Verdict · 2026-10-18

Sakellaridis’s theorem in both characteristics

The theorem and its proof

The characteristic-zero statement and the equal-characteristic argument now share this workspace.

Statement and proof → · Characteristic-p proof →

proved

Sakellaridis's Theorem 8.0.2 holds for $E_6/F_4$ over $\mathbb F_q(\!(t)\!)$, $p$ odd, by transfer from ramified $p$-adic fields, and over every $p$-adic field; his Axiom 2.2.1, relabelled (A3a), holds for $E_6/F_4$ over every $\mathcal O$ with 2 invertible. So (SG) fails over $\mathbb F_q(\!(t)\!)$ and over every $p$-adic field, $p$ odd. Audited in the run and again by eight fresh reviewers (Summary 7.5).

1 · The answer

Yes, for $E_6/F_4$, and in a form that covers symmetric varieties. Sakellaridis proves Theorem 8.0.2 for local fields of characteristic 0 only (§1.1). The run restates it for $F=\mathbb F_q(\!(t)\!)$, $p$ odd, and proves it: $$C_c^\infty(X(F))^K\ \simeq\ \mathbb C[\delta(X)^{1/2}A_X^*]^{W_X},\qquad \mathbf 1_{X(\mathcal O)}\longmapsto1,$$ compatibly with $\mathcal H_K\simeq\mathbb C[A^*]^W$ acting through the restriction $r$.

Table 2. part of Proposition VI and statement and status
part of Proposition VI	statement	status
VIa	Theorem 8.0.2 in characteristic 0, under (A1)–(A5)	cited; for ramified $F$ the normalizations are checked (Audit 4.7) and [Sa08] covers every characteristic-0 field (Audit 7.3); the two gaps of the printed proof are closed (Remark S1.4)
VIb	the same over $\mathbb F_q(\!(t)\!)$ for symmetric $X$ under (T1)–(T5)	proved, audited (Proposition 4.9, Audit 5.7); directly under the list of Theorem S1.5
VIc	$E_6/F_4$ satisfies the hypotheses in both characteristics, (A3a) included	proved, audited (Theorem 2.8, Summary 6.9; Summary 3.12, Summary 5.13; Statement 6.3.1 in characteristic $p$ by Audit 5.19; (T1)–(T5) by the list in Proposition 4.9)
VId	for $E_6/F_4$, (SG) holds iff $r$ is onto, over $\mathbb F_q(\!(t)\!)$ and every $p$-adic $F$	proved (Theorem 4.8 over $\mathbb F_q(\!(t)\!)$; over $p$-adic $F$ from VIa with (A1)–(A5) over $\mathcal O_F$); with Proposition VII, (SG) fails

Relabelled hypotheses. (A1) and (A2) as before; (A1ₚ) is (A1) with $F=\mathbb F_q(\!(t)\!)$; the paper's three axioms are (A3a) rank-one models over $\mathcal O$ (Axiom 2.2.1), (A3b) Cartan decomposition (Axiom 2.4.1), (A3c) Iwahori orbits (Axiom 2.4.2); (A4), (A5) as before. The transfer's hypotheses (T1)–(T5) are stated in Proposition 4.9.

2 · (A3a) for E6/F4

(A3a) for $E_6/F_4$ over every $\mathcal O$ with $2\in\mathcal O^\times$ (Theorem 2.8, audited Summary 6.9).

Tori. Let $T'$ be a $\theta$-stable maximal torus, $a=\dim T'_+$. Counting eigenspaces of $\theta$ on $\mathfrak g$ ($\dim\mathfrak g^\theta=52$, $\dim\mathfrak g^{-\theta}=26$) gives $i_c-i_n=32-2a$ for the compact and noncompact imaginary roots; for $a\le3$ there are at most 12 imaginary roots, so $a=4$ and $T'$ is $F_4$-conjugate to $T$ (Proposition 2.3, repaired in Audit 6.3).
Orbits. Every Borel orbit meets $N_G(T)x_0$; the orbits are the 45 orbits $B\dot wx_0$, $ww_Lw^{-1}\in\mathcal C$, all of maximal rank, in every characteristic $\ne2$, $p=3$ included (Proposition 2.3).
Orbit schemes. $B_x=(B\cap\theta_xB)^{\theta_x}$ is smooth over $\mathbb Z[1/2]$ of fibre dimension $16+c_x$, and $B/B_x\to X$ is an immersion with smooth image $Y_x$ (Lemma 2.4).
Rank-one quotients. $YP_\alpha/R(P_\alpha)$ is $\operatorname{Spec}\mathcal O=\mathrm{PGL}_2\backslash\mathrm{PGL}_2$ when $\alpha\in w(\Phi_L)$ (90 pairs) and $B_2\backslash\mathrm{PGL}_2=\mathbb P^1_{\mathcal O}$ otherwise (180 pairs); types $T$, $N$ never occur (Proposition 2.5, Computation 2.6).
Sakellaridis's scheme structure. By induction on closures from the closed orbit $x=w_0$, the flat closure of each generic orbit has the expected special fibre (Lemma 2.7, repaired in Audit 6.6); so his $Y$ is the orbit scheme and the complement clause holds for all 270 pairs, read inside closures, the only satisfiable reading (Audit 6.7).

3 · Characteristic p by close local fields

Characteristic $p$ by close local fields (Theorem 4.8, Proposition 4.9; audited Audit 5.1–Audit 5.7).

Close rings. For $F'_e=\mathbb Q_q(p^{1/e})$, $\mathcal O'/\varpi'^m\simeq\mathbb F_q[t]/(t^m)$ for $m\le e$ (Lemma 4.2).
Finite determinacy. Membership in $K$, in a Cartan double coset or in an Iwasawa cell is read from valuations of matrix coefficients in the fundamental Weyl modules, so every count below is decided modulo $\varpi^D$, $D$ independent of the field (Lemma 4.2).
Orbits. Counting $K$-orbits in $Kt^\lambda X(\mathcal O)$ carries the Cartan decomposition from $F'_e$ to $F$: $X(F)=\bigsqcup_{\kappa\in\Lambda_S^+}Kx_\kappa$ (Proposition 4.5). A second, direct proof uses Delorme–Sécherre over $F$ (Theorem 3.8).
Structure constants. $(T_\lambda\ast\mathbf 1_{Kx_\mu})(x_\nu)$ and the Satake coefficients of $T_\lambda$ agree over $F$ and $F'_e$ once $e\ge D(\lambda,\mu)$ (Proposition 4.6, with the support bound of Objection 5.4).
Characteristic 0. Over $F'_e$, Theorem 8.0.2 holds: the paper's "unramified" simplification only fixes $\psi$ and $dx$, which §1.7 fixes with conductor $\mathcal O$ (Audit 4.7); [Sa08] covers every characteristic-0 field (Audit 7.3); (A3a) holds over $\mathcal O'_e$ (Theorem 2.8).
Conclusion. $\mathbf 1_{Kx_\kappa}\mapsto P_\kappa$, the images over $F'_e$, is an $\mathcal H_K$-isomorphism over $F$; it is the unique one with $\varphi_0\mapsto1$ (Theorem 4.8). For a symmetric $X$ the same argument needs exactly (T1)–(T5) (Proposition 4.9).

4 · The hypotheses over F_q((t))

The hypotheses over $\mathbb F_q(\!(t)\!)$ (Summary 3.12, audited Audit 5.8–Audit 5.12). Every characteristic-0 citation of the $p$-adic check is replaced: Kneser by Bruhat–Tits III for $H^1(F,\mathrm{Spin}_8)=H^1(F,F_4)=1$ (Lemma 3.3); Vust–Richardson and Helminck–Wang by a direct proof that the maximal $\theta$-split tori form one $H(F)$-class (Proposition 3.4); Knop's transport of weight lattices by the $\theta$-stable-torus description of the orbits (Proposition 3.6, with the converse of Audit 5.10); and the colour weights by the diagram automorphism of $E_6$ (Objection 5.11). The integral model, open orbit and integral points need only that 2 is invertible (Proposition 3.2).

5 · The direct rerun

The direct rerun (Theorem 1.9). A1 reran Sakellaridis's proof over $\mathbb F_q(\!(t)\!)$ for symmetric $X$, $p$ odd: an inventory of his 27 external inputs (Strategy 1.1); smooth stabilizers and the Levi of the open orbit without Brion–Luna–Vust (Lemma 1.3); finitely many $B(F)$-orbits (Lemma 1.4); the open-orbit functionals without [Sa08] (Proposition 1.5); the functional equations and the sum formula with $W_X'=N_W(\mathfrak a)\cap[W/W_L]$ instead of Knop's action (Proposition 1.6); and for $E_6/F_4$ the $L$-factor $\beta$, the constant $c=(1+q^{-4})(1+q^{-4}+q^{-8})$ and triangularity (Computation 1.7). This route does not use (A3a). Its one open input for $E_6/F_4$ is Statement 6.3.1 in characteristic $p$, which the level lemma of Dead end 1.8 reduces to ramified $p$-adic fields. The audit (Audit 5.14–Summary 5.21) confirmed every step for $E_6/F_4$, and found that Statement 6.3.1 in characteristic $p$ follows from (A3a) through Theorem 8.0.2 at the ramified fields (Audit 5.19). For a general symmetric $X$, the statement needs three further hypotheses (Objection 5.20): H1, the triangularity (7.5) for the actual $\Theta$; H2, $F$-rational adapted data; H3, no type $N$. Sakellaridis's own argument for (7.5) on p. 62 is off by a factor 2. A1's argument proves (7.5) when $\Theta^+\subseteq\check\Phi_X^+$ has no repeated coroot, which covers $E_6/F_4$ (Audit 5.17). The fresh audit found one more gap in the general statement: it used non-zero leading coefficients of the images $P_\lambda$ without proof (Objection 5.20, Summary 5.21). The salvage proves them by positivity of the open-orbit integrals (Lemma S1.3). It also shows that H1 is equivalent to (U), one set of colours for each simple spherical root, which gives (7.5) for every $\Theta$ (Lemma S1.1). The complete list is Theorem S1.5. For $E_6/F_4$ this route rests on (A3a) through the transfer (Audit 5.19), so it confirms the answer but does not prove it independently.

6 · Audit

Audit. A6 audited A2 with an independent recomputation of the orbit table (Audit 6.1–Summary 6.9). A5 audited A4 and A3 (Audit 5.1–Summary 5.13), then A1 (Audit 5.14–Summary 5.21). No conclusion failed; one step, step 3 of Proposition 3.7, failed as written and was repaired (Objection 5.11). The repairs: two Borel subgroups share a maximal torus (Audit 6.3); a fibre-dimension bound (Audit 6.5); closedness of $P_\alpha Z$ over $\mathcal O$ (Audit 6.6); the valuation cone behind "Axiom 2.4.1" (Audit 5.3); a field-independent bound on the Satake support (Objection 5.4); the converse for the orbit action (Audit 5.10); the colour weights (Objection 5.11). The root checked the one citation the auditors could not open (Audit 7.3). A second, fresh audit followed (section 8).

7 · Team and timeline

Team and timeline. Six agents (four searchers, two auditors) and the root, 01:50–04:42 UTC on 18 October 2026. The statement was fixed at 02:04 and round 1 launched at 02:07. A4 returned at 02:58, A3 at 03:18, A2 at 03:33. A1's first attempt ended without output when one response exceeded the output limit, and it was relaunched at 03:28. A6 returned at 04:08 and A5 at 04:14. The root checked [Sa08] at 04:15, and A1 returned at 04:15. Audit pass, 05:27–07:16 UTC in a new session: eight fresh reviewers from 05:27 to 06:07; one salvage agent from 06:12 to 06:53; re-checks by the two reviewers who had found the errors, which ended at 07:16; the final verdict Summary 7.5.

8 · The audit pass

Eight fresh reviewers graded all 76 steps, twice the usual number of reviewers. The in-run audits and the root's steps were included (Review G1 to Review G8). 52 steps are correct and 21 have minor defects that change no claim. The proof of the main result held: (A3a) for $E_6/F_4$, the close-field transfer and the [Sa08] check. Three steps were errors, all in the hypothesis list of the direct route for a general symmetric variety (Theorem 1.9, Objection 5.20, Summary 5.21). One salvage agent repaired them (Lemma S1.1, Lemma S1.3, Theorem S1.5, Summary S1.6), and the two reviewers who had found them re-checked every repair: all hold (Re-check V1, Re-check V2). Of the 16 inputs of the verdict, 14 held and 2 were salvaged; none was lost. The answer is unchanged. Review Review G8 also corrected eleven small mismatches between the published texts and the steps; the corrected verdict is Summary 7.5.

Problem

Question (VI in characteristic $p$). Proposition VI quotes Sakellaridis's Theorem 8.0.2, which he proves only for local fields of characteristic $0$ (§1.1). The $E_6/F_4$ divergence of the verdict compares sheaves over $\overline{\mathbb F}_q$ with functions; to put both on one field it needs the theorem over $F=\mathbb F_q(\!(t)\!)$: $$C_c^\infty(X(F))^K\ \simeq\ \mathbb C[\delta(X)^{1/2}A_X^*]^{W_X},\qquad \mathbf 1_{X(\mathcal O)}\longmapsto1,\qquad F=\mathbb F_q(\!(t)\!),\ p\ \text{odd}.$$ (Q1) State and prove it for symmetric $X=G/G^\theta$, at least for $E_6/F_4$, with exactly the hypotheses the characteristic-$p$ proof needs. (Q2) Prove (A3a), Sakellaridis's Axiom 2.2.1 (rank-one models over $\mathcal O$), for the $E_6/F_4$ model over every complete discrete valuation ring of odd residue characteristic. (Q3) Verify the other hypotheses for $E_6/F_4$ over $\mathbb F_q(\!(t)\!)$.

Setting and conventions

$p$ an odd prime, $q$ a power of $p$; $F$ a non-archimedean local field with ring of integers $\mathcal O$, uniformizer $\varpi$, residue field $\mathbb F_q$. Either $\operatorname{char}F=0$ ($p$-adic) or $F=\mathbb F_q(\!(t)\!)$, $\mathcal O=\mathbb F_q[\![t]\!]$.
$G$ split reductive over $\mathcal O$ with connected fibres and simply connected derived group, $B\supseteq A$, $K=G(\mathcal O)$, $J$ the Iwahori subgroup; $\mathcal H_K\simeq\mathbb C[A^*]^W$ (Satake).
$X=H\backslash G$ an $\mathcal O$-scheme with open $B_F$-orbit and base point $x_0\in\mathring X(\mathcal O)$ (Sakellaridis's right action; the pages write $G/H$); invariants $A_X$, $\Lambda_X^+$, $A_X^+$, $W_X$, $\delta(X)$, $\Delta_X$, colours and $\Theta$ as in §§2, 6, 7.
$C_K=C_c^\infty(X(F))^K$, $\varphi_0=\mathbf 1_{X(\mathcal O)}$, (SG): $\mathcal H_K\ast\varphi_0=C_K$; $r:\mathbb C[A^*]^W\to S=\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}$ the restriction (8.2).
Hypotheses, relabelled: (A1) characteristic-$0$ field and split group with simply connected derived group; (A1ₚ) the same with $F=\mathbb F_q(\!(t)\!)$; (A2) integral model with smooth open orbit; (A3a) rank-one models over $\mathcal O$ (Axiom 2.2.1); (A3b) Cartan decomposition (Axiom 2.4.1); (A3c) Iwahori orbits (Axiom 2.4.2); (A4) spherical roots and colours (§6 opening, §7, Statements 6.3.1, 7.1.5); (A5) homogeneous, affine, one open $B(F)$-orbit (§8).
$E_6/F_4$: $G_{\mathbb Z}$ split simply connected of type $E_6$, $\theta_0=\operatorname{Int}(\sigma_L)\circ C$ of type E IV over $\mathbb Z$, the model $\mathcal M_\theta=(G_{\mathbb Z}/G_{\mathbb Z}^{\theta_0})\otimes\mathcal O$, $p$ odd; $\Lambda_X=\mathbb Ze_1\oplus\mathbb Ze_6$, $W_X=S_3$, 45 Borel orbits of maximal rank. Over $p$-adic $F$ every hypothesis but (A3a) was verified in the salvage audit (S4, S4b).

What counts as a solution

A complete answer gives the characteristic-$p$ theorem with its exact hypotheses and a proof in which every step is proved or cited with its number and checked to hold over $\mathbb F_q(\!(t)\!)$; (A3a) for $\mathcal M_\theta$ over every complete discrete valuation ring of odd residue characteristic, mixed and equal; and the hypotheses verified for $\mathcal M_\theta$ over $\mathbb F_q(\!(t)\!)$. Then, for $E_6/F_4$ over $\mathbb F_q(\!(t)\!)$ and over $p$-adic $F$, $p$ odd, (SG) holds iff $r$ is onto, so (SG) fails. Insufficient: "the proof carries over verbatim" without checking each characteristic-$0$ input; a $p$-adic result presented as a characteristic-$p$ one; (A3a) checked only over $\overline F$, only on one fibre, or at almost every place; a close-field transfer that assumes Theorem 8.0.2 for ramified $p$-adic fields without checking the paper's normalizations; numerical checks for small $q$.

Sources

References

Proposition VI · Sakellaridis's theorem with every hypothesis, the E6/F4 verification, the open (A3a)
Proposition VII · r is not onto, for every q
Satake generation at two levels · the verdict whose E6/F4 row this run serves
Sakellaridis, Spherical functions on spherical varieties, arXiv:0905.4244v3 · §1.1, §1.7, Axioms 2.2.1, 2.4.1, 2.4.2, §§3–8, Theorem 8.0.2
Salvage returns S4, S4b · the p-adic verification for E6/F4 (S4b: ../satake-two-levels/audit/S4b.json)
Close-field transfer (IId run) · failures at a label pass from highly ramified p-adic fields to F_q((t))
E6/F4 family page · root data of the symmetric pair

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★ key step

part of Proposition VI	statement	status
VIa	Theorem 8.0.2 in characteristic 0, under (A1)–(A5)	cited; for ramified \(F\) the normalizations are checked (Audit 4.7) and [Sa08] covers every characteristic-0 field (Audit 7.3); the two gaps of the printed proof are closed (Remark S1.4)
VIb	the same over \(\mathbb F_q(\!(t)\!)\) for symmetric \(X\) under (T1)–(T5)	proved, audited (Proposition 4.9, Audit 5.7); directly under the list of Theorem S1.5
VIc	\(E_6/F_4\) satisfies the hypotheses in both characteristics, (A3a) included	proved, audited (Theorem 2.8, Summary 6.9; Summary 3.12, Summary 5.13; Statement 6.3.1 in characteristic \(p\) by Audit 5.19; (T1)–(T5) by the list in Proposition 4.9)
VId	for \(E_6/F_4\), (SG) holds iff \(r\) is onto, over \(\mathbb F_q(\!(t)\!)\) and every \(p\)-adic \(F\)	proved (Theorem 4.8 over \(\mathbb F_q(\!(t)\!)\); over \(p\)-adic \(F\) from VIa with (A1)–(A5) over \(\mathcal O_F\)); with Proposition VII, (SG) fails