: in Sakellaridis's proof they enter only the proof of Statement 6.3.1 (Theorem 6.3.2). The proof also cites, from the characteristic \(\neq2\) literature: Springer 1985 (finiteness of \(B\)-orbits; a point with \(\theta_y\)-stable torus in \(B\)), Richardson 1982 (\(N_W(\mathfrak a)/W_L\) is the restricted Weyl group; not needed for \(M_\theta\)), Conrad–Gabber–Prasad A.8.10 and 2.1.10, Bernstein–Zelevinsky, and Bernstein's principle.

Corollary for \(M_\theta\). For \(X=M_\theta\) over \(\mathbb F_q((t))\), \(p\) odd, the hypotheses reduce to (A2), (A3b′), (A3c) and \((\mathrm{S6.3.1})_p\). Here \(W_X'\cong W(A_2)\), \(\Theta^+=\check\Phi_X^+\), (E) holds, and \(c=(1+q^{-4})(1+q^{-4}+q^{-8})\) (Computation 1.7); \(X̊(F)=x_0B(F)\) by Lemma 1.4(d), and \(X(O)=x_0K\) by Hensel and Lang. Under these, Proposition VII (\(r\) not onto, every \(q\)) gives that (SG) fails for \(E_6/F_4\) over \(\mathbb F_q((t))\). \((\mathrm{S6.3.1})_p\) is open (Dead end 1.8); (A3b′) and (A3c) over \(\mathbb F_q((t))\) are (Q3).

Proof. Combine the nodes as follows.

Table 2. Sakellaridis's step and reason it holds over \(\mathbb F_q((t))\), \(p\) odd
Sakellaridis's step	reason it holds over \(\mathbb F_q((t))\), \(p\) odd
§2.1 invariants \(P(X),L(X),A_X,\Lambda_X,\delta(X)\); (3.5)	Lemma 1.3(c),(d) (replaces [BLV86, Thm 3.4])
§3.3 differential forms, Lemma 3.4.2	smooth \(H\) and \(B_y\): Lemma 1.3(a),(e)
(3.2) convergence and rationality; Theorem 3.2.2	Proposition 1.5 (convergence; generic uniqueness via Lemma 1.4; Bernstein's principle). Theorem 3.2.2 itself is not needed
Theorem 3.7.1, Cor. 3.7.2	Proposition 1.6(c): uniqueness for \(w\in W_X'\); incompatibility with \(X̊\) otherwise
§3.8 [BK98], (4.8)–(4.9) [Ca80]	Proposition 1.6(b): normalized \(T_w\), rank-one \(SL_2\) integrals, \(\psi\) of conductor \(O\)
Lemma 3.9.1	(A3c) only
Prop. 4.2.1, Thm 4.2.2, (6.1)–(6.2)	Proposition 1.6(b),(d)
§5 rank-one constants, §6 Statement 6.3.1	hypothesis (A4)(iii)\(_p\); for \(M_\theta\) this is \((\mathrm{S6.3.1})_p\), open (Dead end 1.8)
Theorem 7.2.1	combinatorial: Sakellaridis's proof with (A4)(iv)\(_p\) and (E) replacing [Kn91, §6]; for \(M_\theta\) Computation 1.7 (including a correction on p. 62)
§8 proof of Theorem 8.0.2	Proposition 1.2, with \(D'=\delta(X)^{-1/2}A_X^*\), \(\kappa\) and \(Q_\lambda\) as in its Remark (i)
(SG) \(\iff\) \(r\) onto	\(\mathcal H_K\mathbf 1_{x_0K}\leftrightarrow r(\mathbb C[A^]^W)\cdot1\)

Steps in detail:

Proposition 1.5 gives the rational family \(S_\chi\), \(\chi\in D'\). Proposition 1.6 gives (6.2) with summation over \(W_X'\).
With (A4)(iii)\(_p\) and (iv)\(_p\), Sakellaridis's proof of Theorem 7.2.1 gives \(B_w=\beta/(\beta\circ w)\) and (7.4). It is pure algebra once (E) replaces Knop's separation, and in it the inequality (7.5) should be justified by the inversion-set argument of Computation 1.7 step 5. Then \(\Omega_\chi(x_0)=c\beta\) with \(c\ne0\), by Lemma 4.2.3, whose proof needs only the invariant measure.
The adjunction \(\hat S(\mathbf 1_{x_\lambda K})(\chi)=\operatorname{vol}(x_\lambda K)\,\Omega_{\chi^{-1}}(x_\lambda)/m(\chi^{-1})\) (with \(m\) the scalar of \(\Phi_{K,\chi}\) against \(\varphi_{K,\chi}\)), followed by \(w_L\), puts us in the situation of Proposition 1.2 with \(\mathcal T=-w_0^X\operatorname{cone}(\Theta^+)\). Proposition 1.2 then gives the isomorphism.

Answer to (Q1), as far as it goes. (1) The analytic half of Sakellaridis's proof (§§3–5 for the open orbit, §8) holds over \(\mathbb F_q((t))\), \(p\) odd, for symmetric \(X\), without any characteristic-0 citation. The only external inputs are characteristic-\(\neq2\) structure results on symmetric varieties. (2) Every characteristic-0 citation of the paper sits inside two places: the proof of Statement 6.3.1 (Theorem 6.3.2: Knop, Brion, Luna, [Sa06], Wasserman, Akhiezer, and Axiom 2.2.1 over \(O\)), and the colour combinatorics of §7.1 ([Kn91, §6], Lemmas 7.1.1–7.1.4). The theorem above keeps both as explicit hypotheses in characteristic \(p\). (3) For \(E_6/F_4\) the colour part is unnecessary and the theorem holds modulo \((\mathrm{S6.3.1})_p\), (A3b′) and (A3c).

Builds on

Proposition 1.2 · Triangular spherical transforms give \(C_K\cong\mathbb C[D]^{W_X}\), any characteristic
Proposition 1.5 · \(S_\chi\) over \(\mathbb F_q((t))\): convergence, generic uniqueness, rationality
Proposition 1.6 · \(F_wS_\chi=b_wS_{w\chi}\) and the sum formula (6.2) over \(\mathbb F_q((t))\)
Computation 1.7 · \(M_\theta\): \(W_X'\cong S_3\), coboundary \(\beta\), formula (7.4), triangularity
Dead end 1.8 · \(B_{w_\gamma}\) for \(M_\theta\) over \(\mathbb F_q((t))\): two routes, missing lemmas

Built on by

Objection 5.20 · Audit of A1-9: the characteristic-p Theorem 8.0.2 and its hypotheses
Lemma S1.1 · (7.5) for every \(\Theta\): H1 is equivalent to one colour set per simple root
Proposition S1.2 · The coefficient \(a_J\) of \(m_\lambda\) in the right side of (7.4)
Remark S1.4 · Sakellaridis's proof of Theorem 8.0.2 also needs \(a_J\neq0\); the p. 62 Remark misses it
Theorem S1.5 · Theorem 8.0.2 over \(\mathbb F_q((t))\) for symmetric \(X\): the complete hypothesis list
Summary S1.6 · Verdict inputs A5-20 and A5-21 re-derived; how R-4(b) and its open items read
Review G1 · Review G1 · Agent 1

Sources

sakellaridis-v3.pdf, Theorem 8.0.2 and Corollary 8.0.4 pp. 64–65, Theorem 7.2.1 pp. 60–62
props/e6-f4-module.html (Proposition VII: r not onto)

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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
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