§9 · Salvage 1 · Leading coefficients · 2026-10-18T06:47:26Z

Lemma S1.3 · Positivity of $\hat S$ forces $c\neq0$ and $a_J/c>0$ over every local field

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 spherical transforms $\hat S_{\mathbf 1_{xK}}(\chi^{-1})$ are Laurent series in $e^\mu(\tilde\chi)$ with non-negative coefficients. The labels of $x_0K$ lie in the colour cone, and $x_\lambda J\subseteq x_\lambda B(O)$ contributes $e^\lambda$ with a positive coefficient. Comparing coefficients of $e^\lambda$ in $\Pi_\lambda\cdot\hat S_{\mathbf 1_{x_0K}}=c\,C_\lambda\,\hat S_{\mathbf 1_{x_\lambda K}}$ gives $c\neq0$ and $a_J/c>0$. So hypothesis (c) of Proposition 1.2 holds for every $\Theta$, in characteristic $p$ as in characteristic 0, and no extra hypothesis H4 is needed.

Lemma S1.3.

Let $F$ be a non-archimedean local field ($\operatorname{char}F=0$, or $F=\mathbb F_q((t))$ with $p$ odd), and $X=G/G^\theta$ as in Theorem 1.9 ($\theta$ over $O$, so $X$ is smooth over $O$ with geometrically connected fibres). Assume (A2), (A5), (A3b′) (representatives $x_\lambda\in A_X^+$ with antidominant labels $\lambda\in\Lambda_X^-$), (A3c), and the hypotheses under which (6.2) and (7.4) hold: Statements 6.3.1 and 7.1.5, (U), (E), H2, H3 (Proposition 1.5, Proposition 1.6, Lemma S1.1, Proposition S1.2). Labels: for $y\in X̊(F)=x_0B(F)$ let $a(y)\in A_X(F)$ be its image and $\nu(y)\in\Lambda_X$ its class, so that $\tilde\chi(a(y))=e^{\nu(y)}(\tilde\chi)$ and $|f_\eta(y)|=|f_\eta(x_0)|\,q^{-\langle\nu(y),\eta\rangle}$ (Proposition 1.5 step 2). For $K$-invariant $\varphi$ put $\hat S_\varphi(\chi):=S_\chi(\varphi)(1)$, $\chi\in D'$, and write $\hat S_\lambda:=\hat S_{\mathbf 1_{x_\lambda K}}$. Then:

(i) $\Omega_\chi(x_\lambda)=m(\chi)\,\hat S_\lambda(\chi^{-1})/\operatorname{vol}(x_\lambda K)$, where $\Phi_{K,\chi}=m(\chi)\varphi_{K,\chi}$ and $m\not\equiv0$.

(ii) On the region of absolute convergence of (3.2) at $\chi^{-1}$, $\hat S_\lambda(\chi^{-1})=\sum_{\mu\in\Lambda_X}V_\lambda(\mu)\,e^{\mu}(\tilde\chi)$ with $V_\lambda(\mu)\ge0$ and $V_\lambda(\lambda)>0$. Moreover $V_0(\mu)=0$ unless $\mu\in\mathcal T_{col}$, the colour cone.

(iii) $c\neq0$, where $c=\Pi_0$ is the constant of (7.1) and (7.4) (Proposition S1.2(a)).

(iv) $\Pi_\lambda(\tilde\chi)\,\hat S_0(\chi^{-1})=c\,C_\lambda\,\hat S_\lambda(\chi^{-1})$, where $C_\lambda=\delta_{P(X)}^{1/2}(x_\lambda)\operatorname{vol}(x_0K)/\operatorname{vol}(x_\lambda K)>0$ and $\Pi_\lambda$ is as in Proposition S1.2.

(v) $a_J=c\,C_\lambda V_\lambda(\lambda)/V_0(0)$ for every antidominant $\lambda$ with stabilizer $W_J$. In particular $a_J\neq0$ and $a_J/c>0$.

Consequently hypothesis (c) of Proposition 1.2 holds: $\kappa\not\equiv0$ and non-zero leading coefficients. It holds with no condition on $\Theta$ beyond those that make (7.4) valid.

(i), adjunction. $\Delta_\chi$ is the adjoint of $S_{\chi^{-1}}$ (Proposition 1.6(d)): $\int_X\Delta_\chi(f)\Phi=\langle f,S_{\chi^{-1}}\Phi\rangle$. Take $f=\varphi_{K,\chi}$ and $\Phi=\mathbf 1_{x_\lambda K}$. Since $\Delta_\chi(\varphi_{K,\chi})$ is $K$-invariant, the left side is $\operatorname{vol}(x_\lambda K)\,\Delta_\chi(\varphi_{K,\chi})(x_\lambda)$. On the right, $S_{\chi^{-1}}(\mathbf 1_{x_\lambda K})=\hat S_\lambda(\chi^{-1})\varphi_{K,\chi^{-1}}$ and $\langle\varphi_{K,\chi},\varphi_{K,\chi^{-1}}\rangle=\operatorname{vol}K=1$. Multiply by $m$: $\Omega_\chi=\Delta_\chi(\Phi_{K,\chi})=m\,\Delta_\chi(\varphi_{K,\chi})$. Moreover $m\not\equiv0$ because $\Omega_\chi\not\equiv0$ (Audit 5.15 step 7). This is the adjunction of Proposition 1.2 Remark (i) and Audit 5.16, before the substitution $\iota$.
(ii), series and signs. By (3.2) and Proposition 1.5 step 4, $\hat S_\varphi(\chi^{-1})=\int_{X̊(F)}\varphi(y)\,\tilde\chi(a(y))\,\nu_0(a(y))\,|\omega_X|(y)$ on the convergence region. Here $\nu_0$ is a fixed positive unramified character of $A_X(F)$, built from $\delta^{1/2}$, $\delta(X)^{1/2}$ and $|c|$ in (3.2). Since the integrand is $\tilde\chi(a(y))$ to the power $+1$, this matches the term $w=1$ of (6.2) and Lemma 3.9.1.

Grouping $y$ by label gives $V_\lambda(\mu)=\nu_0(\mu(\varpi))\,|\omega_X|\big(x_\lambda K\cap\{\nu(y)=\mu\}\big)\ge0$. The series converges absolutely there (Proposition 1.5(a)) and represents the rational function of Proposition 1.5(c). $V_\lambda(\lambda)>0$: by (A3c), $x_\lambda J\subseteq x_\lambda B(O)$. Right translation by $B(O)$ changes $a(\cdot)$ by an element of $A_X(O)$, so all of $x_\lambda J$ has label $\lambda$, and $|\omega_X|(x_\lambda J)>0$. Support of $V_0$. Let $f_\eta$ be a regular $B$-eigenfunction of weight $\eta$. Since $B$ is split, the weights over $F$ and over $\bar F$ agree. Scale $f_\eta$ to be primitive in $O[X]$; then its reduction $\bar f_\eta$ is a non-zero $B_{\mathbb F_q}$-eigenfunction on $X_{\mathbb F_q}$. If $\bar f_\eta(\bar x_0)=0$, then $\bar f_\eta$ would vanish on $\bar x_0B_{\mathbb F_q}=X̊_{\mathbb F_q}$ (A2). That set is open and dense in the integral scheme $X_{\mathbb F_q}$, so $\bar f_\eta=0$, a contradiction. Hence $|f_\eta(x_0)|=1$. On the other hand $|f_\eta(y)|\le1$ on $X(O)\supseteq x_0K$. So $\langle\nu(y),\eta\rangle\ge0$ for every weight $\eta$ of $F[X]^{(B)}$, i.e. $\nu(y)$ lies in the dual of the weight cone. For affine $X$ that dual is $\mathcal T_{col}$, because a $B$-eigenfunction is regular iff it has no pole on any colour. $\mathcal T_{col}$ is strictly convex because the weights span $\mathcal X(X)$ (Proposition 1.5 step 1).

(iii). At $\lambda=0$, (i) and (7.4) give $m(\chi)\hat S_0(\chi^{-1})/\operatorname{vol}(x_0K)=\Omega_\chi(x_0)=\beta(\tilde\chi)\Pi_0=c\,\beta(\tilde\chi)$, using $\Pi_0=c$ (Proposition S1.2(a)). For real $\tilde\chi$ in the region, the series in (ii) has non-negative terms and $V_0(0)>0$, so $\hat S_0(\chi^{-1})>0$. Since also $m\not\equiv0$, the left side is a non-zero rational function, hence $c\neq0$. This replaces '$c\neq0$ by Lemma 4.2.3' in Theorem 1.9 step 2, whose proof needs (4.5), i.e. [CS80]. It also replaces '$c\neq0$ follows from $\Omega_\chi\not\equiv0$' in Objection 5.20 and Audit 5.17, which is a non sequitur: $\Omega_\chi\not\equiv0$ on $A_X^+$ says nothing about the single value at $x_0$. It is the repair proposed by review G6, using the positivity already behind Dead end 1.8(i).
(iv). Divide '(i) and (7.4) at $\lambda$' by '(i) and (7.4) at $0$'; $\beta$, $m$ and $\hat S_0$ are non-zero rational functions. The result is the displayed identity of rational functions. On the region both $\hat S$ are absolutely convergent Laurent series in the $e^\mu(\tilde\chi)$. The region depends only on $|\tilde\chi|$ (Proposition 1.5 step 5), so Laurent coefficients there are unique (Cauchy integrals over compact tori), and multiplying by the Laurent polynomial $\Pi_\lambda$ acts termwise.
(v), the coefficient of $e^\lambda$. On the left it is $\sum_{\mu'+\mu''=\lambda}p_{\mu'}V_0(\mu'')$, where $p_{\mu'}$ are the coefficients of $\Pi_\lambda$. By Proposition S1.2(a), $\mu'\in\lambda+\operatorname{cone}(\Theta^+)$; by (ii), $\mu''\in\mathcal T_{col}$. Then $-\mu''=\mu'-\lambda\in\operatorname{cone}(\Theta^+)\subseteq\mathcal T_{col}$, so $\mu''\in\mathcal T_{col}\cap-\mathcal T_{col}=\{0\}$. The left side is therefore $p_\lambda V_0(0)=a_JV_0(0)$ (Proposition S1.2(a)). The right side is $c\,C_\lambda V_\lambda(\lambda)$. Hence $a_J=c\,C_\lambda V_\lambda(\lambda)/V_0(0)$, a non-zero multiple of $c$ with the same sign.
Consequence for Proposition 1.2(c). By Proposition 1.2 Remark (i) and Audit 5.16, $Q_\lambda\propto P_\lambda\circ\iota$ with $P_\lambda=\Pi_\lambda/c$, and $\kappa$ is $c\beta/m$ transported. Step 3 gives $\kappa\not\equiv0$. The leading coefficient of $Q_\lambda$ is a non-zero multiple of $a_J/c>0$. Nothing used depends on $\operatorname{char}F$ or on the shape of $\Theta$: repeated weights, weights that are not coroots, shared type-$T$ colours, $\sigma=-$ and $r=\frac12$ are all allowed. So the list H1–H3 of Objection 5.20 needs no fourth hypothesis.
Consistency checks. For $E_6/F_4$, $a_J/c\in\{1/c,(1+t)/c,1\}$ with $c>0$. For $T\backslash PGL_2$, $c=1-q^{-1}>0$ and $a_\emptyset=1$. For Example 7.2.4, $c=1-q^{-2}$ and $a_J\in\{1,1-q^{-1}\}$ (Proposition S1.2). All signs agree with (v).

Builds on

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)
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 $\mathbb F_q((t))$
Dead end 1.8 · $B_{w_\gamma}$ for $M_\theta$ over $\mathbb F_q((t))$: two routes, missing lemmas
Audit 5.15 · Audit of A1-6: F_w S_χ = b_w S_{wχ}, the sum over W_X′, and the open-orbit route
Audit 5.16 · Audit of A1-2: triangular spherical transforms give C_K ≅ C[D]^{W_X}
Audit 5.17 · Audit of A1-7: W_X′ ≅ S_3, the coboundary β, the constant c and triangularity for M_θ

Built on by

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
Re-check V2 · Re-check by G6
Summary 7.5 · Verdict after the audit: unchanged for $E_6/F_4$; the general direct form completed

Sources

sakellaridis-v3.pdf: (3.2) and the definition of the index, pp. 12-13; Lemma 3.9.1 p. 24; Theorem 4.2.2 and (4.2) p. 26; Lemma 4.2.3 and (4.5) p. 27; Theorem 7.2.1 and (7.3)-(7.4) p. 60; proof of Theorem 8.0.2 p. 64
run steps Proposition 1.5 (steps 2, 4, 5), Proposition 1.6 (d), Audit 5.15 (step 7), Proposition 1.2 Remark (i), Audit 5.16

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

Lemma S1.3 · Positivity of \(\hat S\) forces \(c\neq0\) and \(a_J/c>0\) over every local field

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