§9 · Root · Registry and synthesis · 2026-10-17T22:15:17Z
Summary 9.6 · Verdict after the repairs: every input held or repaired; \(E_6/F_4\) still the one divergence
partial ★ key step
Four fresh agents (X1–X4) repaired every error the run had left unrepaired, and fresh reviewers re-checked each repair: of the twelve statements the verdict rests on, six held and six were repaired, none lost. The answer is unchanged; the \(E_6/F_4\) function cell is sharper: \(C_K\) needs two generators, under Sakellaridis\'s theorem, all of whose hypotheses but one now hold.
Summary 9.6.
Answer (status partial). Unchanged by the repairs: \(E_6/F_4\) is the one divergence, and it compares two fields.
- Every row other than \(E_6/F_4\) gets the same answer at both levels. Both statements fail for \(\mathrm{SL}_2/\mathrm{SO}_2\) (functions now for both \(\mathbb F_q\)-forms, every odd \(q\): Proposition X2.6), \(\mathrm{GL}_n/\mathrm O_n\) and \(\mathrm{PGL}_2/\mathrm{PO}_2\); both hold for every other row. For \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) the function cell is the pair model over every non-archimedean field; in residue characteristic \(2\) the involution model's basic function does not generate, so the row must name its model (Proposition X3.5, Summary X3.7).
- \(E_6/F_4\). Sheaves: \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) over \(\overline{\mathbb F}_q\), \(p\ne2\), under D0–D2The standing inputs of the sheaf–function dictionary over F̄_q, p ≠ 2: D0 Chen–Yi's orbit geometry; D1 traces of Frobenius and the perverse t-structure; D2 decomposition of Satake objects into shifted IC sheaves; D3 Lang's theorem; D4 unitriangularity of Satake traces; D5 comparison of F̄_q with C., and in the Betti category through D5D5 · Comparison with C: “IC_μ is a summand of V_λ ∗ IC_0” has the same answer over F̄_q and over C (BBD §6.1, for large p). (Summary 8.10). Functions, over \(p\)-adic \(F\) with \(p\) odd: the image of Sakellaridis's restriction map has codimension one, a fact of pure algebra (Proposition X4.7); under his Theorem 8.0.2, \(C_K\) is not a cyclic Hecke module and needs exactly two generators, so (SG) fails (Corollary X4.8). For the model \(G/G^{\theta_0}\), every hypothesis of that theorem now holds, some through cited inputs, except his Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. For E6/F4 it holds over every O with 2 invertible (VI run). as \(\mathcal O\)-schemes (Proposition X4.13, Summary X4.14).
- No reverse case, as before (Theorem 4.6).
The run's inputs after the repairs: 12 inputs; 6 held, 6 repaired, 0 lost.
| input | what the run used it for | status | after the repairs |
|---|---|---|---|
| Summary 1.9 | the status table, \(\mathrm{GL}_4/\mathrm{Sp}_4\) both hold | repaired | holds with a changed reason: the translates reaching \((1,1,1,1)\) are those of Correction X1.1 |
| Summary 2.9 | \(\mathrm{SL}_2/\mathrm{SO}_2\) fails at both levels | held | — |
| Summary 3.11 | the symplectic verdict | held | its error was repaired inside the run |
| Theorem 4.6 | functions \(\Rightarrow\) sheaves over \(\mathbb F_q(\!(t)\!)\) | held | — |
| Summary 4.13 | one direction is a theorem | repaired | holds; the higher symplectic cells now cite Theorem 5.7 (Re-derivation X1.5, Summary X2.8) |
| Theorem 5.7 | (SG) for \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) | repaired | holds as stated: it never used the wrong table entries (Correction X1.2) |
| Theorem 6.6 | \(E_6/F_4\): \(r\) not onto | held | sharpened: \(C_K\) needs exactly two generators (Proposition X4.7, Corollary X4.8) |
| Summary 6.10 | spheres agree; \(E_6/F_4\) fails for functions | held | — |
| Summary 7.19 | the first audit's five conclusions | repaired | holds; the codimension formula now rests on the stabilizer count of Proposition X1.4 |
| Summary 7.25 | every row but \(E_6/F_4\) decided | repaired | holds; add the correction of the pair-model support bound (Summary X3.7) |
| Summary 7.32 | \(E_6/F_4\) is the one divergence | repaired | holds once the \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) row names the pair model (Summary X3.7); \(E_6/F_4\) cell sharpened (Summary X4.14) |
| Summary 8.10 | \(E_6/F_4\): the Satake span is everything | held | in the conditional form of Summary 7.32, as the run already corrected it |
What was done. The second audit had found 15 errors among the run's 110 steps (Summary 9.5). Eight of them, in agents' steps, had no repair inside the run; they fell into three clusters. Three fresh agents repaired them: the symplectic counts (Correction X1.1, Correction X1.3, Proposition X1.4), the \(\mathrm{SL}_2/\mathrm{SO}_2\) trace argument (Theorem (repair of A4-11) X2.4, Correction (repair of A7-16) X2.5) and the pair model in residue characteristic \(2\) (Proposition X3.2, Correction X3.6). A fourth agent, added at the user's request, made Sakellaridis's theorem self-contained and asked whether \(C_K\) is cyclic at \(E_6/F_4\) (Proposition X4.7, Proposition X4.13). A fresh reviewer re-checked each agent's steps (Re-check V1 to Re-check V5). One step, Gap X4.6, failed the re-check because it left open a hypothesis that a published decomposition settles; its agent's second attempt (Proposition X4.12, Proposition X4.13) passed.
The root's own errors. Strategy 9.1 declared the only possible divergence too early, and Summary 9.4 asserted a span failure over \(\mathbb F_q(\!(t)\!)\) that is open. Both were corrected by the second audit (Correction 10.1, Summary 9.5); nothing here changes them.
What remains open.
- His Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. For E6/F4 it holds over every O with 2 invertible (VI run). as \(\mathcal O\)-schemes for \(E_6/F_4\): 45 maximal-rank Borel orbits, 6 simple roots, types \(G\) and \(U\) only (Proposition X4.13).
- The cited inputs of the \(E_6/F_4\) function cell: Delorme–Sécherre's decomposition, Kneser, Edixhoven and Steinberg, Alper, and the paper's remark on ramified \(F\) (Summary X4.14).
- The comparison D5D5 · Comparison with C: “IC_μ is a summand of V_λ ∗ IC_0” has the same answer over F̄_q and over C (BBD §6.1, for large p). between \(\overline{\mathbb F}_q\) and \(\mathbb C\).
- Sakellaridis's theorem in characteristic \(p\), which would put both halves of \(E_6/F_4\) on one field.
Builds on
- Summary 9.5 · Verdict after the second audit: \(E_6/F_4\) is still the one divergence
- Re-derivation X1.5 · (SG) for GL₂ₘ/Sp₂ₘ re-derived from Lemmas 2–3 alone; A4-13's open cells now decided
- 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
- Correction X1.1 · GL₄/Sp₄: every T_λ nonzero on the orbit (1,1,1,1), exact values, step 2 corrected
- Correction X1.2 · GL₆/Sp₆: Lemmas 1–3 re-proved; the |b|=4 table proved for all q; |b|≤5 computed
- Correction X1.3 · GL₆/Sp₆ at Σμ=4: the twelve values of R1–R3, e₂∉span, one larger-box translate
- Proposition X1.4 · L(GL₂ₘ/Sp₂ₘ): codim = ⟨ρ,μ−ν⟩ in every component, char ≠ 2; σ reaches only m | s
- Lemma X2.1 · \(\sigma_n\) is translation by \(h\in L^+\mathrm{SL}_2(\overline{\mathbb F}_q)\) with \(h^{-1}\mathrm{Fr}^n(h)=-I\)
- Lemma X2.2 · Weil inertia lemma: \(\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}\circ\sigma_n=\chi(-1)\,\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}\)
- Theorem (repair of A4-11) X2.4 · Repair of A4-11: \(\mathrm{IC}_{\mathrm{sgn},\mu}\notin\mathcal C_K^{\mathrm{Sat}}\) in both \(\mathbb F_q\)-forms, \(q\) odd
- Correction (repair of A7-16) X2.5 · Chen–Yi §8.2 holds for the \(\mathrm{SO}_2\)-form iff \(q\equiv1\bmod4\); Lang’s count holds in both
- Proposition X3.2 · Support bound and top orbit for the pair model, every dominant \(\lambda\), every \(F\)
- Proposition X3.3 · Cartan label of \(\tau(x)\) is \(\mu((\ell(x)-e)^+)\), \(e=\operatorname{val}2\)
- Theorem X3.4 · Corrected A5-8 for the three families, and (SG) for the pair model without \(\tau\)
- Correction X3.6 · Correction of A7-24: its two objections stand; its claim that A5-8 holds needs a new proof
- Proposition X4.3 · \(E_6/F_4\): type-\(G\) spherical roots, two colours; Statements 6.3.1, 7.1.5 hold
- Proposition X4.4 · \(E_6/F_4\), \(p\) odd: the \(\mathcal O\)-model meets §1.7 and §8
- Proposition X4.5 · \(E_6/F_4\), \(p\) odd: Axiom 2.4.2Sakellaridis's Axiom 2.4.2, relabelled (A3c): Iwahori orbits at the antidominant torus points., half of Axiom 2.4.1Sakellaridis's Axiom 2.4.1, relabelled (A3b): the Cartan decomposition of X(F) into K-orbits., types in 2.2.1
- Proposition X4.7 · \(\operatorname{im}r=\{g:g(y_1)=g(\sigma y_1)\}\), of codimension one, every \(q\)
- Corollary X4.8 · Under Theorem 8.0.2, \(C_K\) is not cyclic and needs exactly two generators
- Audit X4.10 · Part 2 assumes no type-\(N\) reflections; this holds for \(E_6/F_4\)
- Proposition X4.11 · An involution of type E IV of \(E_6\) defined over \(\mathbb Z\)
- Proposition X4.13 · \(E_6/F_4\), \(p\) odd: Axiom 2.4.1Sakellaridis's Axiom 2.4.1, relabelled (A3b): the Cartan decomposition of X(F) into K-orbits. holds; only Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. For E6/F4 it holds over every O with 2 invertible (VI run). over \(\mathcal O\) open
- Summary X4.14 · \(E_6/F_4\) function cell: two generators, modulo Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. For E6/F4 it holds over every O with 2 invertible (VI run). over \(\mathcal O\)
Built on by
- —
Sources
- audit/S1.json
- audit/S2.json
- audit/S3.json
- audit/S4.json
- audit/S4b.json
- audit/recheck.json