Summary 1.10 · Verdict of the audit: rows 1–3 agree, the rest reduce to one module computation

Summary 1.10.

1. Rows 1–3 (\(H\) under \(H\times H\) with \(W_H\ne1\); \(\mathbb A^n\); \(\mathbb G_m\)), over \(\overline{\mathbb F}_q\) and over \(\mathbb C\): I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III)., I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant., Frobenius-stability of all finite-level orbits (\(n=1\)), a finite-length perverse heart compatible with the colimit, and the decomposition of \(\mathrm{IC}_w\ast_I\mathrm{IC}_0\) with no cancellation all hold (Lemma 1.2, Lemma 1.3, Lemma 1.4, Lemma 1.6). \(M_I=c_0\not\ni e\) for the group; \(M_I\) is every simple object for \(\mathbb A^n\) and \(\mathbb G_m\). \(L^+X\) is one \(K\)-orbit for rows 1 and 3, not for \(\mathbb A^n\), where only \(\operatorname{tr}\mathrm{IC}_0=\pm\varphi_0\) is used.
2. Both levels agree on rows 1–3 (and on \(\mathrm{Spin}_4/\mathrm{Spin}_3\), the group \(\mathrm{SL}_2\)): sheaves false / (IG) fails for every \(F\); sheaves true / (IG) holds for every \(F\). No divergence is proved for any row.
3. Other rows (GL, symplectic, sphere families; \(E_6/F_4\)), over \(\overline{\mathbb F}_q\), \(p\) odd: I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III)., PH\(_I\), DT\(_I\), the local-system classification and \(L^+X=K x_0\) hold (Chen–Yi); I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant. holds for the GL, symplectic and sphere families and is open for \(E_6/F_4\) (Proposition 1.5); Frobenius-stability of every \(I\)-orbit for the given \(q\) is open (Lemma 1.6, D6).
4. Deciding statement. With \(M\) Chen–Yi's module, \(m_0=[\mathrm{IC}_0]\), \(C'_w=[\mathrm{IC}_w]\): sheaves hold iff every IC-basis vector lies in \(\bigcup_w\operatorname{supp}(C'_w m_0)\); (IG) over \(\mathbb F_q((t))\) holds iff the specializations of \(C'_wm_0\) span \(M\otimes\overline{\mathbb Q}_\ell\) (Audit 1.7). Span implies support, so a divergence over \(\mathbb F_q((t))\) can only be "sheaves hold, (IG) fails"; for \(\mathrm{GL}_2/T\), where (IG) fails for every \(F\), a divergence occurs iff its support is full.