§9 · Root · Registry and synthesis · 2026-10-17T07:23:13Z

Summary 9.5 · Verdict after the second audit: \(E_6/F_4\) is still the one divergence

partial ★ key step

Seven independent auditors re-checked all 110 steps of the run and found 15 errors: false side remarks, wrong table entries, an unchecked input, a circular step, results moved across fields, and syntheses that claimed more than was proved. Each is now corrected, inside the run or by one of eight correction steps, and none changes the answer: \(E_6/F_4\) is the only symmetric space on the pages where the two levels disagree, and no row has the reverse.

Summary 9.5.

Answer (status partial: the conditions named below are not all verified).

Every row other than \(E_6/F_4\) gets the same answer at both levels. Both statements fail for \(\mathrm{SL}_2/\mathrm{SO}_2\), \(\mathrm{GL}_n/\mathrm O_n\) and \(\mathrm{PGL}_2/\mathrm{PO}_2\); both hold for every other row of Table 1. Where (SG) is marked "every \(F\)" it holds over every non-archimedean local field without Sakellaridis's hypotheses; for \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) this is the pair model, which is the pages' model when \(q\) is odd. Apart from the group case, every "equal" sheaf cell is proved \(\ell\)-adically over \(\overline{\mathbb F}_q\), \(p\ne2\), and reaches the Betti category of the pages through D5.
\(E_6/F_4\) is a divergence across fields. \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) over \(\overline{\mathbb F}_q\), \(p\ne2\), under D0–D2 and Chen–Yi's Prop. 64, and in the Betti category through D5 (Summary 8.10); (SG) fails over every \(p\)-adic field under Sakellaridis's (A1)–(A5), of which (A2)–(A4) are unverified (Theorem 6.6). Whether (SG) holds over \(\mathbb F_q(\!(t)\!)\) is open (Correction 10.1).
No reverse case. Over \(\mathbb F_q(\!(t)\!)\), \(q\) odd, function-level generation implies categorical generation for every split symmetric space with \(G^\theta\) and \(Z_{G^\theta}(S)\) connected, under D0–D3 (Theorem 4.6); for every field the table excludes the reverse case row by row: wherever (SG) is known, \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) holds too.

What was checked. Every step was re-derived by an auditor who had not written it, in seven groups: the literature audit and \(\mathrm{SL}_2/\mathrm{SO}_2\); the symplectic family; the sheaf–function dictionary; function-level generation without (A1)–(A5); the restriction map for \(E_6/F_4\); the \(E_6/F_4\) sheaf proof; the synthesis and the verdict page. The auditors recomputed the finite computations with their own programs, by methods different from the run's, checked every citation against its source, and checked that the hypotheses of each cited theorem hold in the case at hand.

Errors and their corrections. A step is an error if a claim in it is false or a proof step fails as written; "carried to" lists the later steps that used the faulty part.

Table 2. step and what was wrong and carried to
step	what was wrong	carried to	corrected by
Proposition 1.2	a false side remark: \(\mathrm{IC}_0\) is not a summand of \(V_{(1,0)}\ast\mathrm{IC}_0\)	Summary 1.9	Objection 7.15
Proposition 1.4	the wrong list of Hecke translates reaching the label \((1,1,1,1)\)	—	Correction 10.3
Summary 1.9	repeated that remark; called a settled question open; stale cells	—	Objection 7.15, Objection 7.17
Summary 3.11	combined a result over \(\mathbb F_q(\!(t)\!)\) with a \(p\)-adic one	—	Audit 7.18, Audit 7.23
Dead end 4.10	four wrong entries in the \(\mathrm{GL}_6/\mathrm{Sp}_6\) table	—	Correction 10.7
Theorem 4.11	a circular step (\(\sigma\)-invariance of the traces)	Summary 4.13	Correction 10.8
Lemma 5.6	two wrong entries in the \(m=3\) table	—	Correction 10.6
Proposition 5.8	an input that fails in residue characteristic 2	Objection 7.24	Correction 10.4
Objection 7.11	a false repair of the Prop. 64 citation	Summary 7.19, Summary 7.25, Summary 9.2, verdict §5	Correction 10.2
Audit 7.16	a false remark on the \(\mathrm{SO}_2\)-model	—	Correction 10.5
Objection 7.24	passed the unchecked input of Proposition 5.8	—	Correction 10.4
Lemma 8.5	a false dimension count	Theorem 8.8, as an assumption	Objection 7.28
Summary 8.10	the divergence stated with too few conditions	—	Objection 7.31
Strategy 9.1	"the only live cell", premature at 02:56	—	Audit 9.3
Summary 9.4	asserted the open span failure over \(\mathbb F_q(\!(t)\!)\)	—	Correction 10.1

In every case the erroneous step's conclusion either still holds by another argument or was never used for a conclusion; the corrections re-prove what was used, and the verdict page now carries the qualifiers the audit asked for.

What remains open. Sakellaridis's (A2)–(A4) for \(E_6/F_4\); D5, the pages' open item 1; and (SG) for \(E_6/F_4\) over \(\mathbb F_q(\!(t)\!)\), which would put both halves of the divergence on one field. Correction 10.1 gives a sufficient criterion for the last.

Builds on

Summary 9.4 · Synthesis: \(E_6/F_4\) is the one divergence (categorical generation holds, functions fail under (A)); no reverse case
Correction 10.1 · Correction: the mechanism of the \(E_6/F_4\) divergence, separated by field
Correction 10.2 · Codimensions of orbits in \(L(\GL_{2m}/\Sp_{2m})\) in every component, without connectedness of \(LX\)
Correction 10.3 · \(\GL_4/\Sp_4\): which Hecke translates reach the label \((1,1,1,1)\), and the corrected cancellation
Correction 10.4 · Correction: the function-level support bound for the pair model of \(\GL_n/(\GL_p\times\GL_q)\) over every local field
Correction 10.5 · The \(\SO_2\)-model of \(\SL_2/\SO_2\) over \(\F_q\) and Chen–Yi's splitness hypothesis
Correction 10.6 · Correction: the \(m=3\) check table for \(\GL_6/\Sp_6\), with the Lagrangians that are graphs
Correction 10.7 · Correction: the in-box Hecke translates of \(\GL_6/\Sp_6\) at \(\sum\mu=4\), and one translate that reaches \((1,1,1,1,0,0)\)
Correction 10.8 · Correction: the square-class automorphism multiplies \(\tr_n\IC_{\chi,\nu}\) by \(\chi(-1)\)

Built on by

Summary 9.6 · Verdict after the repairs: every input held or repaired; \(E_6/F_4\) still the one divergence

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