C-V1 · 2026-10-21T15:12:12Z
Re-check by G1
S2-1 (holds): The two conjugate lines give the required counterexample, including the connected torus action and rational fixed intersection point. Uniqueness of the geometric component through a rational point is a valid sufficient criterion. The repair preserves the blocked status and correctly separates full IC multiplicity spaces from arbitrary fibre cohomology. No new split counterexample is claimed.
S2-2 (holds): The full-torus radial tree recurrence proves (SG) independently of every categorical transfer. For the norm-one torus, unit norm surjectivity gives the rational two-orbit obstruction, while the explicitly calculated stabilizer components and central -I sign local system give the geometric obstruction in the same odd characteristic. The deleted full-torus categorical claim is unnecessary. Finite orbit-support of the sign extension is compatible with the degree-two finite etale map GL2/SO2→GL2/O2 to the symmetric quotient.
S2-4 (holds): The dominant-lift proof closes the earlier ineffective-factor reduction sketch. Injectivity forces the connected theta-fixed torus into the action kernel; connected T^theta gives integral surjectivity of 1-theta on cocharacters. The reduced connected action kernel is normal reductive, and its Weyl group can make a lift dominant inside its ineffective root factors without changing the other simple-root pairings or its image in S. The proof that S centralizes the kernel uses theta-fixedness and surjectivity of squaring on S, valid in the stated setting. The conclusion remains conditional on the original triangular/orbit inputs.
S2-5 (holds): The affected scope statements follow from S2-1/S2-2 and S2-4. The cyclicity convention is now explicit, label twisting is distinguished from multiplicity-space Frobenius, and the no-induced-Weil-structure sentence is restricted to the designated non-even IC target. The split nonsymmetric categorical half and the independent categorical-trace cluster retain their stated obligations. This check does not certify branches delegated to G2/G3.
S1-1 (holds): Excision below the actual top dimension and Poincare duality give H_c^{2d} as the top-component permutation representation with Tate twist -d. The intersecting conjugate-lines example has H^0 trace 1 and H^2 trace 0 at the first power, so it correctly disproves the inference to a complete trace. The repair now requires a separately proved multiplicity/top-cohomology identification and absence of other perverse degrees before computing a full coefficient. The quotient-Weil-ring information-loss argument is preserved.
S1-5 (holds): The scoped rederivation uses the independently checked group-case generation and injective rational top-label map a→2a. Non-even dominant labels exist in every nonzero reductive cocharacter lattice. The odd/even extension distinction and multiplicity-one PGL2 tensor-square argument do not use the erroneous component or global-cone assertions. The IC/indicator comparison uses the separately checked local-cone repair. The full-(R) question and the categorical Hochschild claim remain explicitly outside this cluster.