§9 · Root · Registry and synthesis · 2026-10-17T03:54:12Z

Audit 9.3 · Registry after the lattice-count audit: every row except \(E_6/F_4\) decided at both levels; the \(E_6\) Weyl conjugacy re-checked

proved audit: minor defect

Minor defect (second audit). The root check is correct: recomputed independently (e6_g7.py), both weights reach (1,1,1,0,1,1) by greedy words of length 18 and 13 (norms 42 and 42); the S_3-orbits of (-5,1) and (-5,4) are disjoint and sigma y_1 = (1,-5) lies in the orbit of y_2; M_(3,0) has exponents {-9,3,6} at y_1 and {-6,-3,9} at y_2, while M_(1,1) agrees at both. The summary of the lattice-count audit matches Audit 7.22/A7-23 (at q=3 one cell of the m=3 table was not run but is forced to vanish by Lemma 3, as Audit 7.22 notes). Defects (missing hypotheses only): the transfer theorem's conclusion needs q odd and D0-D3The 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.; 'GL_n/(GL_p x GL_q) ... residue characteristic 2 included' holds for the pair model, which differs from the pages' J-model when q is even (Audit 7.21); and 'only in the direction categorical holds, functions fail' / 'excludes the reverse direction' hold over F_q((t)) (D0-D3The 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.), while at E6/F4 the p-adic reverse case is excluded only 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.. Also: "the \(m=3\) table recomputed" overstates Audit 7.22, which recomputed two of its five rows; two of the others were wrong (corrected in Correction 10.6).

The audited reverse-dominance triangularity gives (SG) for the symplectic families over every local field, and the transfer theorem turns this into \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) over \(\overline{\mathbb F}_q\). An independent recomputation confirms the Weyl-group coincidence behind the \(E_6/F_4\) function-level failure, so only \(E_6/F_4\) can still diverge.

Audit 9.3.

  1. (SG) holds over every non-archimedean local field for \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) (Theorem 5.7, audited in Audit 7.22 and Audit 7.23), and the transfer theorem Theorem 4.6 turns this into \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) over \(\overline{\mathbb F}_q\) for those rows. The spheres and \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) hold at function level over every \(F\), residue characteristic 2 included (Audit 7.20, Audit 7.21).
  2. In fundamental-weight coordinates of \(E_6\) (Bourbaki labels), \(\rho-9\omega_1-3\omega_6=(-8,1,1,1,1,-2)\) and \(\rho-9\omega_1=(-8,1,1,1,1,1)\) are both carried by simple reflections to the dominant weight \(\rho-\omega_4=(1,1,1,0,1,1)\), by words of length 18 and 13; hence they are \(W(E_6)\)-conjugate. In logarithmic coordinates the \(S_3\)-orbits of \(y_1=(-5,1)\) and \(y_2=(-5,4)\) are disjoint, and the orbit sum \(M_{3\omega_1^\vee}\) takes the values \(q^{-9}+q^{3}+q^{6}\) at \(y_1\) and \(q^{-6}+q^{-3}+q^{9}\) at \(y_2\). So the \(E_6/F_4\) function-level failure 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. (Theorem 6.6) stands.
  3. The only row where the two levels can still disagree is \(E_6/F_4\), and only in the direction "categorical holds, functions fail"; the transfer theorem excludes the reverse direction in every row with \(Z_{G^\theta}(S)\) connected.
  1. Reverse dominance. For \(b=(b_1\ge\dots\ge b_m\ge0)\) put \(\lambda^*(b)=(b_1,\dots,b_m,0,\dots,0)\) and \(N(\lambda,b')=(T_\lambda\ast\varphi_0)(x_{b'})\), the value of a Hecke translate of the basic function on the orbit of type \(b'\). Lemma 3 of Lemma 5.6 states \(N(\lambda^*(b),b')\ne0\Rightarrow b'\ge_{\mathrm{dom}}b\). It comes from the perfect pairing \(\Lambda_0^\#/\Lambda''\times\Lambda''/\Lambda_0\to F/\mathcal O\) induced by the form, which makes the two quotients dual, together with the bound on the order of a \(k\)-generated submodule of a finite \(\mathcal O\)-module. It was checked line by line, and the table for \(m=3\) was recomputed over \(\mathbb F_2[\![t]\!]\) and \(\mathbb F_3[\![t]\!]\) (Audit 7.22).
  2. Induction. Since \(N(\lambda^*(b),b)\ge1\), descending induction along dominance inside the finite sets \(\{b:\ b_m\ge2c,\ \sum_ib_i=d\}\), using central translates, produces every orbit indicator (Audit 7.23).
  3. Sheaves. The transfer theorem applies: \(G^\theta=\mathrm{Sp}_{2m}\) and \(Z_{G^\theta}(S)=\mathrm{SL}_2^m\) are connected, and (SG) holds over \(\mathbb F_q(\!(t)\!)\) for every \(q\) (Audit 7.23).
  4. The \(E_6\) check. A dominant representative is reached by applying, at a negative coordinate \(c_j\), the simple reflection \(s_j(c)=c-c_j\,(\text{row }j\text{ of the Cartan matrix})\); two weights are \(W\)-conjugate exactly when they reach the same dominant weight. The restricted Weyl group \(S_3\) acts on \((\log_qu,\log_qv)\) by \((a,b)\mapsto(-a,a+b)\) and \((a,b)\mapsto(a+b,-b)\), where \(u,v\) are the values of a point of \(A_X^*\) on the two simple restricted coroots. The orbit sum is \(M_{3\omega_1^\vee}=u^2v+u^{-1}v+u^{-1}v^{-2}\).
  5. State. Steps 1–4 and the updated audit verdict Summary 7.25 give the registry; the \(E_6/F_4\) sheaf level at the labels \(3k\omega_1^\vee\) was still awaited.

Verified by computation: e6check.py; recomputed independently in e6_g7.py.

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Original record, as the agent wrote it at 03:54 UTC

A5 audited: every Table 1 row except E6/F4 now decided at both levels; root re-checks the E6 conjugacy

What changed. A5's reverse-dominance triangularity (Lemma 3: \(N(\lambda^*(b),b')\ne0\Rightarrow b'\ge_{\mathrm{dom}}b\), from the perfect pairing \(\Lambda_0^\#/\Lambda''\times\Lambda''/\Lambda_0\to F/\mathcal O\)) survives line-by-line audit, with the \(m=3\) table recomputed over \(\mathbb F_2[\![t]\!]\) and \(\mathbb F_3[\![t]\!]\) (A7-22). So (SG) holds over every non-archimedean \(F\) for \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\), and Theorem 3 (A4-6) turns this into \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) over \(\overline{\mathbb F}_q\) for those rows (A7-23). The spheres and \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) hold at function level over every \(F\), residue characteristic 2 included (A7-20, A7-21).

Root check. Recomputed independently (scratchpad/solve/R/e6check.py): \(\rho-9\omega_1-3\omega_6\) and \(\rho-9\omega_1\) both reflect to \(\rho-\omega_4=(1,1,1,0,1,1)\) (word lengths 18 and 13, matching A6-6); the two \(S_3\)-orbits of \(y_1=(-5,1)\), \(y_2=(-5,4)\) (log coordinates) are disjoint; \(M_{3\omega_1^\vee}\) takes the values \(q^{-9}+q^3+q^6\) and \(q^{-6}+q^{-3}+q^9\). The \(E_6/F_4\) function-level failure under (A) stands.

State. The only row where the two levels can still disagree is \(E_6/F_4\), and only in the direction 'categorical holds, functions fail'. Theorem 3 excludes the reverse direction in every row with connected \(Z_{G^\theta}(S)\). Waiting on A8 (sheaf level at \(3k\omega_1^\vee\)).

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