S1-4 · 2026-10-21T15:11:19Z
Re-derived split input after removing the parity claim
For \(X=\mathrm{SL}_2^3\backslash(\mathrm{Sp}_4\times\mathrm{Sp}_4)\) and odd \(q\), the independently checked function argument proves failure of \((\mathrm{SG})_n\) for every \(n\). Given (O1)–(O3) of A5-10, the checked geometry yields \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) and
$$m^{(n)}=D_n c D'_n,$$
where \(D_n,D'_n\) are invertible diagonal matrices and \(c\) is the same branching matrix for every \(n\). The conclusion of A5-12 is held with these explicit qualifications; it is not an unconditional split counterexample.
- The function-side proof A5-5 is uniform over non-archimedean local fields of odd residue characteristic and hence applies over every \(\mathbb F_{q^n}((t))\). It does not use A5-11's erroneous parity remark or the unproved categorical formalism.
- Under (O1)–(O3), A5-9's semismall support-and-cosupport argument identifies each relevant multiplicity space with \(H_c^{2f}(F^\lambda_\nu)\). The geometric dimension \(f\), rather than the expected dimension of an irrelevant fibre, is used here. The extension-field counts established in A5-8 and S1-3 give \(A=q^f\operatorname{id}\). All irrelevant pairs have zero multiplicity. This yields the stated diagonal rescaling of the branching matrix.
- Every row of \(c\) is nonzero by the explicit relevance constructions in A5-8, step 6. Thus every simple IC occurs as a direct summand and the assumed recollement/perverse formalism gives thick generation. This uses no claim that one nonzero trace entry implies spanning.
- The failure of column spanning is concrete. In the dual-group coordinates of A3-3, put \(x=(2,3,2)\), \(y=(3,2,3)\). Their images in \(\mathrm{SO}_5^2\) are Weyl conjugate: the torus map is \((a,b,c)\mapsto((ab,a/b),(bc,b/c))\), and inversion of the second coordinate in each \(\mathrm{SO}_5\) factor exchanges these images. Thus evaluation at \(x\) minus evaluation at \(y\) vanishes on every restricted character, i.e. every column of \(c\). On the middle adjoint character \(V_{(0,2,0)}\) it equals
$$3^2+3^{-2}-2^2-2^{-2}=175/36\ne0.$$
This exact choice also satisfies all the corrected genericity exclusions \(a\ne\pm b^{\pm1}\); no genericity assertion or numerical near-zero test is needed. The row corresponding to that character is the label \((d_1,d_2,e)=(0,0,2)\). It is excluded from the column span of \(c\), and diagonal rescaling excludes the corresponding basis vector from the span of every \(m^{(n)}\).
- The explicit restriction at \(\lambda=((1,1),0)\) is the sum of the outer adjoint, middle adjoint, and outer-by-middle standard tensor representations, of dimensions \(3+3+4\). Consequently the IC with label \((0,0,2)\) is a geometric retract. With the inherited Weil structure its trace is a nonzero scalar multiple of the canonical IC trace (the multiplicity eigenvalue is \(q^f\)); literal equality requires rescaling the Weil structure or the inserted endomorphism. Either version is excluded from the Hecke span. This is the precise retract-versus-whole-object obstruction, and no parity argument enters.
- The remaining obligation remains (O1) the positive-characteristic degeneration/local-structure input, (O2) compatibility of the proposed dimension theory, and (O3) the required perverse/decomposition/recollement transfer. None is discharged by these trace corrections.