C-V2 · 2026-10-21T15:12:39Z
Re-check by G2
S1-3 (holds): Holds. The three-cycle counterexample and the distinction between one multiplicity summand, an entry, and an entire row remove the false parity inference. For the point-count assertion, Lang-Weil makes the normalized leading term the number of fixed f-dimensional components up to a vanishing error; periodicity and arbitrarily large n congruent to 0 and 1 modulo the permutation order force exactly c components, all fixed. The c>0 hypothesis excludes the irrelevant-fibre case. Multiplicity identification remains a separate conditional support argument.
S1-4 (holds): Holds with its explicit conditional scope. A5-5 proves the all-extension function failure independently. Given O1-O3, expected-dimensional H_c multiplicities and S1-3 give the same branching matrix up to invertible row/column scaling. At (2,3,2) and (3,2,3) ambient Weyl conjugacy is explicit, and the middle adjoint difference is exactly 175/36. Since a diagonal row scaling only rescales a basis vector, that vector stays outside every numerical column span. The inherited Weil retract trace is correctly stated only up to nonzero scalar. No unsupported parity or unconditional geometric transfer remains.
S2-3 (holds): Holds. Semisimple conjugacy in SL2^3 modulo simultaneous sign gives precisely a=+/-b or a=+/-b^{-1}; equality of quotient torus points requires a=+/-b. Inverting each ambient second coordinate proves fusion, and the explicit inverse-sign example indeed gives conjugate but unequal points. Factoring the character difference through a^2 and b^2 verifies its vanishing exactly on the four exceptional loci. This fully repairs the A5-2 equality/conjugacy defect without changing generic fusion.