§7 · Agent 7 · Adversarial audit · 2026-10-17T04:11:59Z

Summary 7.32 · Final audit verdict: the five conclusions survive; \(E_6/F_4\) is the one divergence, under (A)

proved audit: minor defect ★ key step

Minor defect (second audit). The five conclusions, both columns and the answer follow from Summary 7.19, Summary 7.25 and Audit 7.26...A7-31, and the divergence is correctly stated as conditional and cross-field (sheaves over F_q-bar under D0-D2 and Chen-Yi Prop. 64, Betti via D5; functions p-adic under (A1)-(A5)). I recomputed the E6 data that the sheaf argument uses (e6_g7.py, e6_fibre_g7.py): uncovered labels exactly {(3k,0),(0,3k): k odd} for a,b<=18; phi(lambda_k)=mu_k', <2rho,lambda_k>=delta(mu_k')=8(3k+1), codimension 8; |W lambda_k|=270 with 8 torus-fixed points over mu_k and 1 over mu_k'; no root vanishes on the black torus; Lie Z_k of dimension 36 and stabilizer Lie algebras of dimension 29. Defects: (1) '(iv) ... superseded by A5 for every F' overstates: the lattice counts cover the Table 1 rows, not every symmetric space satisfying Proposition III's two conditions; (2) 'equivalently, exhibit a 2-dimensional space of K-invariant H_K-eigenfunctions' is not an equivalence with proving Theorem 8.0.2 over F_q((t)): such a space is a sufficient criterion for (SG) to fail there (a cyclic module has at most one-dimensional Hom to each character), and Theorem 8.0.2 would produce it.

With the \(E_6/F_4\) sheaf-level argument audited and repaired, the sheaf column is decided in every row, and \(E_6/F_4\) is the unique row where \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) while (SG) fails. The divergence holds with sheaves over \(\overline{\mathbb F}_q\) (Betti via D5) and functions over \(p\)-adic fields under Sakellaridis's hypotheses; two upgrades would make it unconditional or put both sides in one field.

Summary 7.32.

Surviving conclusions.

\(\mathrm{SL}_2/\mathrm{SO}_2\): both levels fail. Sheaves in the Betti category, by central inertia (Theorem 2.4); functions for every \(q\) and every \(F\).
\(\mathrm{GL}_4/\mathrm{Sp}_4\): both levels hold. Functions over every \(F\); sheaves over \(\overline{\mathbb F}_q\) under D0–D2, Betti via D5.
(SG) over \(\mathbb F_q(\!(t)\!)\) implies \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) when \(Z_{G^\theta}(S)\) is connected, under D0–D3. The converse is not a theorem.
Proposition III's conditions imply (SG) over \(\mathbb F_q(\!(t)\!)\); this is superseded by the lattice-count results, valid for every \(F\).
\(E_6/F_4\): (SG) fails under (A). Spheres: (SG) holds, unconditionally.

Sheaf column, now decided in every row. \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) for the group, the spheres, \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), \(\mathbb G_m\), \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) (lattice counts and the transfer theorem Theorem 4.6), and \(E_6/F_4\) (Summary 8.10, audited, with the fibre over the covering orbit repaired in Objection 7.28). \(\mathcal C_K^{\mathrm{Sat}}\ne\mathcal C_K\) for \(\mathrm{SL}_2/\mathrm{SO}_2\), \(\mathrm{GL}_n/\mathrm O_n\) and \(\mathrm{PGL}_2/\mathrm{PO}_2\).

Function column. Every row is decided unconditionally over every \(F\), except \(E_6/F_4\), where (SG) fails under (A) for \(p\)-adic \(F\).

Answer. In every row except one the two levels agree. \(E_6/F_4\) is the unique case with \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) but (SG) failing:

the sheaf side is proved over \(\overline{\mathbb F}_q\), \(p\ne2\), under the standing inputs D0–D2 (Chen–Yi's geometry, proper base change, the decomposition theorem) and Chen–Yi's Prop. 64, and in the Betti category over \(\mathbb C\) via D5;
the function side is proved only for \(p\)-adic \(F\), under Sakellaridis's (A1)–(A5).

No row has (SG) holding and \(\mathcal C_K^{\mathrm{Sat}}\ne\mathcal C_K\).

Two upgrades would make the divergence unconditional or same-field: verify (A2)–(A4) for \(E_6/F_4\) at the given place; and prove Theorem 8.0.2 over \(\mathbb F_q(\!(t)\!)\) — the computation of \(r\) in Theorem 6.6 does not depend on the field — equivalently, exhibit a two-dimensional space of \(K\)-invariant \(\mathcal H_K\)-eigenfunctions at the character of \(q^{\rho-\omega_4}\). Then the divergence lives inside the trace dictionary and refutes "\(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\Rightarrow\)(SG)" in a single field.

Notation as in Summary 8.10: for \(k\) odd, \(\mu_k=3k\omega_1^\vee\), \(\mu_k'=\mu_k+\varphi(\alpha_6^\vee)\), \(\lambda_k=\tfrac{3k-1}2\omega_1^\vee+\omega_6^\vee\), \(F_k\) and \(F_k'\) the fibres of the convolution map \(m:\mathrm{Gr}^{\le\lambda_k}\widetilde\times L^+X\to LX\) over \(x_{\mu_k}\) and \(x_{\mu_k'}\), and \(\delta\) the dimension function on orbits.

Labels. The labels \(\mu_k\) have no dominant lift; the interval \(\{\nu:\mu_k\le\nu\le\mu_k'\}\) is \(\{\mu_k,\mu_k'\}\), of codimension \(\langle\rho,\mu_k'-\mu_k\rangle=8\); and \(\lambda_k\) is a dominant coweight with \(\varphi(\lambda_k)=\mu_k'\) and \(\dim\mathrm{Gr}^{\lambda_k}=\delta(\mu_k')\) (Audit 7.26).
Open cell. Both fibres lie in the open cell \(\mathrm{Gr}^{\lambda_k}\): a smaller cell \(\mathrm{Gr}^{\lambda''}\) meeting them would need \(\varphi(\lambda'')\in\{\mu_k,\mu_k'\}\), and neither value is possible. So the stalks of \(V_{\lambda_k}\ast\mathrm{IC}_0\) at \(x_{\mu_k},x_{\mu_k'}\) are the cohomology of \(F_k,F_k'\) shifted by \(\delta(\mu_k')\), and no stalk of the intersection complex of the singular \(\mathrm{Gr}^{\le\lambda_k}\) enters (Audit 7.27).
The covering fibre is a point. The dimension count in Lemma 8.5 fails as written. Instead: the black torus fixes \(x_{\mu_k'}\), no root of \(E_6\) vanishes on it, so its fixed points in \(\mathrm{Gr}\) are the \(t^\nu\); exactly one of them, \(t^{\lambda_k}\), lies in \(F_k'\); and a positive-dimensional torus-stable projective variety has at least two fixed points. Hence \(F_k'=\{t^{\lambda_k}\}\) (Objection 7.28).
The special fibre is large. \(F_k\) has 8 torus-fixed points; through each passes an orbit of \(Z_{G^\theta}(\mu_k)^\circ\) (type \(B_4\), dimension 36) whose stabilizer has Lie algebra of dimension 29. Hence \(\dim F_k\ge7\) (Audit 7.29).
Detection. At \(x_{\mu_k}\) only \(\mathrm{IC}_{\mu_k}\) and \(\mathrm{IC}_{\mu_k'}\) contribute, and \(\mathrm{IC}_{\mu_k'}\) occurs once, unshifted, by step 3. By the support condition its stalk on \(\mathcal O_{\mu_k}\) vanishes in degrees \(\ge-\delta(\mu_k)\), while the top class of \(F_k\) sits in degree \(2\dim F_k-\delta(\mu_k')=-\delta(\mu_k)+(2\dim F_k-8)\ge-\delta(\mu_k)\). So \(\mathrm{IC}_{\mu_k}[-(2\dim F_k-8)]\) is a summand of \(V_{\lambda_k}\ast\mathrm{IC}_0\) (Audit 7.30).
Mirror labels and generation. The labels \(3k\omega_6^\vee\) follow by rerunning the computation with nodes 1 and 6 exchanged (transport by the diagram automorphism would first require normalizing it to preserve \(G^\theta\)); covered labels follow from dominant lifts; the summand criterion then gives \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) over \(\overline{\mathbb F}_q\), \(p\ne2\) (Objection 7.31).
Field caveats. The function-side failure is \(p\)-adic and conditional on (A1)–(A5), with (A2), (A3) at the given place and (A4) beyond "\(L_\gamma\) classical" unverified (Lemma 6.9); (SG) over \(\mathbb F_q(\!(t)\!)\) is undecided for \(E_6/F_4\), so the two levels have not been compared inside the trace dictionary (Objection 7.31, Audit 7.16).
Assembly. Steps 1–7 with the earlier verdicts Summary 7.19 and Summary 7.25 give the statement.

Builds on

Summary 7.19 · First audit verdict: conclusions (i)–(v) survive with their hypotheses; the only candidate divergence is \(E_6/F_4\)
Summary 7.25 · Updated audit verdict: every row of Table 1 except \(E_6/F_4\) is decided at both levels
Audit 7.26 · Audit of the orbit-label lemma and the Hecke label: both hold
Audit 7.27 · Audit of the open-cell lemma: support bound, monotonicity and the black-cone step hold
Objection 7.28 · Objection to the finiteness lemma: the \(K_N\) dimension count is wrong as written; \(F_k'\) is a point by a fixed-point argument
Audit 7.29 · Audit of the fixed-point computation: 8 fixed points, \(\dim Z_k=36\), stabilizer algebra of dimension \(29\), so \(\dim F_k\ge7\)
Audit 7.30 · Audit of the detection lemma and the summand theorem: both hold, independently of the normalization
Objection 7.31 · Objection to the mirror step and the verdict: generation holds, \(\sigma\) needs normalizing, and the divergence needs its hypotheses stated

Built on by

Re-derivation X1.6 · The verdict inputs downstream of A1-4, A4-10, A5-6, A7-11: what changes
Summary X2.8 · Re-derivation of Summary 4.13, Summary 7.19, Summary 7.25, Summary 7.32 from the repaired steps
Summary X3.7 · A7-25, A7-32 re-derived; the \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) row
Summary X4.9 · \(E_6/F_4\) function cell: two generators, modulo (E) and (I)
Summary X4.14 · \(E_6/F_4\) function cell: two generators, modulo Axiom 2.2.1 over \(\mathcal O\)
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 04:11 UTC

Final verdict: all five conclusions survive; E6/F4 is the one divergence, under (A)

Surviving conclusions.

(i) \(\mathrm{SL}_2/\mathrm{SO}_2\): both levels fail. Sheaves: Betti, by central inertia (A2-4). Functions: every \(q\), every \(F\).
(ii) \(\mathrm{GL}_4/\mathrm{Sp}_4\): both levels hold. Functions: every \(F\). Sheaves: \(\overline{\mathbb F}_q\) under D0–D2, Betti via D5.
(iii) (SG) over \(\mathbb F_q(\!(t)\!)\Rightarrow\mathcal C^{\mathrm{Sat}}=\mathcal C\) for connected \(Z_{G^\theta}(S)\), under D0–D3. The converse is not a theorem.
(iv) Prop. III's conditions \(\Rightarrow\) (SG) over \(\mathbb F_q(\!(t)\!)\); superseded by A5 for every \(F\).
(v) \(E_6/F_4\): (SG) fails under (A). Spheres: (SG) holds, unconditionally (A5).

Sheaf column now decided in every row.

Holds: group, spheres, \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), \(\mathbb G_m\), \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) (A5+A4-6), and \(E_6/F_4\) (A8, audited; A8-5 repaired by A7-28).
Fails: \(\mathrm{SL}_2/\mathrm{SO}_2\), \(\mathrm{GL}_n/\mathrm O_n\), \(\mathrm{PGL}_2/\mathrm{PO}_2\).

Function column. Every row is decided unconditionally over every \(F\) except \(E_6/F_4\), which fails under (A) for \(p\)-adic \(F\).

Answer to the team question. In every row except one, the two levels agree. \(E_6/F_4\) is the unique case with \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) but (SG) failing.

Sheaf side: proved over \(\overline{\mathbb F}_q\), \(p\ne2\), under the workspace's standing inputs (D0–D2: Chen–Yi geometry, proper base change, decomposition theorem), plus Prop. 64. Betti over \(\mathbb C\) via D5.
Function side: proved only for \(p\)-adic \(F\), under Sakellaridis's (A1)–(A5).
'Case with (SG) holding but \(\mathcal C^{\mathrm{Sat}}\ne\mathcal C\)': none.

Two upgrades would make the divergence unconditional or same-field.

Verify (A2)–(A4) for \(E_6/F_4\) at the place.
Prove Theorem 8.0.2 over \(\mathbb F_q(\!(t)\!)\). A6-6's computation of \(r\) does not depend on the field; equivalently, exhibit a 2-dimensional space of \(K\)-invariant \(\mathcal H_K\)-eigenfunctions at the character \(q^{\rho-\omega_4}\).

Then the divergence lives inside the trace dictionary, refuting '\(\mathcal C^{\mathrm{Sat}}=\mathcal C\Rightarrow\)(SG)' in a single field.

Sources

Summary 7.19, Summary 7.25 … Objection 7.31
A8.json
A6.json Theorem 6.6
A5.json

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 7.32 · Final audit verdict: the five conclusions survive; \(E_6/F_4\) is the one divergence, 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.

Builds on

Built on by

Final verdict: all five conclusions survive; E6/F4 is the one divergence, under (A)

Sources

Summary 7.32 · Final audit verdict: the five conclusions survive; \(E_6/F_4\) is the one divergence, under (A)