§6 · Agent 6 · E₆/F₄: restriction map · 2026-10-17T02:32:12Z

Summary 6.10 · Verdict for this family: the spheres hold at both levels; \(E_6/F_4\) fails at the function level under (A), sheaf level open

proved audit: minor defect

Minor defect (second audit). Table and the recommended Table 1 entries are correct and correctly qualified ('under (A)'). The 'Answers' paragraph drops the qualifier carried by the function column: answer 1 for E6/F4 ('SG fails') and the 'only at E6/F4' part of answer 2 (which uses 'SG holds' for the spheres) rest on Theorem E, i.e. hold under (A1)-(A5) for p-adic F; likewise 'a divergence would occur iff C^Sat = C' is an (A)-conditional, p-adic statement. The qualifier is recoverable from the table two lines above and did not propagate (Strategy 9.1, Objection 7.17 and Summary 7.19 keep it).

Collects the two rows. For the spheres both statements hold: sheaves by Proposition III, functions under (A) by Proposition 6.3. For \(E_6/F_4\), (SG) fails under (A) by Theorem 6.6 while the sheaf statement is undecided, so this row is the only place in the family where a divergence (sheaves equal, functions fail) can occur.

Summary 6.10.

Table 2. \(X\) and sheaves (Summary tab) and functions
\(X\)	sheaves (Summary tab)	functions	divergence?
\(\Spin_{2n}/\Spin_{2n-1}\), \(n\ge2\)	\(\Csat=\cC_K\) (Proposition III)	(SG) holds under (A): \(r\) onto, Proposition 6.3	no
\(E_6/F_4\)	unknown (H5 fails at \(3k\omega_1^\vee,3k\omega_6^\vee\), \(k\) odd)	(SG) fails under (A): \(r\) not onto, Theorem 6.6	not decidable yet; a divergence (sheaves equal, functions fail) would occur iff \(\Csat=\cC_K\) turns out true

Answers to the team question from this family:

"SG holds but \(\Csat\ne\cC_K\)" is excluded for both rows (sphere: both hold; \(E_6/F_4\): SG fails).
"\(\Csat=\cC_K\) but SG fails" is possible only at \(E_6/F_4\) among these rows, and is exactly as open as the sheaf-level statement there. The function-level obstruction sits at the label \(3\omega_1^\vee\), the first label without a dominant lift, and (heuristically, Conjecture 6.8) it is one-dimensional: the Hecke character of \(q^{\rho-\omega_4}\).

Table 1 of the Summary tab and of the classical page should read: \(\Spin_{2n}/\Spin_{2n-1}\), all \(n\ge2\): holds under (A), \(r(m_{\varepsilon_1+\varepsilon_2})=c_n(q)(u+u^{-1})+\mathrm{const}\); \(E_6/F_4\): fails under (A), the \(W\)-conjugacy of \(q^{\rho-9\omega_1-3\omega_6}\) and \(q^{\rho-9\omega_1}\) glues \(y_1=(q^{-5},q)\) to \(\sigma y_1\), and \(M_{3\omega_1^\vee}\notin\operatorname{im}r\).

Sheaf column. From the Summary tab: the spheres satisfy both conditions of Proposition III, so \(\Csat=\cC_K\). For \(E_6/F_4\), \(Z_{G^\theta}(S)=\Spin_8\) is connected, but the labels \(3k\omega_1^\vee\) and \(3k\omega_6^\vee\) with \(k\) odd have no dominant lift, so H5 fails there, Proposition I does not apply, and the sheaf statement is undecided.
Function column. Both entries come from Theorem E: surjectivity of \(r\) for the spheres (Proposition 6.3) and its failure for \(E_6/F_4\) (Theorem 6.6); both are therefore statements under (A1)–(A5), whose status is recorded in Lemma 6.9.
Divergence column. For the spheres the two columns agree. For \(E_6/F_4\) the function column is negative, so whether the row diverges is the sheaf-level question.
The answers follow by reading the table row by row: a row with "SG holds, \(\Csat\ne\cC_K\)" would need a positive function entry together with a negative sheaf entry, which neither row has; a row with "\(\Csat=\cC_K\), SG fails" needs a negative function entry, which only \(E_6/F_4\) has.
Where the obstruction sits. The label \(3\omega_1^\vee\) is the smallest label without a dominant lift on the sheaf side, and the orbit sum \(M_{3\omega_1^\vee}=M_{(3,0)}\) is the generator missing from \(\operatorname{im}r\) on the function side (Computation 6.4, Theorem 6.6). The one-dimensionality of the cokernel is the conjecture of Conjecture 6.8.

Builds on

Proposition 6.3 · \(\Spin_{2n}/\Spin_{2n-1}\): \(r\) is onto for every \(n\ge2\), so (SG) holds under (A1)–(A5)
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)
Conjecture 6.8 · Conjecture: the image of \(r\) is exactly the codimension-one subalgebra of invariants that agree at \(y_1\) and \(\sigma y_1\)
Lemma 6.9 · Status of Sakellaridis's hypotheses (A1)–(A5) for \(E_6/F_4\) and the spheres

Built on by

Objection 7.17 · Objection: three cells of the status classification disagree with other results and must be updated
Strategy 9.1 · Registry after three first-round returns: \(E_6/F_4\) declared the only live divergence; an auditor and an \(E_6/F_4\) sheaf attack are launched

Original record, as the agent wrote it at 02:32 UTC

Verdict: spheres hold at both levels; E6/F4 fails at function level, sheaf level open

Table 3. \(X\) and sheaves (summary.html) and functions (this report)
\(X\)	sheaves (summary.html)	functions (this report)	divergence?
\(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\), \(n\ge2\)	\(\mathcal C^{\mathrm{Sat}}=\mathcal C\) (Prop. III)	(SG) holds under (A): \(r\) onto, A6-3	no
\(E_6/F_4\)	unknown (H5 fails at \(3k\omega_1^\vee,3k\omega_6^\vee\), \(k\) odd)	(SG) fails under (A): \(r\) not onto, A6-6	not decidable yet; a divergence (sheaves equal, functions fail) would occur iff \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) turns out true

Answers to the team question from this family: (1) 'SG holds but \(\mathcal C^{\mathrm{Sat}}\ne\mathcal C\)' is excluded for both rows (sphere: both hold; \(E_6/F_4\): SG fails). (2) '\(\mathcal C^{\mathrm{Sat}}=\mathcal C\) but SG fails' is possible only at \(E_6/F_4\) among these rows and is exactly as open as the sheaf-level statement there; the function-level obstruction sits at the label \(3\omega_1^\vee\), the first label without a dominant lift, and (heuristically, A6-8) is one-dimensional: the Hecke character of \(q^{\rho-\omega_4}\). Table 1 of summary.html / classical.html should read: \(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\), all \(n\ge2\): holds under (A), \(r(m_{\varepsilon_1+\varepsilon_2})=c_n(q)(u+u^{-1})+\mathrm{const}\); \(E_6/F_4\): fails under (A), \(W\)-conjugacy of \(q^{\rho-9\omega_1-3\omega_6}\) and \(q^{\rho-9\omega_1}\) glues \(y_1=(q^{-5},q)\) to \(\sigma y_1\), \(M_{3\omega_1^\vee}\notin\operatorname{im}r\).

Sources

summary.html Table 1
classical.html §7 Table 1
Proposition 6.3
Theorem 6.6

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 6.10 · Verdict for this family: the spheres hold at both levels; \(E_6/F_4\) fails at the function level under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies., sheaf level open

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Verdict: spheres hold at both levels; E6/F4 fails at function level, sheaf level open

Sources

Summary 6.10 · Verdict for this family: the spheres hold at both levels; \(E_6/F_4\) fails at the function level under (A), sheaf level open