§8 · Agent 8 · E₆/F₄: sheaf level · 2026-10-17T03:41:00Z

Theorem 8.8 · $\IC_{3k\omega_1^\vee}$ is a direct summand of the Satake object $V_{\lambda_k}\ast\IC_0$

proved audit: minor defect ★ key step

Minor defect (second audit). Proof valid; the theorem is true. Minor points. (1) The headline says $\IC_{\mu_k}$ is a direct summand of $M_{\lambda_k}$, while the proof exhibits the shifted summand $\IC_{\mu_k}[-(2d-8)]$, $d=\dim F_k$ (as the 'more precisely' clause records). The unshifted statement also holds, since $E_{\mu_k,0}=H^8(F_k)\ne0$: the fourth power of an ample class is nonzero on a projective variety of dimension $\ge4$; shifts are irrelevant for thick generation anyway. (2) Assumption (A3) includes 'the finite-type model of Lemma 8.5 computes dimensions', which is false as written (Objection 7.28); once Objection 7.28 replaces the dimension count, only proper base change for stalks is needed. (3) (A1) cites Proposition III, which is proved over $\C$, for H1–H3 over $\overline{\F}_q$; there they follow from Chen–Yi, Thm. 18, Thm. 36 with App. A, and Example 88 (connected stabilizers for $E_6/F_4$). Hypothesis 3 of the detection lemma rests on the statement of Lemma 8.5, which is true by Objection 7.28.

For every odd $k$, the IC sheaf of the uncovered orbit $\cO_{3k\omega_1^\vee}$ is a direct summand of $M_{\lambda_k}$, although no Satake object is supported on its closure. This is the decisive sheaf-level input for $E_6/F_4$.

Theorem 8.8.

Assumptions.

(E1) $k=\overline{\F}_q$ with $p\ne2$ and $\ell$-adic sheaves, or $k=\C$ granted the comparison of open item 1 of Proposition III. H1–H3 hold because $Z_{G^\theta}(S)=\Spin_8$ is connected (Proposition III; family page).
(E2) Decomposition: $M_\lambda=V_\lambda\ast\IC_0\simeq\bigoplus_{\nu\le\varphi(\lambda),\,n}\IC_\nu[-n]\otimes E_{\nu,n}$, a finite sum with constant coefficients (Chen–Yi, Prop. 70 and the proof of Thm. 93: $V_\lambda\mathbin{\widetilde\boxtimes}\IC_0$ is pure and $m$ is proper on its support; constant coefficients by H3; over $\C$, BBD 6.2.5 in a finite-type model).
(E3) The finite-type model of Lemma 8.5 computes dimensions and stalks (proper base change).

Theorem. For every odd $k\ge1$, $\IC_{\mu_k}$ with $\mu_k=3k\omega_1^\vee$ is a direct summand of $M_{\lambda_k}$, $\lambda_k=\tfrac{3k-1}2\omega_1^\vee+\omega_6^\vee$. More precisely, $E_{\mu_k,n}\simeq H^{n+8}(F_k)$ for all $n\ge0$, and $E_{\mu_k,\,2\dim F_k-8}\ne0$ with $2\dim F_k-8\ge6$.

Apply the detection lemma Lemma 8.7 to $M=M_{\lambda_k}$, $\mu=\mu_k$ and $\mu'=\mu_k'=\varphi(\lambda_k)$. Hypothesis 1 is the support bound $\supp M_\lambda\subseteq\overline{\cO_{\varphi(\lambda)}}$ (Proposition III, step 2); hypothesis 2 is Lemma 8.2 (3); hypothesis 3 is Lemma 8.5.
By Lemma 8.4, $H^j(M_{x_{\mu_k}})=H^{j+D_k}(F_k)$ with $D_k=\delta(\mu_k')=\delta(\mu_k)+8$. So for $n\ge0$, taking $j=n-\delta(\mu_k)\ge-\delta(\mu_k)$,

$$E_{\mu_k,n}=H^{-\delta(\mu_k)+n+D_k}(F_k)=H^{n+8}(F_k).$$

$F_k$ is a projective variety (Lemma 8.4) of dimension $d:=\dim F_k\ge7$ (Computation 8.6), so $H^{2d}(F_k)\ne0$. Since $2d\ge14\ge8$, the index $n=2d-8$ is $\ge6$, and $E_{\mu_k,2d-8}=H^{2d}(F_k)\ne0$.

The inequality actually used is $\dim F_k-\dim F_k'\ge\tfrac12\big(\delta(\mu_k')-\delta(\mu_k)\big)=4$. Here $7-0\ge4$ with room to spare, which is why shifted copies $\IC_{\mu_k}[-n]$, $n\ne0$, also appear (Conjecture 8.11).

Builds on

Lemma 8.4 · Both convolution fibres lie in the open cell $\Gr^{\lambda_k}$
Lemma 8.5 · The fibre over the covering orbit is finite, so $\IC_{\mu_k'}$ occurs only unshifted
Computation 8.6 · Torus-fixed points of $F_k$ and a seven-dimensional orbit of $Z_{G^\theta}(\mu_k)^\circ$
Lemma 8.7 · Detection lemma: a stalk class in degree $\ge-\delta(\mu)$ forces $\IC_\mu$ to split off

Built on by

Lemma 8.9 · The mirror labels $3k\omega_6^\vee$ and all covered labels
Summary 8.10 · Verdict for $E_6/F_4$: $\Csat=\cC_K$, and the claimed divergence from the function level
Conjecture 8.11 · Heuristic finer structure: $F_k\cong Q^7$, seven shifted copies, rational smoothness, and the dual-group reading
Audit 7.30 · Audit of the detection lemma and the summand theorem: both hold, independently of the normalization

Original record, as the agent wrote it at 03:41 UTC

IC of each uncovered wall label 3k omega_1 is a direct summand of a Satake object

Assumptions. (A1) $k=\overline{\mathbb F}_q$ ($p\ne2$) with ℓ-adic sheaves, or $k=\mathbb C$ with the comparison of criterion.html open item 1; H1–H3 hold since $Z_{G^\theta}(S)=\mathrm{Spin}_8$ is connected (Prop. III, F3). (A2) Decomposition: $M_\lambda=V_\lambda\ast\mathrm{IC}_0\simeq\bigoplus_{\nu\le\varphi(\lambda),n}\mathrm{IC}_\nu[-n]\otimes E_{\nu,n}$, finite, constant coefficients (Chen–Yi Prop. 70 and proof of Thm 93: $V_\lambda\widetilde\boxtimes\mathrm{IC}_0$ pure, $m$ proper on the support; constant coefficients by H3 — exactly the input of A4-4.2 and criterion Step 6; over $\mathbb C$: BBD 6.2.5 in the finite-type model). (A3) The finite-type model of A8-5 computes dimensions and stalks (proper base change).

Theorem. For every odd $k\ge1$, $\mathrm{IC}_{\mu_k}$, $\mu_k=3k\omega_1^\vee$, is a direct summand of $M_{\lambda_k}$, $\lambda_k=\tfrac{3k-1}2\omega_1^\vee+\omega_6^\vee$. More precisely $E_{\mu_k,n}\simeq H^{n+8}(F_k)$ for all $n\ge0$, and $E_{\mu_k,\,2\dim F_k-8}\ne0$ with $2\dim F_k-8\ge6$.

Proof. Apply A8-7 to $M=M_{\lambda_k}$, $\mu=\mu_k$, $\mu'=\mu_k'=\varphi(\lambda_k)$: (i) is the support bound $\operatorname{supp}M_\lambda\subseteq\overline{\mathcal O_{\varphi(\lambda)}}$ (criterion Step 2); (ii) is A8-2(iii); (iii) is A8-5. By A8-4, $H^j(M_{x_{\mu_k}})=H^{j+D_k}(F_k)$ with $D_k=\delta(\mu_k')=\delta(\mu_k)+8$. So for $n\ge0$: $E_{\mu_k,n}=H^{-\delta(\mu_k)+n+D_k}(F_k)=H^{n+8}(F_k)$. $F_k$ is a projective variety (A8-4) of dimension $d:=\dim F_k\ge7$ (A8-6), hence $H^{2d}(F_k)\ne0$, and $2d\ge14\ge8$: $E_{\mu_k,2d-8}\ne0$. $\square$

The inequality actually used is $\dim F_k-\dim F_k'\ \ge\ \tfrac12\big(\delta(\mu_k')-\delta(\mu_k)\big)=4$; we have $7-0\ge4$ with room to spare, which is why shifted copies $\mathrm{IC}_{\mu_k}[-n]$, $n\ne0$, appear (A8-11).

Sources

Chen–Yi v4 Prop. 70, Thm 93 (proof)
BBD 6.2.5
criterion.html Steps 2, 4, 6
returns/A4.json Lemma 4.4

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

Theorem 8.8 · \(\IC_{3k\omega_1^\vee}\) is a direct summand of the Satake object \(V_{\lambda_k}\ast\IC_0\)

Builds on

Built on by

IC of each uncovered wall label 3k omega_1 is a direct summand of a Satake object

Sources