§4 · Agent 4 · Sheaf–function dictionary · 2026-10-17T02:40:19Z

Summary 4.13 · Verdict: one direction is a theorem, the other fails formally; Table 1 rows decided under D0–D5

partial audit: minor defect

Minor defect (second audit). All implications are correctly qualified by field (F_q((t)) versus p-adic) and by D0–D5, and every table entry matches its source node. Imprecisions: (1) 'under (A) otherwise' is inaccurate for the group and GL2/Sp2 (and Spin4/Spin3, GL2/T), which Table 1 already lists as holding unconditionally; (2) 'strictly weaker than (SG)' and 'not a consequence of the dictionary' inherit Proposition 4.7's wording (no example is exhibited); (3) it relies on Theorem 4.11, whose step 2 needs the lemma of Correction 10.8, and on Dead end 4.10's blocked status, whose displayed values are wrong (Correction 10.7) — neither affects an entry. Later in the run the GL2m/Sp2m, SL2n/Sp2n rows were settled by Theorem 5.7 + Theorem 4.6 (Audit 7.23), and the E6/F4 sheaf row by Summary 8.10.

Collects the track's results: (SG) over \(\F_q(\!(t)\!)\) implies \(\Csat=\cC_K\) when \(Z_{G^\theta}(S)\) is connected, the converse is not a consequence of the dictionary, the function level holds over \(\F_q(\!(t)\!)\) under Proposition III's conditions, \(\GL_4/\Sp_4\) holds at both levels and \(\SL_2/\SO_2\) fails at the sheaf level. No row exhibits a divergence; the \(p\)-adic entries remain incomparable.

Summary 4.13.

Theorems (symmetric \(X\) with \(G^\theta\) connected over \(\F_q\), \(q\) odd, \(T\) split; modulo D0–D4 of Setup 4.1, plus D5 for statements over \(\C\)):

Theorem 4.6: \(Z_{G^\theta}(S)\) connected and (SG) over \(\F_{q^n}(\!(t)\!)\) for one \(n\) \(\Rightarrow\) \(\Csat=\cC_K\).
Proposition 4.7: \(\Csat=\cC_K\Rightarrow\) every \(\IC_\mu\) is detected by some Hecke translate over some \(\F_{q^n}(\!(t)\!)\); this is strictly weaker than (SG), and "\(\Csat=\cC_K\Rightarrow\)(SG)" is not a consequence of the dictionary (retract closure versus the Grothendieck group; the shape of Sakellaridis's Example 8.0.1).
Lemma 4.8: the two conditions of Proposition III \(\Rightarrow\) (SG) over \(\F_q(\!(t)\!)\), without (A).
Theorem 4.9: \(\GL_4/\Sp_4\): (SG) over \(\F_q(\!(t)\!)\) (explicit; machine-checked for \(q=3,5\), proved for all \(q\)), hence \(\Csat=\cC_K\).
Theorem 4.11: \(\SL_2/\SO_2\): \(\Csat\ne\cC_K\).

Table 1 updates (sheaf column; function column for \(F=\F_q(\!(t)\!)\)):

Table 2. \(X\) and sheaves and functions over \(\F_q(\!(t)\!)\)
\(X\)	sheaves	functions over \(\F_q(\!(t)\!)\)
group, spheres (all \(n\)), \(\GL_n/(\GL_p\times\GL_q)\), \(\GL_2/\Sp_2\), \(\SL_4/\Sp_4\)	equal (I, III; unchanged)	holds (Lemma 4.8; was open for spheres \(n\ge4\), under (A) otherwise)
\(\GL_4/\Sp_4\)	equal (Theorem 4.9 and Theorem 4.6; was unknown)	holds (Theorem 4.9)
\(\GL_{2m}/\Sp_{2m}\) (\(m\ge3\)), \(\SL_{2n}/\Sp_{2n}\) (\(n\ge3\))	unknown; decided by (SG) over \(\F_q(\!(t)\!)\) via Theorem 4.6	open over \(\F_q(\!(t)\!)\) (Dead end 4.10, Objection 4.12); (A) is \(p\)-adic only
\(E_6/F_4\)	unknown; same reduction	open
\(\SL_2/\SO_2\)	not equal (Theorem 4.11; was undecided)	fails (Proposition J; re-derived from stabilizers, Lemma 4.2)
\(\GL_n/\mathrm O_n\), \(\PGL_2/\mathrm{PO}_2\)	not equal (unchanged)	fails (unchanged)

Answer to the question. For symmetric \(X\) on these pages with \(Z_{G^\theta}(S)\) connected, no divergence of the form "(SG) over \(\F_q(\!(t)\!)\) holds but \(\Csat\ne\cC_K\)" is possible (Theorem 4.6). A divergence of the form "\(\Csat=\cC_K\) but (SG) fails" is not excluded by the dictionary (Proposition 4.7), but no row exhibits it: wherever the sheaf level is known to be equal, the function level over \(\F_q(\!(t)\!)\) holds (Lemma 4.8, Theorem 4.9); where (SG) fails (\(\SL_2/\SO_2\), disconnected \(G^\theta\)), the sheaf level fails too (Theorem 4.11, split forms). The \(p\)-adic "(SG) under (A)" entries remain incomparable with the sheaf level (Objection 4.12). Gap: D0–D5 of Setup 4.1, and the finite computations of Dead end 4.10.

The theorem direction. A nonzero Frobenius trace coefficient forces a nonzero multiplicity space, so spherical generation over \(\F_q(\!(t)\!)\) puts every \(\IC_\mu\) into the thick span (Theorem 4.6).
The converse. It would need a statement in the Grothendieck group — for each uncovered label, a Hecke combination supported in its lower set — which closure under retracts cannot supply (Proposition 4.7).
The decided rows. \(\GL_4/\Sp_4\) is decided at both levels in the same field by two lattice counts and the transfer theorem (Theorem 4.9); \(\SL_2/\SO_2\) by the square-class automorphism read through the dictionary (Theorem 4.11); the rows satisfying Proposition III's conditions gain an (A)-free function level over \(\F_q(\!(t)\!)\) (Lemma 4.8).

Builds on

Theorem 4.6 · Theorem: (SG) over \(\F_q(\!(t)\!)\) implies \(\Csat=\cC_K\) over \(\overline{\F}_q\)
Proposition 4.7 · The converse fails in general: \(\Csat=\cC_K\) gives detection by Hecke translates, not spherical generation
Lemma 4.8 · Under the two conditions of Proposition III, (SG) holds over \(\F_q(\!(t)\!)\) without Sakellaridis's hypotheses
Theorem 4.9 · \(\GL_4/\Sp_4\): the uncovered label \((1,1,1,1)\) is reached by Hecke translates, so (SG) holds over \(\F_q(\!(t)\!)\) and \(\Csat=\cC_K\)
Dead end 4.10 · \(\GL_6/\Sp_6\) at the label \((1,1,1,1,0,0)\): the three in-box translates do not suffice; a larger box is needed
Theorem 4.11 · \(\SL_2/\SO_2\): the sign-twisted IC sheaves are not in the Satake span, so \(\Csat\ne\cC_K\) over \(\overline{\F}_q\)
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

Objection 7.17 · Objection: three cells of the status classification disagree with other results and must be updated
Audit 7.18 · Audit of the remaining checklist items: heuristic labels are honest, and two verdict sentences need the field caveat
Re-derivation X1.5 · (SG) for GL₂ₘ/Sp₂ₘ re-derived from Lemmas 2–3 alone; A4-13's open cells now decided
Summary X2.8 · Re-derivation of Summary 4.13, Summary 7.19, Summary 7.25, Summary 7.32 from the repaired steps
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 02:40 UTC

Verdict: one direction is a theorem, the other fails formally; Table 1 rows decided under D0-D5

Theorems (all for symmetric \(X\) with \(G^\theta\) connected over \(\mathbb F_q\), \(q\) odd, \(T\) split, modulo D0–D4, plus D5 for statements over \(\mathbb C\)).

(A4-6) \(Z_{G^\theta}(S)\) connected and (SG) over \(\mathbb F_{q^n}(\!(t)\!)\) for one \(n\) \(\Rightarrow\mathcal C^{\mathrm{Sat}}=\mathcal C\).
(A4-7) \(\mathcal C^{\mathrm{Sat}}=\mathcal C\Rightarrow\) every \(\mathrm{IC}_\mu\) is detected by some Hecke translate over some \(\mathbb F_{q^n}(\!(t)\!)\); this is strictly weaker than (SG), and the implication "\(\mathcal C^{\mathrm{Sat}}=\mathcal C\Rightarrow\)(SG)" is not a consequence of the dictionary (retract-closure vs \(K_0\); Example 8.0.1 shape).
(A4-8) Prop. III's two conditions \(\Rightarrow\) (SG) over \(\mathbb F_q(\!(t)\!)\), no (A).
(A4-9) \(\mathrm{GL}_4/\mathrm{Sp}_4\): (SG) over \(\mathbb F_q(\!(t)\!)\) (explicit, machine-checked for \(q=3,5\), proved for all \(q\)), hence \(\mathcal C^{\mathrm{Sat}}=\mathcal C\).
(A4-11) \(\mathrm{SL}_2/\mathrm{SO}_2\): \(\mathcal C^{\mathrm{Sat}}\ne\mathcal C\).

Table 1 updates (sheaf column; function column for \(F=\mathbb F_q(\!(t)\!)\)).

Table 3. \(X\) and sheaves and functions over \(\mathbb F_q(\!(t)\!)\)
\(X\)	sheaves	functions over \(\mathbb F_q(\!(t)\!)\)
group, spheres (all \(n\)), \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), \(\mathrm{GL}_2/\mathrm{Sp}_2\), \(\mathrm{SL}_4/\mathrm{Sp}_4\)	equal (I, III; unchanged)	holds (A4-8; was open for spheres \(n\ge4\), under (A) otherwise)
\(\mathrm{GL}_4/\mathrm{Sp}_4\)	equal (A4-9 + A4-6; was unknown)	holds (A4-9)
\(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) \(m\ge3\), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) \(n\ge3\)	unknown; decided by (SG) over \(\mathbb F_q(\!(t)\!)\) via A4-6	open over \(\mathbb F_q(\!(t)\!)\) (A4-10, A4-12.2); (A) is \(p\)-adic only
\(E_6/F_4\)	unknown; same reduction	open
\(\mathrm{SL}_2/\mathrm{SO}_2\)	not equal (A4-11; was undecided)	fails (J; re-derived from stabilizers, A4-2)
\(\mathrm{GL}_n/\mathrm O_n\), \(\mathrm{PGL}_2/\mathrm{PO}_2\)	not equal (unchanged)	fails (unchanged)

Answer to the team question. On these pages, for symmetric \(X\) with \(Z_{G^\theta}(S)\) connected, no divergence of the form "(SG) over \(\mathbb F_q(\!(t)\!)\) holds but \(\mathcal C^{\mathrm{Sat}}\neq\mathcal C\)" is possible (A4-6). A divergence of the form "\(\mathcal C^{\mathrm{Sat}}=\mathcal C\) but (SG) fails" is not excluded by the dictionary (A4-7) but no row exhibits it: wherever the sheaf level is known to be equal, the function level over \(\mathbb F_q(\!(t)\!)\) holds (A4-8, A4-9); where (SG) fails (\(\mathrm{SL}_2/\mathrm{SO}_2\), disconnected \(G^\theta\)), the sheaf level fails too (A4-11, split-forms). The \(p\)-adic "(SG) under (A)" entries remain incomparable with the sheaf level (A4-12.2). Gap: D0–D5 as listed in A4-1, and the finite computations of A4-10.

Sources

/Users/miltonlin/Projects/Trees/subpages/case-study-1-ramifiedrelative/proof_graphs/spherical-generation/summary.html Table 1, §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