The agent audited the second proofs of the first rows of Table 1. For the group, the set-up holds with one repair: a level must be a finite union of double cosets, so the levels must be truncated in \(\Omega\). The pages' own Example B has infinitely many pieces once \(\Omega\) is infinite, as for \(\mathrm{GL}_n\) ( (Audit 8.1)).
It rechecked the sign-character computation with the quadratic relation \((T_s-q)(T_s+1)=0\) ( (Audit 8.3)) and the purity of the multiplicity spaces ( (Audit 8.4)). It confirmed that no \(\Delta_\omega\) is a constituent of a generator over \(\overline{\mathbb F}_q\), and that over \(\mathbb C\) this needs a cited comparison ( (Audit 8.5)). The function half is the pages' sign-character argument, re-derived over \(\mathbb F_q(\!(t)\!)\) from the sheaf half by the bridge.
For \(\mathbb A^n\) it confirmed the orbit classification, the length-zero element \(\tau\), the simple objects and the convolution \(\delta_{\tau^k}\ast_I\mathrm{IC}_0=S_k[k]\) ( (Audit 8.8), (Audit 8.9)). It also confirmed (IG) ( (Audit 8.10)) and \(\mathbb G_m\) ( (Audit 8.11)). It found that a general sign-obstruction lemma needs a Frobenius-orbit repair ( (Objection 8.12)). Its verdict: the second proof of rows 1–2, and of \(\mathbb G_m\), survives, and uses neither the Whittaker functor nor the 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.
Result as the agent reported it proved
No claimed step of Claim III fails; see Summary 8.13. (a) Group X=H, W≠1: over F̄q (any p, H split) no Δ_ω (ω∈Ω) is a perverse constituent of any generator, so Δ_e ∉ C_I^Iwa; over C conditional on (T_C); (IG) fails over every non-archimedean F; M_I ∩ Ω = ∅, and M_I = W~∖Ω in type A_1; levels must be Ω-truncated. (b) A^n under GL_n: over C and F̄_q, C_I^Iwa = C_I = Tri{δ{τ^k}∗IC_0}, and (IG) holds over every non-archimedean F. (c) G_m: both levels hold over C and F̄_q. No argument uses the pages' Proposition II or IV.1.
Gap
(1) Group over C: the Betti statement needs the good-model comparison (T_C) (BBD 6.1.9–6.1.10 / Bezrukavnikov–Finkelberg Prop. 5) compatible with m_! and perverse truncation; it is cited, not proved. (2) The lowest-cell description M_I = c_0Ω and IC_s ∉ C_I^Iwa in rank ≥ 2 rest on Lusztig 1985 and Shi 1987–88, which are not local; they are consistent with an affine-A_2 computation up to length 6. (3) The A^n reverse route is relative to Proposition 5.7 (not audited). (4) Symmetric rows other than G_m and the group: (L_X) for Chen–Yi's presentation remains open, and (∗_X) is undecided. (5) Proposition 2.10(e) needs the Frobenius-orbit repair of Objection 8.12.
Blocked routes
- Direct Betti proof (over C, without Whittaker) of Δ_e ∉ C_I^Iwa for the group: needs a mixed-Hodge-module version of the weight argument (multiplicity spaces pure of weight w+n−ℓ(z), and a ring homomorphism on K_0 of the mixed Hecke category that is compatible with convolution), not available in the local sources.
- Local proof of M_I = c_0Ω for the group in rank ≥ 2: needs Lusztig's a-function properties for affine Weyl groups (z ≤_LR w ⇒ a(z) ≥ a(w); c_0 = {a = ν}), or Shi's theorem; finite KL computations can only give evidence.
Brief as sent
Audit target: The file /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/b7fae8cb-6986-4266-9b6a-db15dbad022f/scratchpad/solve/A8/claims.md contains claimed statements with the proofs exactly as their authors wrote them. Audit every claimed step in it against the adversarial checklist below and against the exact meaning of each term. Do not trust the authors' status labels; check each proof yourself, step by step, and recompute what can be recomputed. The file gives independent versions of Claim III: (a) the group H under H×H (W_H ≠ 1): Δ_e ∉ C_I^{Iwa}, (IG) fails, the summand set M_I; (b) A^n under GL_n: every simple object of C_I is a Hecke translate of IC_0 up to shift, so C_I^{Iwa} = C_I, and (IG) holds; (c) G_m. Check in particular: the sign-character computations and the quadratic relation used; that the vanishing of a trace cannot hide a cancellation (weights over all F_{q^m}, purity of multiplicity spaces); the Ω-components and length-zero elements; the claims about the lowest two-sided cell and what is cited versus proved; for A^n the orbit classification, the length-zero element τ, the convolution IC_{τ^k} ∗_I IC_0, the shift and normalization conventions, that no object is supported at 0; the function-level identities with vol(I) = 1; that none of these arguments uses the Iwahori-generation pages' Proposition II (Whittaker) or IV.1 (I3 dévissage); the transfer to C.
For each claimed step return one or more nodes of kind check or objection whose parents list the claimed step ids audited, and whose statement begins with the outcome: holds · holds with a minor repair (give the repair) · needs hypothesis H (name H and where it is used) · fails at step k (with a counterexample or the exact missing lemma). Where a step fails, say whether the claim survives through another version of it in the file or through a repair you prove. End with a verdict node: which claims survive, with exactly which hypotheses, over which fields.