C-G3 · 2026-10-21T15:04:12Z
Review G3 · Agent 4 and root
A4-1 (error): Proof step 6 asserts that every Frobenius-stable maximal torus of Res_{Fq2/Fq} H has invariant cocharacter rank rk(H). This equality is false: H=GL2 has the maximal torus Res_{Fq4/Fq} Gm in Res_{Fq2/Fq} GL2, whose Frobenius is a four-cycle and whose invariant rank is 1, not 2. The required no-split-maximal-torus conclusion follows from the upper bound <=rk(H)<2rk(H), so the stated forms and counterexample survive. The restriction of D2 purity to Frobenius-stable labels is necessary and correctly stated.
A4-8 (error): Proof step 4 identifies the projective Schubert variety Gr_{<=alpha-coroot} for SL2 with the affine nilpotent cone in sl2. They cannot be isomorphic: one is projective and the other is positive-dimensional affine. The nilpotent A1 cone is the transverse/local singularity at the closed stratum. The rational-smoothness conclusion, trace(IC_1)=1_1+1_0, the column (q^2,q,1), and all nonvanishing/spanning conclusions remain true. Independently, Frobenius-fixed blocks are the only blocks contributing trace, and Clebsch-Gordan for PGL2 gives one line at every c between 0 and 2a.
A4-15 (minor): The concrete orbit-category trace computation re-derives: bounded equivariant complexes on one connected stabilizer are Perf C*(BL); polynomial even cohomology admits a free commutative model, with Frobenius homotopy on generators. The twisted Koszul sequence (1-q^d)v has quotient Q_l in degree 0. For a Frobenius-stable recollement, the cyclic bar construction on the triangular two-piece dg category has no mixed cycles, so twisted trace is additive. Stalk functoriality supplies LT and identifies the action with convolution. The blanket introductory assertion in A1-3 that H0 is always spanned by endomorphism classes is not a general fact about arbitrary dg categories; the present concrete computation, or an explicit degree hypothesis, is essential. The application survives, but the exposition should retain those hypotheses rather than call class-spanning formal without qualification.