§9 · Root · Registry and synthesis · 2026-10-17T04:16:49Z

Summary 9.4 · Synthesis: $E_6/F_4$ is the one divergence (categorical generation holds, functions fail under (A)); no reverse case

partial audit: error

Error. The Answer and the partial Status follow from Summary 7.32, Summary 8.10, Theorem 6.6, Theorem 5.7, Summary 2.9, Summary 3.11 and Theorem 4.6 with the hypotheses stated, and are correctly conditional and cross-field. Error: the Mechanism paragraph asserts as fact that 'at the given q the columns [of the multiplicity matrix] do not span'. The multiplicity matrix (m_{nu,lambda}) is the trace matrix of the dictionary over F_q((t)) (Lemma 4.4, Proposition 4.7), and its columns spanning is exactly (SG) over F_q((t)), which is open for E6/F4. Summary 9.4's own Status paragraph says putting both sides on F_q((t)) needs Theorem 8.0.2 in characteristic p. The gluing of y_1 and sigma y_1 kills the span only for p-adic F under (A1)-(A5), through Theorem E. So a p-adic result is moved into the F_q((t)) dictionary without the bridge. The same paragraph also says 'every column of the multiplicity matrix is nonzero' where C^Sat = C gives 'every row is nonzero' (each IC label occurs; Proposition 4.7's convention), using 'columns' for both conditions. Minor: 'the one row with disconnected Z_{G^theta}(S), SL2/SO2' is true only within the standing setting G^theta connected; GL_n/O_n and PGL_2/PO_2 also have disconnected Z_{G^theta}(S) (and G^theta), but both levels fail there, so 'no reverse case' stands. The transfer theorem also needs q odd, T split, G^theta connected and D0-D3. The rewrite keeps the mechanism claim as made, with the row/column wording clarified; the repair is Correction 10.1. The conclusion still holds by another argument.
Corrected by: Correction 10.1 · Correction: the mechanism of the $E_6/F_4$ divergence, separated by field, Summary X4.9 · $E_6/F_4$ function cell: two generators, modulo (E) and (I).

Every symmetric space on the pages gets the same answer at both levels except $E_6/F_4$, where the Satake span is the whole category while the Hecke translates of the basic function do not span, under Sakellaridis's hypotheses over $p$-adic fields. The result is partial: it rests on unverified hypotheses at the function level, on the standing inputs at the sheaf level, and compares two different fields.

Summary 9.4.

Answer. Every symmetric space on the pages gets the same answer at both levels except $E_6/F_4$, where $$\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\qquad\text{but}\qquad\mathcal H_K\cdot\varphi_0\subsetneq C_K ,$$ the first at sheaf level (Summary 8.10, audited), the second at function level (Theorem 6.6), under (A), for $p$-adic $F$. No case has (SG) holding and $\mathcal C_K^{\mathrm{Sat}}\ne\mathcal C_K$: over $\mathbb F_q(\!(t)\!)$ such a case is impossible whenever $Z_{G^\theta}(S)$ is connected (the transfer theorem Theorem 4.6), and in the one row with disconnected $Z_{G^\theta}(S)$, $\mathrm{SL}_2/\mathrm{SO}_2$, both levels fail.

Status: partial. The divergence is proved modulo (1) Sakellaridis's (A2), (A3) at the given place and (A4) beyond "$L_\gamma$ classical" for $E_6/F_4$ (Lemma 6.9); and (2) the standing inputs D0–D2 of Proposition III for the sheaf side, with the Betti category reached via D5. The two sides live over different fields: putting both on $\mathbb F_q(\!(t)\!)$ needs Sakellaridis's Theorem 8.0.2 in characteristic $p$.

Mechanism. At $3k\omega_1^\vee$ the intersection complex is a retract of $V_{\lambda_k}\ast\mathrm{IC}_0$ with $\varphi(\lambda_k)$ strictly above $3k\omega_1^\vee$, so every label occurs in the multiplicity matrix of the Satake objects; but the Satake-shifted torus glues $y_1$ and $\sigma y_1$, so at the given $q$ the multiplicity vectors of the Satake objects do not span. The categorical statement sees the whole grading; the function statement evaluates it at $q$.

Notation: D0–D5 are the obligations of the trace dictionary (Setup 4.1); "under (A)" means under Sakellaridis's hypotheses (A1)–(A5); $\lambda_k=\tfrac{3k-1}2\omega_1^\vee+\omega_6^\vee$ for $k$ odd, and $y_1,\sigma y_1$ are the points of $A_X^*$ of Theorem 6.6.

The rows other than $E_6/F_4$. By the final audit verdict Summary 7.32, every row except $E_6/F_4$ is decided at both levels and the answers agree: both statements hold for the group, the spheres, $\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)$, $\mathbb G_m$ and the symplectic families (Theorem 5.7 with Theorem 4.6; Summary 3.11 for $\mathrm{GL}_4/\mathrm{Sp}_4$); both fail for $\mathrm{SL}_2/\mathrm{SO}_2$ (Summary 2.9) and for $\mathrm{GL}_n/\mathrm O_n$, $\mathrm{PGL}_2/\mathrm{PO}_2$.
$E_6/F_4$, sheaves. Every $\mathrm{IC}_\mu$, $\mu\in\Lambda_S^+$, is a direct summand of some $V_\lambda\ast\mathrm{IC}_0$: covered labels by dominant lifts, the labels $3k\omega_1^\vee$ and $3k\omega_6^\vee$ by the fibre computation (Summary 8.10, audited in Audit 7.26 to Objection 7.31). The summand criterion gives $\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K$ over $\overline{\mathbb F}_q$.
$E_6/F_4$, functions. Under (A), Theorem E identifies $\mathcal H_K\cdot\varphi_0$ with the image of the restriction map $r$, which misses the orbit sum $M_{3\omega_1^\vee}$ because $q^{\rho_L}y_1$ and $q^{\rho_L}\sigma y_1$ are $W(E_6)$-conjugate while $y_1$ and $\sigma y_1$ are not $W_X$-conjugate (Theorem 6.6; recomputed in Audit 9.3).
No reverse case. Over $\mathbb F_q(\!(t)\!)$, the transfer theorem forbids "(SG) holds, $\mathcal C_K^{\mathrm{Sat}}\ne\mathcal C_K$" when $Z_{G^\theta}(S)$ is connected; the row $\mathrm{SL}_2/\mathrm{SO}_2$ fails at both levels.
Mechanism. By step 2 the summand $\mathrm{IC}_{3k\omega_1^\vee}$ of $V_{\lambda_k}\ast\mathrm{IC}_0$ lies on a strictly larger support, so it is detected without being isolated; by step 3 the gluing of $y_1$ and $\sigma y_1$ kills the span at the given $q$.
Budget. Eight agents were used, six in the first round and two in the second; the auditor was continued three times. Every candidate proof was audited, and the corrections are listed on the verdict page.

Builds on

Summary 7.32 · Final audit verdict: the five conclusions survive; $E_6/F_4$ is the one divergence, under (A)
Summary 8.10 · Verdict for $E_6/F_4$: $\Csat=\cC_K$, and the claimed divergence from the function level
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)
Theorem 5.7 · $(\mathrm{SG})$ for $\GL_{2m}/\Sp_{2m}$ (all $m$) and $\SL_{2n}/\Sp_{2n}$ (all $n$), every non-archimedean local field
Summary 2.9 · Verdict for $\SL_2/\SO_2$: both levels fail, and the sheaf argument is the geometric form of Proposition J(2)
Summary 3.11 · Verdict for the symplectic family: no disagreement between the levels; $\GL_4/\Sp_4$ settled at both
Theorem 4.6 · Theorem: (SG) over $\F_q(\!(t)\!)$ implies $\Csat=\cC_K$ over $\overline{\F}_q$

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Correction 10.1 · Correction: the mechanism of the $E_6/F_4$ divergence, separated by field
Summary 9.5 · Verdict after the second audit: $E_6/F_4$ is still the one divergence

Original record, as the agent wrote it at 04:16 UTC

Synthesis: E6/F4 is the one divergence (categorical holds, functions fail under (A)); no reverse case

Answer. Every symmetric space on the pages gets the same answer at both levels except $E_6/F_4$, where $$\mathcal C^{\mathrm{Sat}}=\mathcal C\quad\text{(sheaves, A8, audited)}\qquad\text{but}\qquad\mathcal H_K\cdot\varphi_0\subsetneq C_K\quad\text{(functions, A6, under (A), }p\text{-adic)} .$$ No case has (SG) holding and $\mathcal C^{\mathrm{Sat}}\ne\mathcal C$. Over $\mathbb F_q(\!(t)\!)$ such a case is impossible whenever $Z_{G^\theta}(S)$ is connected (Theorem 3), and in the one row with disconnected $Z_{G^\theta}(S)$, $\mathrm{SL}_2/\mathrm{SO}_2$, both levels fail.

Status: partial. The divergence is proved modulo (1) Sakellaridis's (A2), (A3) at the place and (A4) beyond "$L_\gamma$ classical" for $E_6/F_4$ (A6-9), (2) the standing inputs D0–D2 of Proposition III for the sheaf side, with Betti via D5. The two sides live over different fields: putting both on $\mathbb F_q(\!(t)\!)$ needs Sakellaridis's Theorem 8.0.2 in characteristic $p$.

Mechanism. At $3k\omega_1^\vee$ the IC sheaf is a retract of $V_{\lambda_k}\ast\mathrm{IC}_0$ with $\varphi(\lambda_k)$ strictly above, so every column of the multiplicity matrix is nonzero; but the Satake-shifted torus glues $y_1$ and $\sigma y_1$, so at the given $q$ the columns do not span. The categorical statement sees the whole grading, the function statement evaluates it at $q$.

Budget. Eight agents (six in round 1, two in round 2); the auditor was continued three times. Every candidate proof was audited; corrections are listed on the verdict page.

Sources

returns/A7c.json
returns/A8.json
returns/A6.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 9.4 · Synthesis: \(E_6/F_4\) is the one divergence (categorical generation holds, functions fail 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.); no reverse case

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Synthesis: E6/F4 is the one divergence (categorical holds, functions fail under (A)); no reverse case

Sources

Summary 9.4 · Synthesis: \(E_6/F_4\) is the one divergence (categorical generation holds, functions fail under (A)); no reverse case