§10 · Audit · Second audit, then re-checks · 2026-10-17T07:19:59Z

Correction 10.8 · Correction: the square-class automorphism multiplies $\tr_n\IC_{\chi,\nu}$ by $\chi(-1)$

proved correction ★ key step

Correction. Corrects Theorem 4.11 · $\SL_2/\SO_2$: the sign-twisted IC sheaves are not in the Satake span, so $\Csat\ne\cC_K$ over $\overline{\F}_q$.

Supplies the missing step of the $\SL_2/\SO_2$ theorem: the automorphism σ is geometrically a translation by an element of $L^+\SL_2$, so it fixes every equivariant simple object up to a scalar, and the scalar is $\chi(-1)$. Hence the traces of the trivial-coefficient IC sheaves span the σ-invariant functions, and the theorem holds as stated.

Correction 10.8.

Let $X=\SL_2/T$ — the split form of $\SL_2/\SO_2$, with $\theta=\operatorname{Ad}\operatorname{diag}(1,-1)$ and $G^\theta=T$ — over $\F_q$, $q$ odd, under the hypotheses of Theorem 4.11. For $n\ge1$ let $u\in\F_{q^n}[\![t]\!]^\times$ be a non-square and $\sigma_n(gT)=sgs^{-1}T$ with $s=\operatorname{diag}(u,1)$. Then for every simple $\IC_{\chi,\nu}$ with its canonical Weil structure (normalised at $x_\nu$), $$\tr_n(\IC_{\chi,\nu})\circ\sigma_n=\chi(-1)\cdot\tr_n(\IC_{\chi,\nu})\qquad(\text{with }\chi(-1):=1\text{ for }\nu=0).$$ Consequently $(C_{K_n})^{\sigma_n}=\operatorname{span}\{\tr_n\IC_{\mathrm{triv},\nu}\}_\nu$, every element of $\cH_{K_n}\varphi_0$ has zero coefficient on each $\tr_n\IC_{\mathrm{sgn},\mu}$, and the conclusion of Theorem 4.11 holds: $\IC_{\mathrm{sgn},\mu}\notin\Csat$ for $\mu\ne0$, under D0–D2 of Setup 4.1 in the form with disconnected stabilizers.

σ is a translation by an arc-group element. Over $k(\!(t)\!)$ write $s=z\,h$ with $z=\sqrt u\cdot I$ central and $h=\operatorname{diag}(\sqrt u,\sqrt u^{-1})\in T(k[\![t]\!])\subseteq L^+\SL_2(k)$. Conjugation by $s$ equals conjugation by $h$, and since $h\in T$, $\sigma_n(gT)=hgh^{-1}T=h\cdot gT$. So $\sigma_n\otimes k$ is the action of $h$ on $LX$.
Geometric invariance. For every $M\in\cC_{K,k}$, the equivariant structure gives an isomorphism $\beta_M:\sigma_n^*M=h^*M\xrightarrow{\sim}M$.
A second Weil structure. $\sigma_n$ is defined over $\F_{q^n}$ and commutes with $\mathrm{Fr}^n$. For a Weil object $(M,\Phi)$ with $\F_{q^n}$-structure $\Phi^{(n)}$, the map $\Phi':=\beta_M\circ\sigma_n^*\Phi^{(n)}\circ(\mathrm{Fr}^{n*}\beta_M)^{-1}$ is another $\F_{q^n}$-structure on $M$, and its trace function is $\tr_n(M,\Phi)\circ\sigma_n$.
A scalar. For $M=\IC_{\chi,\nu}$ one has $\operatorname{Aut}(M)=\overline{\Q}_\ell^\times$ (D1), so $\Phi'=c\,\Phi^{(n)}$ for some $c\in\overline{\Q}_\ell^\times$, and $\tr_n(\IC_{\chi,\nu})\circ\sigma_n=c\cdot\tr_n(\IC_{\chi,\nu})$ on all of $X(F_n)$.
The scalar is $\chi(-1)$. At $x_\nu$, $\tr_n(\IC_{\chi,\nu})(x_\nu)=(-1)^{\delta(\nu)}\ne0$. For $\nu\ne0$, the point $\sigma_n(x_\nu)=hx_\nu$ has Lang cocycle $h^{-1}\mathrm{Fr}^n(h)=-I$, because $\mathrm{Fr}^n(\sqrt u)=-\sqrt u$ ($u$ is not a square in $\F_{q^n}[\![t]\!]$), and $-I$ represents the nontrivial element of $\pi_0\Stab(x_\nu)=\{\pm I\}$. On $\cO_\nu$, $\IC_{\chi,\nu}=\mathcal L_\chi[\delta(\nu)]$ with $\mathcal L_\chi=L^+G\times^{\Stab}\chi$, and at a rational point with cocycle $c$ the Frobenius acts on the stalk by $\chi(c)$ relative to its action at $x_\nu$. Hence $\tr_n(\IC_{\chi,\nu})(hx_\nu)=\chi(-1)(-1)^{\delta(\nu)}$, and $c=\chi(-1)$. For $\nu=0$, $\cO_0(\F_{q^n})$ is a single $K_n$-orbit, so $c=1$.
Conclusion. The $\tr_n\IC_{\chi,\nu}$ form a basis of $C_{K_n}\otimes\overline{\Q}_\ell$ (Theorem 4.11, step 2), the trivial ones σ-invariant and the sign ones σ-anti-invariant by step 5; so $(C_{K_n})^{\sigma_n}$ is the span of the trivial ones. Since $\cH_{K_n}\varphi_0\subseteq(C_{K_n})^{\sigma_n}$ (Theorem 4.11, step 1), every coefficient $m^{(n)}_{(\mathrm{sgn},\mu),\lambda}$ vanishes, for all $n$, $\lambda$ and $\mu\ne0$. Steps 4–5 of Theorem 4.11 then apply unchanged: weight separation (Lemma 4.4, part 4(ii)) gives $E_{(\mathrm{sgn},\mu),i}(\lambda)=0$ for all $i$, and the Serre-subcategory argument (Lemma 4.5, part 2) gives $\IC_{\mathrm{sgn},\mu}\notin\Csat$. $\square$

Remark. Steps 4–5 are the Weil-sheaf shadow of the central inertia of Theorem 2.4: the central element $-I$ acts on $\IC_{\chi,\nu}$ by the scalar $\chi(-1)$. That argument decides the Betti category over $\C$ directly, with no traces and no purity.

Builds on

Theorem 4.11 · $\SL_2/\SO_2$: the sign-twisted IC sheaves are not in the Satake span, so $\Csat\ne\cC_K$ over $\overline{\F}_q$
Lemma 4.2 · $K$-orbits on $X(\F_q(\!(t)\!))$ via Lang's theorem: $\Lambda_S^+$ exactly when $Z_{G^\theta}(S)$ is connected
Lemma 4.3 · The trace of $V_\lambda\ast\IC_0$ is $\tr(V_\lambda)\ast\varphi_0$, and the traces of the $V_\lambda$ form a basis of $\cH_K$
Lemma 4.4 · Expansion of $\tr(V_\lambda\ast\IC_0)$ in the IC basis: the coefficients are traces of Frobenius on multiplicity spaces
Lemma 4.5 · $\Csat=\cC_K$ if and only if every $\IC_\mu$ is a direct summand of a Satake object

Built on by

Lemma X2.1 · $\sigma_n$ is translation by $h\in L^+\mathrm{SL}_2(\overline{\mathbb F}_q)$ with $h^{-1}\mathrm{Fr}^n(h)=-I$
Lemma X2.2 · Weil inertia lemma: $\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}\circ\sigma_n=\chi(-1)\,\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}$
Theorem (repair of A4-11) X2.4 · Repair of A4-11: $\mathrm{IC}_{\mathrm{sgn},\mu}\notin\mathcal C_K^{\mathrm{Sat}}$ in both $\mathbb F_q$-forms, $q$ odd
Summary 9.5 · Verdict after the second audit: $E_6/F_4$ is still the one divergence

Table 1. Page production by session
	tokens	US $
This conversation: the search, the audit, the pages and propositions, and the VI characteristic-p run01:44–05:19 UTC	802 M	$490
Setup pages16:17–19:42 UTC	37 M	$19
Shareable animation17:16–17:22 UTC	1 M	$1
Proposition IIb: figures and the linear-algebra lemma19:48–20:05 UTC	14 M	$7
Audit pass: the repairs by Agents X1–X4, their re-checks and Propositions VI–VII (with the audit skill)19:35–02:12 UTC	275 M	$113
Proposition IX: converse search, verdict and trace formalism19:56–02:12 UTC	199 M	$106
Audit pass on the VI characteristic-p run; the simplified VIb page and the VI corrections05:21–07:33 UTC	268 M	$107
Removing the detail switch; Proposition IX restated; the hub and verdict reviewed and simplified05:34–15:56 UTC	38 M	$20
C_K for the categorical statements, in the Iwahori run's conversation18:27–18:32 UTC	9 M	$3
Total	1,642 M	$866

Correction 10.8 · Correction: the square-class automorphism multiplies \(\tr_n\IC_{\chi,\nu}\) by \(\chi(-1)\)

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