C-V3 · 2026-10-21T15:11:55Z
Re-check by G3
S3-1 (holds): Projection identifies the fixed lattice with ker(vu-1) in one rank-r summand, proving the sufficient upper bound. The four-cycle GL2 torus explicitly refutes the old equality. The Galois action is correctly distinguished from the Frobenius isogeny scalar.
S3-2 (holds): The graph fixed group and diagonal theta-split torus give the connected centralizer geometrically; the Fprime=theta F Lang conjugation produces a rational stable Borel pair. The corrected split-rank bound excludes a split maximal torus. The purity qualification is retained and the group-case finite-level geometry supplies the stated D-inputs.
S3-3 (holds): Each geometric IC comes from (c,0), whereas odd-degree rational Cartan labels have distinct doubled tops 2a. A maximal leading label in a finite combination cannot cancel, so non-even singleton labels are absent. Adding an even regular dominant coroot makes a non-even lattice element dominant without changing parity. Splitting in even degree restores all (c,0) operators. Thus A4-12 is rederived with precisely its outside-(R) scope.
S1-2 (holds): Checked the precise primary locators in Zhu: Lemma 2.1.14 and Example 2.1.16 identify the projective conic cone; Example 2.1.8 and Lemma 2.3.12 give the nilpotent-cone open embedding, translated by t^-1 I to the SL2 component. For p!=2 the local cone is A2/{+/-1}; finite pushforward, the invariant summand and Verdier duality show that the constant shifted sheaf is the intermediate extension. Thus rational smoothness and trace(IC_1)=1_1+1_0 follow without the old false global affine identification. This fully repairs the A4-8 error.