§8 · Agent 8 · E₆/F₄: sheaf level · 2026-10-17T03:47:13Z

Summary 8.10 · Verdict for $E_6/F_4$: $\Csat=\cC_K$, and the claimed divergence from the function level

proved audit: error

Error. The sheaf theorem is correct: with Theorem 8.8 (whose input 3 is re-proved by Objection 7.28), Lemma 8.9 and Lemma 4.5 (iii), every $\IC_\mu$ is a summand and $\Csat=\cC$ over $\overline{\F}_q$, $p\ne2$, under (A1)–(A3); for the Betti category over $\C$ it needs the comparison of open item 1 of Proposition III, as the node says. The divergence verdict is not proved as stated. Its status is 'proved modulo (A1)–(A3)', i.e. modulo the sheaf-side assumptions of Theorem 8.8 only, while the function-side input (Theorem 6.6, through Sakellaridis's Theorem 8.0.2, Theorem E(2) on the classical page) holds only under Sakellaridis's (A1)–(A5), of which (A2), (A3) at the place and (A4) beyond '$L_\gamma$ classical' are unverified for $E_6/F_4$, and only for $p$-adic $F$. The two sides also live over different fields: sheaves over $\overline{\F}_q$ (Betti only via the comparison), functions over a $p$-adic field, with (SG) over $\F_q(\!(t)\!)$ undecided, so no trace-dictionary bridge relates them. The 'mechanism' sentence likewise feeds the sheaf multiplicities $E_{\mu_k,\bullet}$ into the $p$-adic restriction matrix of Computation 6.5 without a bridge. (Minor: H1–H3 over $\overline{\F}_q$ are attributed to Proposition III, which is proved over $\C$; over $\overline{\F}_q$ they follow from Chen–Yi.) Correct form: $E_6/F_4$ is a divergence provided Sakellaridis's (A1)–(A5) hold for $E_6/F_4$ at a $p$-adic place, with the sheaf side over $\overline{\F}_q$ (and over $\C$ granted the transfer). This is exactly how Objection 7.31 (d), Summary 7.32 and Summary 9.4 state it. The conclusion does not hold as stated.
Corrected by: Objection 7.31 · Objection to the mirror step and the verdict: generation holds, $\sigma$ needs normalizing, and the divergence needs its hypotheses stated, Summary 7.32 · Final audit verdict: the five conclusions survive; $E_6/F_4$ is the one divergence, under (A), Summary 9.4 · Synthesis: $E_6/F_4$ is the one divergence (categorical generation holds, functions fail under (A)); no reverse case.

Every IC sheaf on $L(E_6/F_4)$ is a direct summand of a Satake object, so the Satake span is the whole category, under the assumptions (E1)–(E3) of the summand theorem. The node then sets this against the failure of (SG), which is known only under Sakellaridis's hypotheses over a $p$-adic field, and calls the row a genuine divergence.

Summary 8.10.

Theorem (under (E1)–(E3) of Theorem 8.8). For $X=E_6/F_4$, every $\IC_\mu$, $\mu\in\Lambda_S^+$, is a direct summand of some $V_\lambda\ast\IC_0$, $\lambda\in\Lambda_T^+$. Consequently $$\Csat=\Thick\{V_\lambda\ast\IC_0\}=\cC_K .$$ For the labels without a dominant lift the generators are explicit: for odd $k$, $$\IC_{3k\omega_1^\vee}\ \text{is a retract of}\ V_{\frac{3k-1}2\omega_1^\vee+\omega_6^\vee}\ast\IC_0,\qquad\IC_{3k\omega_6^\vee}\ \text{is a retract of}\ V_{\omega_1^\vee+\frac{3k-1}2\omega_6^\vee}\ast\IC_0,$$ with $\varphi(\lambda)>\mu$ strictly. So H5 fails at these labels while generation holds: the hypotheses of Proposition I are sufficient but not necessary.

Two levels. By Theorem 6.6, under Sakellaridis's hypotheses (A1)–(A5), (SG) fails for $E_6/F_4$ over a $p$-adic field. Hence the $E_6/F_4$ row is a genuine divergence: $\Csat=\cC_K$ at the sheaf level, (SG) false at the function level.

Status. Proved modulo (E1)–(E3) of Theorem 8.8.

Every IC sheaf is a summand. For $\mu$ with a dominant lift this is Lemma 8.9 (2); for $\mu=3k\omega_1^\vee$ it is Theorem 8.8; for $\mu=3k\omega_6^\vee$ it is Lemma 8.9 (1). By Lemma 8.9 (3) these cases exhaust $\Lambda_S^+$.
Generation. The summand criterion Lemma 4.5 (iii) applies. Its hypotheses are H1–H3, which hold by Proposition III because $Z_{G^\theta}(S)=\Spin_8$ is connected, and the decomposition (E2). A summand is a retract, so every $\IC_\mu$ lies in $\Csat$, and the $\IC_\mu$ generate $\cC_K$.
Strictness. $\varphi(\lambda_k)=\mu_k'=\mu_k+e_6>\mu_k$, so the Satake object that splits off $\IC_{\mu_k}$ is supported on $\overline{\cO_{\mu_k'}}$ and not on $\overline{\cO_{\mu_k}}$. H5 asks for the latter, and fails (Proposition III).
Comparison with functions. By Theorem 6.6, under Sakellaridis's (A1)–(A5) and over a $p$-adic field, $\cH_K\varphi_0$ misses the orbit sum $M_{3\omega_1^\vee}$. The node concludes that the $E_6/F_4$ row is a genuine divergence, and reads it as the mechanism predicted in Proposition 4.7: the retract $\IC_{\mu_k}\subset M_{\lambda_k}$ lives inside $\supp M_{\lambda_k}=\overline{\cO_{\mu_k'}}\not\subseteq\overline{\cO_{\mu_k}}$, so it does not produce a Hecke translate supported in the lower set of $\mu_k$, and the graded multiplicities $E_{\mu_k,\bullet}$ enter the function-level matrix with $q$-weights that admit the $\sigma$-odd linear relation of Computation 6.5.
Status. Proved modulo (E1)–(E3) of Theorem 8.8, the same inputs the workspace already uses for Proposition III and the summand criterion (base field $\overline{\F}_q$ with Chen–Yi's theory; the comparison to $\C$ is open item 1 of Proposition III). No analogue is used: the computation is intrinsic to the root data of $E_6$.

The hypotheses under which the comparison with functions holds are the subject of Objection 7.31.

Builds on

Theorem 8.8 · $\IC_{3k\omega_1^\vee}$ is a direct summand of the Satake object $V_{\lambda_k}\ast\IC_0$
Lemma 8.9 · The mirror labels $3k\omega_6^\vee$ and all covered labels
Lemma 4.5 · $\Csat=\cC_K$ if and only if every $\IC_\mu$ is a direct summand of a Satake object
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)

Built on by

Remark 8.12 · Exactly what the $E_6/F_4$ argument assumes, and what would sharpen it
Objection 7.31 · Objection to the mirror step and the verdict: generation holds, $\sigma$ needs normalizing, and the divergence needs its hypotheses stated
Correction 10.1 · Correction: the mechanism of the $E_6/F_4$ divergence, separated by field
Summary 9.4 · Synthesis: $E_6/F_4$ is the one divergence (categorical generation holds, functions fail under (A)); no reverse case

Original record, as the agent wrote it at 03:47 UTC

Verdict: the Satake span is the whole category for E6/F4, a genuine divergence from the function level

Theorem (under (A1)–(A3) of A8-8). For $X=E_6/F_4$ every $\mathrm{IC}_\mu$, $\mu\in\Lambda_S^+$, is a direct summand of some $V_\lambda\ast\mathrm{IC}_0$, $\lambda\in\Lambda_T^+$ (A8-8, A8-9). By (F1) = A4-5(iii) (hypotheses: H1–H3, which hold by Prop. III since $Z_{G^\theta}(S)=\mathrm{Spin}_8$ is connected, and the decomposition (A2)), $$\mathcal C^{\mathrm{Sat}}=\operatorname{Thick}\{V_\lambda\ast\mathrm{IC}_0\}=\mathcal C .$$ The explicit generators for the labels without a dominant lift are $$\mathrm{IC}_{3k\omega_1^\vee}\ \text{retract of}\ V_{\frac{3k-1}2\omega_1^\vee+\omega_6^\vee}\ast\mathrm{IC}_0,\qquad\mathrm{IC}_{3k\omega_6^\vee}\ \text{retract of}\ V_{\omega_1^\vee+\frac{3k-1}2\omega_6^\vee}\ast\mathrm{IC}_0\quad(k\ \text{odd}),$$ with $\varphi(\lambda)>\mu$ strictly, so H5 still fails at these labels (Prop. III) while generation holds: Proposition I's hypotheses are sufficient, not necessary.

Two levels. By (F3) = A6-6, under Sakellaridis's hypotheses (A1)–(A5) the function-level statement (SG) fails for $E_6/F_4$ over a $p$-adic field: $\mathcal H_K\varphi_0$ misses the orbit sum $M_{3\omega_1^\vee}$. Hence the $E_6/F_4$ row is a genuine divergence: $\mathcal C^{\mathrm{Sat}}=\mathcal C$ at the sheaf level, (SG) false at the function level. This is the mechanism A4-7 predicted: the retract $\mathrm{IC}_{\mu_k}\subset M_{\lambda_k}$ lives inside $\operatorname{supp}M_{\lambda_k}=\overline{\mathcal O_{\mu_k'}}\not\subseteq\overline{\mathcal O_{\mu_k}}$, so it does not produce a Hecke translate supported in the lower set of $\mu_k$, and the graded multiplicities $E_{\mu_k,\bullet}$ enter the function-level matrix with $q$-weights that admit the σ-odd linear relation of A6-5.

Status. Proved modulo (A1)–(A3), all of which are the same inputs the workspace already uses for Proposition III and A4-4/A4-5 (field $\overline{\mathbb F}_q$ with Chen–Yi's theory; the comparison to $\mathbb C$ is criterion.html's open item 1). No analogue was used: the computation is intrinsic to $E_6$ root data.

Sources

returns/A4.json Lemma 4.5, Proposition 4.7
returns/A6.json Theorem 6.6
criterion.html
pdf/proposition-III.tex

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

Summary 8.10 · Verdict for \(E_6/F_4\): \(\Csat=\cC_K\), and the claimed divergence from the function level

Builds on

Built on by

Verdict: the Satake span is the whole category for E6/F4, a genuine divergence from the function level

Sources