§9 · Salvage 1 · Leading coefficients · 2026-10-18T06:50:14Z

Theorem S1.5 · Theorem 8.0.2 over $\mathbb F_q((t))$ for symmetric $X$: the complete hypothesis list

proved correction ★ key step

Correction. Corrects Theorem 1.9 · Theorem 8.0.2 over $\mathbb F_q((t))$, $p$ odd, symmetric $X$: hypotheses, Objection 5.20 · Audit of A1-9: the characteristic-p Theorem 8.0.2 and its hypotheses.

The direct characteristic-$p$ form of Theorem 8.0.2 for symmetric $X$ holds under the hypotheses of Theorem 1.9 plus H1 in the form (U) (one colour set per simple spherical root, equivalent to (7.5)), H2 (adapted data over $F$) and H3 (no type $N$). The non-vanishing of $c$ and of every leading coefficient $a_J$ is a consequence, by positivity, and not a fourth hypothesis. The $E_6/F_4$ corollary is unchanged.

Theorem S1.5.

Theorem. Let $F=\mathbb F_q((t))$ with $p$ odd, $G$ as in (A1), $\theta$ an involution of $G$ over $O$, $X=G/G^\theta$, $x_0=eG^\theta$. 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 $\Lambda_X^-$. Let $\Theta$, $\Theta^+$ and the types be as in Sakellaridis §§6.3, 7.1, read over an algebraic closure of $F$. Assume:

(A2), (A5): $X$ affine and homogeneous with the integral model of (A2), and $X̊(F)=x_0B(F)$;
(A3b′): $X(F)=\bigsqcup_{\lambda\in\Lambda_X^-}x_\lambda K$ with $x_\lambda\in A_X^+$;
(A3c): $xJ\subseteq xB(O)$ for $x\in A_X^+$;
H2: there is a $\theta$-stable split maximal $F$-torus $T\subseteq B$ containing a maximal $\theta$-split torus, with $B$ adapted and $x_0\in X̊$ (the data over $F$ used by Lemma 1.4, Proposition 1.5);
H3: every $\gamma\in\Delta_X$ has type $G$, $T$ split or $T$ non-split (no type $N$);
(A4)(iii)$_p$: Statement 6.3.1 holds for the cocycle $B_{w_\gamma}$ of Proposition 1.6(d), for every $\gamma\in\Delta_X$;
(A4)(iv)$_p$: Statement 7.1.5 holds for $\Theta$;
H1 = (U): for every $\gamma\in\Delta_X$ the virtual colours belonging to $\gamma$ are exactly those of Statement 6.3.1, each counted once, i.e. $\sum_{\theta\in D_\gamma}\check\theta=\check\gamma$. By Lemma S1.1 this is equivalent to (7.5);
(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 (SG) $\iff$ $r$ is onto. No hypothesis on the leading coefficients is needed: $c\neq0$ and $a_J\neq0$ for all $J$ follow from the others (Lemma S1.3). Axiom 2.2.1 ((A3a)) and '$L_\gamma$ classical' ((A4)(ii)) are not used. 'No type $N$' ((A4)(i)) is used, as H3.

Corollary ($\mathcal M_\theta$, unchanged). For $X=\mathcal M_\theta$ ($E_6/F_4$) the hypotheses H1, H2, H3, (E) and (A4)(iv)$_p$ hold or are not needed: $\Theta^+=\check\Phi_X^+$ with multiplicity one, both simple roots of type $G$, adapted data explicit (S4-11), and $\beta$ built directly (Computation 1.7). Also $a_J\in\{1,1+t,c\}$ explicitly (Proposition S1.2(b)). So the theorem holds modulo (A2), (A3b′), (A3c) and $(\mathrm{S6.3.1})_p$, as in Theorem 1.9 and Objection 5.20.

Proof sketch. Each line names the step that proves it.

Invariants and local structure ($P(X)$, $L(X)$, $A_X$, $\Lambda_X$, $\delta(X)$; smooth $H$ and $B_y$): Lemma 1.3, over $\bar F$, characteristic $\neq2$. Finitely many $B(F)$-orbits, only $\mu_2$-torsion: Lemma 1.4, with H2.
Open-orbit functionals $S_\chi$, $\chi\in D'=\delta(X)^{-1/2}A_X^*$: convergence, generic uniqueness, rationality: Proposition 1.5, with H2.
Functional equations and the sum formula (6.2) over $W_X'$: Proposition 1.6, using (A3c) and the invariant measure.
Coboundary. For $\gamma\in\Delta_X$, Sakellaridis's computation (p. 61) gives $\beta/(\beta\circ w_\gamma)=-e^{\check\gamma}\prod_{\theta\in D_\gamma}\frac{1-\sigma_\theta q^{-r_\theta}e^{-\check\theta}}{1-\sigma_\theta q^{-r_\theta}e^{\check\theta}}$. Here $D_\gamma=N_\Theta(w_\gamma)$ by (A4)(iv)$_p$, and $w_\gamma$ permutes $\Theta^+\smallsetminus D_\gamma$. H3 gives every simple root a formula in Statement 6.3.1. (U) makes the product exactly the factor of (6.3)–(6.5), so (A4)(iii)$_p$ gives $B_{w_\gamma}=\beta/(\beta\circ w_\gamma)$; this is the step that review G1 found to rest on Lemma 7.1.4. Both sides are cocycles of the same type (Audit 5.15(c)), so $B_w=\beta/(\beta\circ w)$ for all $w\in W_X'$, and symmetrizing gives (7.4).
Triangularity. By Lemma S1.1, (7.5) holds with $\mathcal T=\operatorname{cone}(\Theta^+)$. By Proposition S1.2(a), the weights of $\Pi_\lambda$ lie in $\lambda+\mathcal T$ and $\Pi_0=c$; the latter uses (E). Here $\mathcal T$ is strictly convex (it lies in the colour cone), $\mathcal R\subseteq\mathcal T$, and $\mathcal T\cap C^-=\{0\}$ by (E).
Non-vanishing. $c\neq0$ and $a_J/c>0$ for every antidominant $\lambda$: Lemma S1.3, using (A2), (A3c), steps 2–5 and the adjunction.
Last step. Apply Proposition 1.2 with $\mathcal T$ as in 5: (a) is (A3b′); (b) is step 2; (c) follows from 5–6 through the adjunction and the substitution $\iota(\psi)=w_L\psi^{-1}$ (Proposition 1.2 Remark (i), Audit 5.16), with leading index $\sigma\lambda=-w_0^X\lambda$ and cone $-w_0^X\mathcal T$, which still meets $C^-$ only in $0$. $P_0=1$ gives $\mathbf 1_{x_0K}\mapsto1$.
(SG). $\mathcal H_K*\mathbf 1_{x_0K}$ maps to $r(\mathbb C[A^*]^W)\cdot1$, so (SG) $\iff$ $r$ onto (Objection 5.20 step 5).

What changed against Theorem 1.9. (i) The general list gains (U), H2 and H3, as in Objection 5.20. (U) replaces the shape condition 'multiplicity-free $\Theta^+\subseteq\check\Phi_X^+$'. (ii) (7.5) is proved by Lemma S1.1, not by the inversion-set argument of Computation 1.7. (iii) '$c\neq0$ by Lemma 4.2.3' is replaced by Lemma S1.3(iii). (iv) The leading coefficients are no longer an unproved input. (v) Review G6 says H2 should follow from (A2) by $H^1(\mathbb Z/2,U''(F))=0$; this is not checked here, so H2 stays a hypothesis. (vi) Characteristic 0 is the same statement with $F/\mathbb Q_p$; there (U) is Lemma 7.1.4 (Remark S1.4).

Builds on

Theorem 1.9 · Theorem 8.0.2 over $\mathbb F_q((t))$, $p$ odd, symmetric $X$: hypotheses
Objection 5.20 · Audit of A1-9: the characteristic-p Theorem 8.0.2 and its hypotheses
Proposition 1.2 · Triangular spherical transforms give $C_K\cong\mathbb C[D]^{W_X}$, any characteristic
Lemma 1.3 · Local structure and smooth Borel stabilizers for $G/G^\theta$ in characteristic $\neq2$
Lemma 1.4 · Finiteness of $B(F)\backslash X(F)$ for symmetric $X$ over $\mathbb F_q((t))$, $p$ odd
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
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)
Lemma S1.3 · Positivity of $\hat S$ forces $c\neq0$ and $a_J/c>0$ over every local field

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Summary S1.6 · Verdict inputs A5-20 and A5-21 re-derived; how R-4(b) and its open items read
Re-check V1 · Re-check by G1
Summary 7.5 · Verdict after the audit: unchanged for $E_6/F_4$; the general direct form completed

Sources

sakellaridis-v3.pdf: Statement 6.3.1 pp. 44-45; Section 7.1 and Theorem 7.2.1 with proof pp. 58-62; Section 8 and Theorem 8.0.2 pp. 63-64
run steps Proposition 1.2 ... Theorem 1.9, Audit 5.15, Audit 5.16, Objection 5.20
S1-1, S1-2, S1-3

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

Theorem S1.5 · Theorem 8.0.2 over \(\mathbb F_q((t))\) for symmetric \(X\): the complete hypothesis list

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