§3 · Agent 3 · Symplectic family: fibres · 2026-10-17T02:48:26Z

Summary 3.11 · Verdict for the symplectic family: no disagreement between the levels; \(\GL_4/\Sp_4\) settled at both

proved audit: error

Error. Point 1 is correct. Point 2's bold claim 'No disagreement can occur' for GL_{2m}/Sp_{2m} (m >= 3) and SL_{2n}/Sp_{2n} (n >= 3) is not established by the argument given. The first exclusion uses Proposition 3.10, whose hypothesis is (SG) over F_q((t)); the second uses Proposition F, a p-adic result under (A1)-(A5), and Proposition 3.10's own Remark (b) says Theorem E is not available over F_q((t)). For no single field are both exclusions available. Over a p-adic F, '(SG) holds and Csat != C' is not excluded: Proposition 3.10 does not apply, and under (A) (SG) does hold p-adically while the sheaf level is only partial. Over F_q((t)), 'Csat = C and (SG) fails' is not excluded. The node's caveat covers the dependence on (A), not this change of field, so the row was undecided at the time: a result moved across fields without the bridge, inside a verdict marked proved. The conclusion is nevertheless true by a later argument. Theorem 5.7 proves (SG) over every non-archimedean F for all m (with the SL corollary), so Proposition 3.10 applies over F_q((t)), q odd, and gives Csat = C over F_q-bar (Betti via the transfer item); both levels hold (Audit 7.23). The conclusion still holds by another argument.
Corrected by: Audit 7.18 · Audit of the remaining checklist items: heuristic labels are honest, and two verdict sentences need the field caveat, Audit 7.23 · Audit of the \(\GL_{2m}/\Sp_{2m}\) theorem: the induction holds (directly, not through Proposition B as stated); the \(\SL\) corollary and the sheaf consequence hold.

Records that \(\GL_4/\Sp_4\) satisfies both statements: sheaves by the closed-orbit retract and the scalar shift, functions unconditionally over every local field. For larger ranks the sheaf level is partial, and the node concludes that no disagreement can occur, combining the trace implication over \(\F_q(\!(t)\!)\) with Sakellaridis's theorem over \(p\)-adic fields under (A1)–(A5).

Summary 3.11.

\(\GL_4/\Sp_4\) (Table 1 row "H5 fails by coverage → unknown"): both levels hold. Sheaves: \(\IC_{(1,1)}=\Delta_{(1,1)}\) is a retract of \(V_{\omega_2}\ast\IC_0\) (Proposition 3.3); all odd \((a,a)\) follow by the scalar shift \(\sigma\) and central translation (Theorem 3.6); covered labels by Proposition III, step 6; so \(\mathcal M=\Lambda_S^+\) and \(\Csat=\cC_K\) by the summand criterion (Proposition 3.4), modulo the inputs Proposition III already uses (decomposition theorem for Satake objects in \(\cC_K\), transfer from ℓ-adic to Betti sheaves). Functions: (SG) holds over every non-archimedean local field with no hypothesis (Theorem 3.9). The Table-1 entry should read "holds (retract, not H5)".
\(\GL_{2m}/\Sp_{2m}\), \(m\ge3\), and \(\SL_{2n}/\Sp_{2n}\), \(n\ge3\). Sheaf level partial: \(\mathcal M\supseteq\varphi(\Lambda_T^+)\cup\{(1^k0^{m-k})+c,\ (2^{k-m}1^{2m-k})+c\}\) (Theorem 3.6); \(\SL\) reduces to component 0 of \(\GL\) (Lemma 3.7); the smallest \(\SL_6\) label \((1,1,-2)\) is open, with a predicted witness (Conjecture 3.8). Function level as on the pages (under (A)). No disagreement can occur: (SG) over \(\F_q(\!(t)\!)\) implies \(\Csat=\cC_K\) (Proposition 3.10), so "(SG) holds, \(\Csat\ne\cC_K\)" is impossible; and "\(\Csat=\cC_K\), (SG) fails" is excluded under (A) by Proposition F (\(r\) surjective for every \(q\)). Unconditionally the second direction remains open for \(m\ge3\), as does the sheaf level for the labels not listed.
Mechanism. For uncovered \(\mu\) the H5-type inputs are unavailable (Proposition 3.4). What works is a retract supported on a proper closed subset of the support of \(V_\lambda\ast\IC_0\), detected by a non-semismall fibre \(LG(n_1,2n_1)\) at the offending odd pairs: exactly the \(\SL_2\)-grading of the relative Satake picture.

Point 1 assembles the sheaf-level results Proposition 3.3 (the closed-orbit retract), Theorem 3.6 (shift stability, so every \((a,a)\) is reached) and Proposition III, step 6 (covered labels), and applies the summand criterion Proposition 3.4(2). The standing inputs are those of Proposition III: the decomposition theorem for Satake objects and the passage from ℓ-adic to Betti sheaves. The function-level half is Theorem 3.9.

Point 2 collects the partial sheaf-level results of Theorem 3.6, Lemma 3.7, Conjecture 3.8. The claim that no disagreement can occur rests on two implications:
- Proposition 3.10: if (SG) holds over \(\F_q(\!(t)\!)\), then \(\Csat=\cC_K\) over \(\overline{\F}_q\), and over \(\C\) modulo the transfer item;
- Proposition F with Theorem E: for \(p\)-adic \(F\) and under (A1)–(A5), \(r\) is surjective for every \(q\), so (SG) holds.
The node notes that without (A) the second direction is open for \(m\ge3\).

Point 3 reads the mechanism off Proposition 3.3 and Theorem 3.6: the retract appears with the extreme shift at the stratum where the fibre \(LG(n_1,2n_1)\) has maximal defect, matching the \(\SL_2\)-weight in Conjecture 3.8.

Builds on

Proposition 3.3 · Decomposition of \(V_{\omega_2}\ast\IC_0\) for \(\GL_4/\Sp_4\): \(\IC_{(1,1)}\) is a retract with multiplicity \(k[2]\oplus k\oplus k[-2]\)
Theorem 3.6 · \(\GL_{2m}/\Sp_{2m}\): the labels \((1^k0^{m-k})\), \((2^{k-m}1^{2m-k})\) and their shifts lie in \(\mathcal M\); \(\GL_4/\Sp_4\) is complete
Lemma 3.7 · \(\SL_{2n}/\Sp_{2n}\) is the degree-zero component of \(\GL_{2n}/\Sp_{2n}\); its smallest uncovered label stays open
Conjecture 3.8 · Relative-Satake prediction of every multiplicity; general uncovered labels and the exact gap
Theorem 3.9 · Explicit \(T_{\omega_k}\ast\varphi_0\) over every local field; (SG) for \(\GL_4/\Sp_4\) without Sakellaridis's hypotheses
Proposition 3.10 · Function level implies sheaf level: (SG) over \(\F_q(\!(t)\!)\) forces \(\mathcal M=\Lambda_S^+\)

Built on by

Audit 7.16 · Audit: the comparability objection is correct — three fields, two bridges, and what each bridge needs
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
Summary 9.2 · Registry after the first audit round: the first-round results survive with their hypotheses, three cosmetic corrections, and \(E_6/F_4\) the only live cell
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 02:48 UTC

Verdict for the symplectic family: no disagreement between the levels; GL_4/Sp_4 settled at both

\(\mathrm{GL}_4/\mathrm{Sp}_4\) (Table 1 row "H5 fails by coverage → unknown"): both levels hold. Sheaves: \(\mathrm{IC}_{(1,1)}=\Delta_{(1,1)}\) is a retract of \(V_{\omega_2}\ast\mathrm{IC}_0\) (A3-3), all odd \((a,a)\) follow by the scalar shift \(\sigma\) and central translation (A3-6), covered labels by Prop. III step 6, so \(\mathcal M=\Lambda_S^+\) and \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) by the summand criterion A3-4 — modulo the inputs Prop. III already uses (decomposition theorem for Satake objects in \(\mathcal C\), transfer from \(\ell\)-adic to Betti). Functions: (SG) holds over every non-archimedean local field with no hypothesis (A3-9). The Table-1 entry should read "holds (retract, not H5)".
\(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(m\ge3\), and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\), \(n\ge3\): sheaf level partial: \(\mathcal M\supseteq\varphi(\Lambda_T^+)\cup\{(1^k0^{m-k})+c,(2^{k-m}1^{2m-k})+c\}\) (A3-6), \(\mathrm{SL}\) reduces to \(\mathrm{GL}\) component 0 (A3-7), the smallest \(\mathrm{SL}_6\) label \((1,1,-2)\) is open with a predicted witness (A3-8). Function level as on the pages (under (A)). No disagreement can occur: (SG) over \(\mathbb F_q(\!(t)\!)\) implies \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) (A3-10), so "(SG) holds, \(\mathcal C^{\mathrm{Sat}}\ne\mathcal C\)" is impossible; and "\(\mathcal C^{\mathrm{Sat}}=\mathcal C\), (SG) fails" is excluded under (A) by Prop. F (surjective \(r\) for every \(q\)). Unconditionally the second direction remains open for \(m\ge3\), as does the sheaf level for the labels not listed.
Mechanism: for uncovered \(\mu\) the H5-type inputs are unavailable (A3-4); what works is a retract supported on a proper closed subset of the support of \(V_\lambda\ast\mathrm{IC}_0\), detected by a non-semismall fibre (\(LG(n_1,2n_1)\)) at the offending odd pairs, exactly the \(\mathrm{SL}_2\)-grading of the relative Satake picture.

Sources

summary.html Table 1 and §4
criterion.html Open
classical.html Prop. F

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