§9 · Salvage 1 · Leading coefficients · 2026-10-18T06:49:02Z

Remark S1.4 · Sakellaridis's proof of Theorem 8.0.2 also needs $a_J\neq0$; the p. 62 Remark misses it

proved audit: correct

Second audit: correct. (a) and (b) are right. The 'usual inductive argument' of p. 64 is Proposition 1.2's steps 4-5 and needs a_J!=0. Since >2 is reflexive, the p. 62 Remark says nothing about the coefficient of e^lambda, and read strictly (a_J=|W_J|) it is false at walls: I computed a{gamma1}=1+q^-4!=2 for E6/F4 and c=1-q^-1!=2 for T\PGL2. (c) In characteristic 0, Lemma 7.1.4 together with the minimality of D_V and the exponent clause of 6.3.1 gives (U), which the coboundary step on p. 61 already presupposes; with S1-1 (checked) and S1-3 (graded by G6), the printed proof is completed. In S1-3's positivity and support argument I found no flaw: it uses only the homogeneous integral model, (A3b'), (A3c), (7.4) and the adjunction, all available in characteristic 0. (d) E6/F4 is unchanged (a_J in {1, 1+t, c} recomputed).

The last step of Sakellaridis's own proof (p. 64) needs '$\check\lambda$ is the lowest weight of $P_{\check\lambda}$', i.e. $a_J\neq0$. His Remark after Theorem 7.2.1 is stated with the non-strict order $\succ_2$ and is then silent on this coefficient; read strictly, it is false at walls. Lemma S1.1 and Lemma S1.3 are characteristic-free and fill this gap and the p. 62 error, so the cited characteristic-0 Theorem 8.0.2 (part (a) of Proposition VI) holds as stated for general $X$. For $E_6/F_4$ nothing changes.

Remark S1.4.

(a) The proof of Theorem 8.0.2 (p. 64) uses: '$\check\lambda$ is the lowest weight of $P_{\check\lambda}$ under $\succ_2$ … by the usual inductive argument … the $P_{\check\lambda}$ generate'. The inductive argument is steps 4–5 of Proposition 1.2. It needs the coefficient of $m_{\check\lambda}$ in $P_{\check\lambda}$, i.e. $a_J/c$ of Proposition S1.2, to be non-zero for every antidominant $\check\lambda$. (b) The Remark after the proof of Theorem 7.2.1 (p. 62) states $\prod_{\Theta^+}(1-\sigma q^{-r}T)s_{\check\lambda}-\sum_{w\in W_X}e^{w\check\lambda}\succ_2\check\lambda$. Here $\succ_2$ is defined on p. 61 as '$\check\lambda-\check\mu\in\mathcal T$', a non-strict order. Read this way, the Remark only says that all weights lie in $\check\lambda+\mathcal T$, which the proof of (7.5) already gives. It gives no information on the coefficient of $e^{\check\lambda}$. Read strictly, it would say that this coefficient equals $|W_J|$ (the multiplicity of $e^{\check\lambda}$ in $\sum_{w\in W_X}e^{w\check\lambda}$). That is false at every wall, e.g. $E_6/F_4$, $W_J=\langle w_{\gamma_1}\rangle$: $a_J=1+q^{-4}\neq2$; $T\backslash PGL_2$, $\check\lambda=0$: $a=c=1-q^{-1}\neq2$. (c) In characteristic 0, under (A1)–(A5), Sakellaridis's Lemma 7.1.4 (uniqueness of the colours of each simple spherical root, proved there by inspection) is (U). So Lemma S1.1 gives (7.5) in place of the p. 62 argument, and Lemma S1.3 (valid for every local field) gives $a_J\neq0$ and $c\neq0$. Theorem 8.0.2 in characteristic 0 therefore holds as stated for general $X$, with the printed proof completed by Lemma S1.1 and Lemma S1.3. In particular part (a) of Proposition VI needs no 'H1 caveat' (review G8's minor remark), because in characteristic 0, H1 is Lemma 7.1.4. (d) For $E_6/F_4$ nothing changes: $a_J\in\{1,1+t,(1+t)(1+t+t^2)\}$, $t=q^{-4}$ (Proposition S1.2(b)), and (7.5) holds by the inversion-set argument of Computation 1.7, a special case of Lemma S1.1.

Where the coefficient is used. In Proposition 1.2, independence (step 4) takes a $\succeq$-minimal leading index and needs $a_{\lambda,\sigma\lambda}\neq0$; spanning (step 5) divides by it. The same two steps are 'the usual inductive argument' on p. 64. So Sakellaridis's statement that $\check\lambda$ is the lowest weight of $P_{\check\lambda}$ includes $a_J\neq0$.
What the Remark says. By the definitions on p. 61, $\check\lambda\succ_1\check\mu$ iff $\check\lambda-\check\mu\in\mathcal R$ and $\check\lambda\succ_2\check\mu$ iff $\check\lambda-\check\mu\in\mathcal T$. Both are reflexive. With this reading the Remark is implied by '$\operatorname{supp}\Pi_{\check\lambda}\subseteq\check\lambda+\mathcal T$' together with $w\check\lambda\in\check\lambda+\mathcal R$. Since $\check\lambda\succ_2\check\lambda$, it does not exclude a cancellation of $e^{\check\lambda}$.

With a strict reading, the coefficient of $e^{\check\lambda}$ in $\Pi_{\check\lambda}-\sum_{W_X}e^{w\check\lambda}$ would be $0$, i.e. $a_J=|W_J|$. For regular $\check\lambda$ this is true ($a_\emptyset=1$). At a wall it fails: Proposition S1.2 gives $a_{\{\gamma_1\}}=1+q^{-4}$ for $E_6/F_4$ and $a_{W_X}=1-q^{-1}$ for $T\backslash PGL_2$. Review G6 made the same point.

(7.5) in characteristic 0. On p. 62, (7.5) is derived from $|\Theta^+|-w|\Theta^+|=\check\rho-w\check\rho$, false by a factor 2 (Computation 1.7, Audit 5.17). Lemma S1.1(d) proves (7.5) from (U). In characteristic 0, (U) holds: Lemma 7.1.4 gives one colour of valuation $\check\gamma$ for type $G$, and at most two colours with valuations summing to $\check\gamma$ for type $T$, and the coboundary step of Theorem 7.2.1 already uses that $D_\gamma$ is exactly the set of Statement 6.3.1. The cone condition (E) is Sakellaridis's '$\mathcal T\cap\mathcal V=\{0\}$', from [Kn91, §6].
$a_J\neq0$ in characteristic 0. The proof of Lemma S1.3 uses only: the integral (3.2) and its convergence, the adjunction, (A2), (A3c), (7.4) and the integral model. All of these are part of Sakellaridis's setting in characteristic 0 (Axioms 2.4.1–2.4.2, §3.2, §4.2, Theorem 7.2.1). So the argument applies verbatim with $F/\mathbb Q_p$ and gives $a_J/c>0$.
Conclusion for Proposition VI (a). The cited Theorem 8.0.2 holds as stated for general $X$ under (A1₀), (A1)–(A5). Its printed proof has two gaps, (7.5) and $a_J\neq0$; both are closed by characteristic-free arguments, so no hypothesis has to be added. For $E_6/F_4$ the values are explicit and nothing changes.

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)
Lemma S1.3 · Positivity of $\hat S$ forces $c\neq0$ and $a_J/c>0$ over every local field
Proposition 1.2 · Triangular spherical transforms give $C_K\cong\mathbb C[D]^{W_X}$, any characteristic
Computation 1.7 · $M_\theta$: $W_X'\cong S_3$, coboundary $\beta$, formula (7.4), triangularity
Theorem 1.9 · Theorem 8.0.2 over $\mathbb F_q((t))$, $p$ odd, symmetric $X$: hypotheses

Built on by

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: orders succ_1, succ_2 p. 61; Lemma 7.2.2, (7.5) and the Remark pp. 61-62; Lemma 7.1.4 p. 59; proof of Theorem 8.0.2 p. 64
review G6 (finding on Objection 5.20)

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

Remark S1.4 · Sakellaridis's proof of Theorem 8.0.2 also needs \(a_J\neq0\); the p. 62 Remark misses it

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