S1-5 · 2026-10-21T15:11:19Z

Re-derived nonsplit and ordinary-trace inputs

The nonsplit counterexample and ordinary function–sheaf/multiplicity-matrix conclusions used in A1-12 and A4-12 are held after S1-1 and S1-2. In particular the unitary twist of the group case for any nontrivial split connected reductive \(H\) has geometric Satake generation and fails \((\mathrm{SG})_n\) for odd \(n\), while \((\mathrm{SG})_n\) holds for even \(n\). This node makes no independent assertion about the unrestricted categorical Hochschild-trace presentation of A1-3; its separate repair must be used in the final synthesis.

  1. The geometric identification \(X_k\simeq H_k\) turns the action into the group case, and the checked convolution isomorphism \(V_{(c,0)}*\mathrm{IC}_0\simeq\mathrm{IC}_c\) gives every simple. This is A4-2's group-case proof and does not involve either of the false assertions repaired above.
  2. Rational Hecke labels in the unitary model are \((a,a)\). The independently checked support statement A4-4 gives a nonzero leading coefficient on label \(2a\) and support below it. The top map \(a\mapsto2a\) is injective. In a finite nonzero linear combination, a maximal chosen \(a\) has a surviving coefficient on \(2a\); removing maximal terms successively proves that an orbit indicator with label \(c\notin2\Lambda_H\) cannot lie in the Hecke-generated span. Such a dominant label exists for every nontrivial \(H\): a primitive nonzero dominant lattice vector is not twice a lattice vector. Thus \((\mathrm{SG})\) fails. For even extensions the group splits and the group-case Hecke module is cyclic; odd extensions retain the same unitary model.
  3. For \(H=\mathrm{SL}_2\), geometric Satake and Clebsch–Gordan for \(\check H=\mathrm{PGL}_2\) give one-dimensional multiplicity spaces for all \(0\le c\le2a\) in \(V_a\star V_a\). Frobenius preserves each such line and is invertible, so every such coefficient is nonzero; nevertheless the even leading labels give a proper column span. The identity \(\operatorname{tr}\mathrm{IC}_1=\mathbf1_1+\mathbf1_0\) follows from S1-2, so the IC-vector and orbit-indicator obstruction agree. The audited transitivity statement for this example does not use the cone computation.
  4. For ordinary traces under the matrix dictionary, thick generation records occurrence of simple summands, whereas \((\mathrm{SG})_n\) records column spanning of the numerical matrix \(m^{(n)}\). The quotient-Weil-ring information loss survives S1-1. Neither occurrence nor the quotient ring forces column spanning; no statement about the complete coefficient is inferred solely from a permutation of fibre components.
  5. Under the split hypotheses (R), (C), the nonsplit label-twisting mechanism is absent. The remaining question remains the spanning of the projected columns on labels without a dominant lift, as in A1-8 and A1-11. Removing the unjustified top-component assertion produces no new proof of this spanning and no new counterexample. The categorical-trace portion of A1-12 must use the independently repaired version of A1-3, rather than be certified by this cluster.

Builds on: S1-1 · S1-2 · A1-12 · A4-12 · A4-2 · A4-4 · A1-8 · A1-11