§10 · Audit · Second audit, then re-checks · 2026-10-17T06:55:02Z

Correction 10.1 · Correction: the mechanism of the $E_6/F_4$ divergence, separated by field

proved correction ★ key step

Correction. Corrects Summary 9.4 · Synthesis: $E_6/F_4$ is the one divergence (categorical generation holds, functions fail under (A)); no reverse case.

The synthesis asserts that at the given $q$ the multiplicity vectors of the Satake objects fail to span. That is the failure of (SG) inside the trace dictionary over $\mathbb F_q(\!(t)\!)$, which is not proved. This node states what is proved over $\overline{\mathbb F}_q$, what is proved $p$-adically under (A1)–(A5), and exactly what remains open, with a sufficient criterion for closing it.

Correction 10.1.

Let $X=E_6/F_4$, with $G$ the split simply connected group of type $E_6$, and use the labels of Summary 8.10: for $k$ odd, $\mu_k=3k\omega_1^\vee$, $\mu_k'=\mu_k+\varphi(\alpha_6^\vee)$ and $\lambda_k=\tfrac{3k-1}2\omega_1^\vee+\omega_6^\vee$. For $\lambda\in\Lambda_T^+$ write $V_\lambda\ast\mathrm{IC}_0\simeq\bigoplus_{\nu,i}\mathrm{IC}_\nu[-i]\otimes E_{\nu,i}(\lambda)$, and over $\mathbb F_q$ put $$m_{\nu\lambda}=\sum_i(-1)^i\operatorname{tr}\big(\mathrm{Fr}\mid E_{\nu,i}(\lambda)\big),$$ a matrix whose rows are indexed by orbit labels $\nu\in\Lambda_S^+$ and whose columns are indexed by $\lambda\in\Lambda_T^+$.

Sheaves (over $\overline{\mathbb F}_q$, $p\ne2$, under D0–D2 and Chen–Yi's Prop. 64). For every $\nu\in\Lambda_S^+$ there is $\lambda$ with $E_{\nu,\bullet}(\lambda)\ne0$; for $\nu=\mu_k$ one may take $\lambda=\lambda_k$, although $\varphi(\lambda_k)=\mu_k'>\mu_k$. Thus every row of the graded multiplicity matrix is nonzero.
Functions over a $p$-adic field (under (A1)–(A5)). For every $q$, the Hecke translates $T_\lambda\ast\varphi_0$, $\lambda\in\Lambda_T^+$, do not span $C_K$.
Functions over $\mathbb F_q(\!(t)\!)$ ($q$ odd, under D0–D3). The columns of $(m_{\nu\lambda})$ span the finitely supported vectors on $\Lambda_S^+$ if and only if (SG) holds over $\mathbb F_q(\!(t)\!)$. Whether they do is open: statement 1 does not decide it, and statement 2 transfers to $\mathbb F_q(\!(t)\!)$ only through Sakellaridis's Theorem 8.0.2 over that field, or through a direct computation of the Hecke action there. If for some character $\chi$ of $\mathcal H_K$ the space $\operatorname{Hom}_{\mathcal H_K}(C_K,\mathbb C_\chi)$ has dimension at least 2 — as Theorem 8.0.2 would give at the character of $q^{\rho-\omega_4}$ — then (SG) fails over $\mathbb F_q(\!(t)\!)$.

Consequently the sentence "at the given $q$ the multiplicity vectors of the Satake objects do not span" in Summary 9.4 holds for the Hecke translates over $p$-adic $F$ under (A1)–(A5), and is open for the trace matrix of the dictionary. The $E_6/F_4$ divergence of Summary 9.4 compares sheaves over $\overline{\mathbb F}_q$, with the Betti category reached via D5, with $p$-adic functions.

Statement 1, covered labels. If $\nu=\varphi(\lambda)$ for a dominant $\lambda$, the restriction of $V_\lambda\ast\mathrm{IC}_0$ to $\mathcal O_\nu$ is a nonzero sum of shifted constant sheaves (Proposition III, step 6, which uses that $Z_{G^\theta}(S)=\mathrm{Spin}_8$ is connected). By the support bound every summand of the decomposition is supported in $\overline{\mathcal O_\nu}$, and among these only the summands $\mathrm{IC}_\nu[-i]$ meet $\mathcal O_\nu$; so $E_{\nu,\bullet}(\lambda)\ne0$.
Statement 1, uncovered labels. The labels without a dominant lift are exactly $3k\omega_1^\vee$ and $3k\omega_6^\vee$, $k$ odd. For $\mu_k$, Summary 8.10 together with the repair Objection 7.28 shows: the fibre of the convolution map over $x_{\mu_k'}$ is the single point $t^{\lambda_k}$; the fibre $F_k$ over $x_{\mu_k}$ lies in the open cell $\mathrm{Gr}^{\lambda_k}$, is projective and has dimension at least 7; and its top cohomology class lies in a degree that $\mathrm{IC}_{\mu_k'}$ does not reach, because $\operatorname{codim}=\langle\rho,\mu_k'-\mu_k\rangle=8\le2\dim F_k$. Hence $E_{\mu_k,\,2\dim F_k-8}(\lambda_k)\ne0$. The labels $3k\omega_6^\vee$ are treated by the same computation with nodes 1 and 6 exchanged.
Statement 2. Under (A1)–(A5), Theorem E (Sakellaridis, Thm. 8.0.2 and Cor. 8.0.4) gives an isomorphism of $\mathcal H_K$-modules $C_K\simeq\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}$ sending $\varphi_0$ to $1$, with $\mathcal H_K\simeq\mathbb C[A^*]^W$ acting through the restriction map $r$; so $\mathcal H_K\cdot\varphi_0$ corresponds to $\operatorname{im}r$. By Theorem 6.6, with $\delta(X)^{1/2}=q^{\rho_L}$, the points $q^{\rho_L}y_1$ and $q^{\rho_L}y_2$ of $A^*$ are $W(E_6)$-conjugate while $y_1,y_2$ are not $W_X$-conjugate. Every element of $\operatorname{im}r$ therefore takes equal values at $y_1$ and $y_2$, while the orbit sum $M_{3\omega_1^\vee}$ takes the values $q^{-9}+q^3+q^6\ne q^{-6}+q^{-3}+q^9$. The weights involved are integral, so this holds for every $q$.
The dictionary. Assume $q$ odd and D0–D3. By Lang's theorem the $K$-orbits on $X(\mathbb F_q(\!(t)\!))$ are the sets $\mathcal O_\nu(\mathbb F_q)$, $\nu\in\Lambda_S^+$ (Lemma 4.2), and the trace functions $\operatorname{tr}\mathrm{IC}_\nu$ are unitriangular in the orbit indicators, hence form a basis of $C_K\otimes\overline{\mathbb Q}_\ell$ (Lemma 4.4). The trace of $V_\lambda\ast\mathrm{IC}_0$ is $\operatorname{tr}(V_\lambda)\ast\varphi_0$, the functions $\operatorname{tr}V_\lambda$ span $\mathcal H_K\otimes\overline{\mathbb Q}_\ell$ (Lemma 4.3), and the coordinate vector of $\operatorname{tr}(V_\lambda\ast\mathrm{IC}_0)$ in the basis $\operatorname{tr}\mathrm{IC}_\nu$ is the column $(m_{\nu\lambda})_\nu$. So the column span is $\mathcal H_K\varphi_0\otimes\overline{\mathbb Q}_\ell$. Since $\mathcal H_K\varphi_0$ is spanned by the integer-valued functions $T_\lambda\ast\varphi_0$ and $C_K$ by orbit indicators, the column span is everything exactly when (SG) holds over $\mathbb F_q(\!(t)\!)$.
Why statement 1 does not decide it. A nonzero row says only that $\mathrm{IC}_\nu$ is detected by some Satake object. For $\nu=\mu_k$, every Satake object containing $\mathrm{IC}_{\mu_k}$ is supported on $\overline{\mathcal O_{\varphi(\lambda)}}$ with $\varphi(\lambda)>\mu_k$, so detection by itself produces no element of $\mathcal H_K\varphi_0$ supported in the lower set of $\mu_k$ and nonzero at $\mu_k$; whether one exists depends on the values of the traces at the given $q$ (Proposition 4.7).
Why statement 2 does not decide it. Theorem E is proved for $\operatorname{char}F=0$ (hypothesis (A1)), and no comparison of $C_K$ as an $\mathcal H_K$-module between a $p$-adic field and $\mathbb F_q(\!(t)\!)$ is available (Objection 4.12). The computation of $r$ in Theorem 6.6 does not depend on the field, so the missing input is exactly Theorem 8.0.2 over $\mathbb F_q(\!(t)\!)$, or a direct count of the Hecke action on the $E_6(\mathcal O)$-orbits of the norm-one Albert hypersurface.
The eigenfunction criterion. If (SG) holds, then $C_K=\mathcal H_K\varphi_0$ is cyclic, an $\mathcal H_K$-module map $C_K\to\mathbb C_\chi$ is determined by the image of $\varphi_0$, and $\dim\operatorname{Hom}_{\mathcal H_K}(C_K,\mathbb C_\chi)\le1$ for every $\chi$. Integration against $K$-invariant functions identifies these maps with $K$-invariant $\mathcal H_K$-eigenfunctions on $X(F)$. Under Theorem 8.0.2 the module $\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}$ is finite over $\mathbb C[A^*]^W$, and its fibre over the point of $A^*/W$ given by $q^{\rho-\omega_4}$ contains the two distinct points $W_Xy_1$ and $W_X\sigma y_1$; so that fibre, and hence the space of maps, has dimension at least 2 (Sakellaridis, remark after Cor. 8.0.4).
Conclusion. Steps 1–7 prove statements 1–3. What no longer follows is only the Mechanism sentence of Summary 9.4 read inside the dictionary; its Answer and its partial status are unchanged, since they already compare $\overline{\mathbb F}_q$-sheaves with $p$-adic functions. The same sentence reappears in §2 and §7 of the verdict page.

Verified by computation: e6_g7.py, e6_fibre_g7.py.

Builds on

Summary 9.4 · Synthesis: $E_6/F_4$ is the one divergence (categorical generation holds, functions fail under (A)); no reverse case
Summary 8.10 · Verdict for $E_6/F_4$: $\Csat=\cC_K$, and the claimed divergence from the function level
Objection 7.28 · Objection to the finiteness lemma: the $K_N$ dimension count is wrong as written; $F_k'$ is a point by a fixed-point argument
Theorem 6.6 · $E_6/F_4$: two points of the shifted torus that are not $W_X$-conjugate are $W(E_6)$-conjugate, so $r$ is not onto and (SG) fails under (A1)–(A5)
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
Proposition 4.7 · The converse fails in general: $\Csat=\cC_K$ gives detection by Hecke translates, not spherical generation
Objection 4.12 · Obstacles: $\C$ versus $\overline{\F}_q$, characteristic $0$ in (A1) versus $\F_q(\!(t)\!)$, Weil versus geometric, eigenvalue cancellation

Built on by

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 10.1 · Correction: the mechanism of the \(E_6/F_4\) divergence, separated by field

Builds on

Built on by