§8 · Agent 8 · Audit: group, 𝔸ⁿ, 𝔾ₘ · 2026-10-18T19:25:24Z

Summary 8.13 · Verdict of the audit of Claim III: group, $\mathbb A^n$, $\mathbb G_m$

proved ★ key step

No claimed step fails. The group failure is proved over F̄_q and conditional on (T_C) over C; A^n and G_m generation is proved over both fields; (IG) is settled for all three; the other rows remain reduced to (∗_X) under (L_X).

Summary 8.13.

Survives.

  1. Group \(X=H\), \(W\ne1\) (Audit 8.1–Audit 8.7). Over \(\overline{\mathbb F}_q\) (any \(p\), \(H\) split): no \(\Delta_\omega\), \(\omega\in\Omega\), is a perverse constituent of any \(\mathrm{IC}_x\ast\mathrm{IC}_0\ast\mathrm{IC}_y\), hence \(\Delta_e\notin\mathcal C_I^{\rm Iwa}\subseteq\mathcal D_{\widetilde W\setminus\Omega}\). Inputs: sign character, trace of convolution, Lang, Gabber purity, BBD 5.4; no decomposition theorem is needed for this half. Over \(\mathbb C\): the same conclusion, conditional on the good-model comparison (T\(_{\mathbb C}\)). (IG) fails over every non-archimedean \(F\); this is the pages' Proposition E argument. \(\mathcal M_I\supseteq\{xw_0y\text{ reduced}\}\), \(\mathcal M_I\cap\Omega=\emptyset\); in type \(A_1\), \(\mathcal M_I=\widetilde W\setminus\Omega\), with the equivariant splitting from Sun Thm 1.2. In general, \(\mathcal M_I=c_0\Omega\) (lowest two-sided cell) is a citation, consistent with the \(\widetilde A_2\) computation. (II\(_I\)) for \(X=H\) is proved and vacuous. Repair: levels must be \(\Omega\)-truncated; the pages' Example B fails when \(\Omega\) is infinite.
  2. \(\mathbb A^n\) under \(\mathrm{GL}_n\) (Audit 8.8–Audit 8.10). Over \(\mathbb C\) and \(\overline{\mathbb F}_q\), with no weights and no decomposition theorem: \(\operatorname{Simple}(\mathcal P_I)=\{S_k=\underline k_{\tau^k\mathcal O^n}[-k]\}\), each \(S_k=(\delta_{\tau^k}\ast_I\mathrm{IC}_0)[-k]\), so \(\mathcal C_I^{\rm Iwa}=\mathcal C_I=\operatorname{Tri}\{\delta_{\tau^k}\ast\mathrm{IC}_0\}\). (IG) holds over every non-archimedean \(F\), already for \(\mathbb C[\Omega]\). The reverse route (functions ⇒ sheaves) is correct relative to A5-7, which was not audited.
  3. \(\mathbb G_m\) (Audit 8.11). Both levels hold over \(\mathbb C\) and \(\overline{\mathbb F}_q\); the row agrees.
  4. Other rows (Audit 8.11, Objection 8.12). Sheaf generation is equivalent to \((\ast_X)\) given (L\(_X\)) and Chen–Yi's hypotheses (\(p\ne2\), \(K\) connected, true for all rows). The sign-obstruction lemma holds after the Frobenius-orbit repair. Nothing is decided for these rows.

Independence. No argument in the file uses the pages' Proposition II (Whittaker) or IV.1 (I3I3 · Triangularity on pieces: for every piece some Hecke translate of IC_0 or of a lower standard object is supported on its closure with nonzero constant restriction (Iwahori-generation pages). dévissage). The group proof is the trace shadow of the Whittaker functor, with weights in place of exactness; the \(\mathbb A^n\) proof shares with IV.1 only the re-proved lattice-translate fact.

  1. Claim III(a) survives in both versions: Theorem 4.5 and Theorem 2.9 are independent and each complete over \(\overline{\mathbb F}_q\). The decisive point (checklist 6) is correct: \(\varepsilon\circ\operatorname{tr}\) vanishes on generators but not a priori on retracts, and the purity weight \(w+n\) of every \(\Delta_\omega\)-multiplicity space fixes the sign of its contribution, at all \(\mathbb F_{q^m}\) (Audit 8.4, Audit 8.5).
  2. Claim III(b) survives in both versions (Theorem 5.5 and Theorem 2.7). The orbit counts, the \(T_{s_0}\)-formula and the KL-side identities were recomputed (Audit 8.8, Audit 8.10).
  3. Claim III(c) survives (Audit 8.11).
  4. Steps relying on material outside the file: A2-1 to A2-5 (used by Proposition 2.6, Theorem 2.7, Proposition 2.8, Theorem 2.9, Proposition 2.10), A5-4 and A5-7 (used by Theorem 5.5, Corollary 5.8), A4-8 (a remark in Proposition 4.7). The A4 and A5 versions are self-contained for the main conclusions, so these dependencies affect only Corollary 5.8 (via A5-7) and Proposition 2.10(b) (via A2-3).
  5. Fields. Group sheaves: \(\overline{\mathbb F}_q\) proved, \(\mathbb C\) conditional on (T\(_{\mathbb C}\)). \(\mathbb A^n\), \(\mathbb G_m\) sheaves: both fields. Functions: any non-archimedean \(F\); trace statements: \(F=\mathbb F_{q^m}(\!(t)\!)\).

Builds on

  • Audit 8.1 · Audit of the group set-up (A4-1, A2-8): pieces, stabilizers, generators, levels
  • Audit 8.2 · Audit of the group summand criterion (A4-2)
  • Audit 8.3 · Audit of the sign character (A4-3): $\varepsilon$ kills IC traces off $\Omega$
  • Audit 8.4 · Audit of the trace expansion (A4-4): $E_{z,n}$ has weight $w+n-\ell(z)$
  • Audit 8.5 · Audit of the main group theorem (A4-5, A2-9): no $\Delta_\omega$ constituent
  • Audit 8.6 · Audit of (IG) and (II$_I$) for the group (A4-6, A2-9 step 8)
  • Audit 8.7 · Audit of $\mathcal M_I$ for the group (A4-7)
  • Audit 8.8 · Audit of $\mathbb A^n$ orbits, levels and simple objects (A5-1, A5-2, A2-6)
  • Audit 8.9 · Audit of $\mathcal C_I^{\rm Iwa}=\mathcal C_I$ for $\mathbb A^n$ (A5-3, A5-5, A2-7)
  • Audit 8.10 · Audit of (IG) for $\mathbb A^n$ and the reverse route (A5-6, A5-8)
  • Audit 8.11 · Audit of $\mathbb G_m$ and the other rows (A2-10 (a)–(d), (f))
  • Objection 8.12 · Objection to the sign-obstruction lemma (A2-10(e))

Built on by

  • Summary 10.2 · Verdict: the criterion and the bridge hold; the decided rows of Table 1 agree; GL₂/T fails at both levels
Original record, as the agent wrote it at 19:25 UTC

Verdict of the audit of Claim III: group, $\mathbb A^n$, $\mathbb G_m$

  1. Claim III(a) survives in both versions: Theorem 4.5 and Theorem 2.9 are independent and each complete over \(\overline{\mathbb F}_q\). The decisive point (checklist 6) is correct: \(\varepsilon\circ\operatorname{tr}\) vanishes on generators but not a priori on retracts, and the purity weight \(w+n\) of every \(\Delta_\omega\)-multiplicity space fixes the sign of its contribution, at all \(\mathbb F_{q^m}\) (Audit 8.4, Audit 8.5).
  2. Claim III(b) survives in both versions (Theorem 5.5 and Theorem 2.7). The orbit counts, the \(T_{s_0}\)-formula and the KL-side identities were recomputed (Audit 8.8, Audit 8.10).
  3. Claim III(c) survives (Audit 8.11).
  4. Steps relying on material outside the file: A2-1 to A2-5 (used by Proposition 2.6, Theorem 2.7, Proposition 2.8, Theorem 2.9, Proposition 2.10), A5-4 and A5-7 (used by Theorem 5.5, Corollary 5.8), A4-8 (a remark in Proposition 4.7). The A4 and A5 versions are self-contained for the main conclusions, so these dependencies affect only Corollary 5.8 (via A5-7) and Proposition 2.10(b) (via A2-3).
  5. Fields. Group sheaves: \(\overline{\mathbb F}_q\) proved, \(\mathbb C\) conditional on (T\(_{\mathbb C}\)). \(\mathbb A^n\), \(\mathbb G_m\) sheaves: both fields. Functions: any non-archimedean \(F\); trace statements: \(F=\mathbb F_{q^m}(\!(t)\!)\).

Sources