The agent audited, from scratch, the eleven steps behind the split candidate \(X=\mathrm{SL}_2^3\backslash(\mathrm{Sp}_4\times\mathrm{Sp}_4)\).
On the function side it recomputed the Mackey analysis of a generic principal series of \(\mathrm{Sp}_4\) restricted to \(\mathrm{Sp}_2\times\mathrm{Sp}_2\): four orbits on the flag variety, their stabilizers, the exact characters \(c_1=\chi_1\chi_2^{-1}\) and \(c_2=\chi_1\chi_2\) of the two \(\mathrm{SL}_2\)-quotients, and the Ext arguments, valid in every characteristic (Audit 5.3). A finite-field version of the computation matched in all 40 test cases (Computation 5.4). It then re-derived the two periods on \(\pi\boxtimes\pi\) and their independence: (SG) fails over \(\mathbb F_q(\!(t)\!)\) for every \(q\) and \(n\), with no characteristic-0 input (Audit 5.5).
On the sheaf side it confirmed the orbit labels, the connected stabilizers, the dimension formula, the convolution fibres, semismallness and the branching multiplicities. The checks were by hand and by brute-force lattice counts over \(\mathbb F_q(\!(t)\!)\) for \(q=3,5,7\) (Computation 5.7). It found that the split agent had mislocated the remaining gap: Chen–Yi's placidity theorem does not use the symmetric-space embedding and extends to spherical varieties (their Remark 37). The true remaining input is Chen–Yi's \(\ell\)-adic framework for this non-symmetric \(X\) in odd characteristic, split into three obligations (Objection 5.10). Its verdict: spherical generation fails, proved; sheaf generation holds modulo those obligations (Summary 5.12).
Result as the agent reported it partial
Audit of Lemma 3.1…Remark 3.11 for Sakellaridis's split X = SL_2^3\(Sp_4×Sp_4). (1) Function side PROVED: Lemma 3.1, Lemma 3.4, Theorem 3.5 hold (minor repairs only), so (SG) and every (SG)n fail over F_q((t)) for every q, with no characteristic-0 input; recomputed c_1 = χ_1χ_2^{-1}, c_2 = χ_1χ_2 exactly, and confirmed them by a finite-field Mackey computation (40/40 cases). (2) Geometric sheaf-side inputs PROVED: Proposition 3.6 (orbits (d_1,d_2,e), connected stabilizers, δ = 3d_1+3d_2+e, finite closures), Computation 3.7 (position formula, counts) and Proposition 3.8 (semismallness, top components, c{νλ} = SO_5^2↓ι(SL_2^3) branching, every label relevant). They were re-derived by hand and confirmed by brute-force elementary-divisor counts at q = 3,5,7 (exact mass identities; exact agreement on 4068 and 31900 pairs) and by Freudenthal branching on 5247 pairs. (3) Corollary 3.9 and Proposition 3.2 hold as conditional statements. (4) Dead end 3.10 misdiagnoses the gap: Chen–Yi's placidity theorem (Thm 36) does not use τ, and their Remark 37 extends it to general spherical varieties. (5) Status of 'X is a split counterexample to IId': proved modulo a named input — Chen–Yi's ℓ-adic framework for this non-symmetric X in odd characteristic, i.e. obligations (O1)–(O3).
Gap
(O1) The Sakellaridis–Wang affine degeneration [SW22, §2.2] and [SW22, Lemma 2.3.10] for X in odd characteristic, so that Chen–Yi Thm 36 / Remark 37 gives placid orbit closures; smoothness of the generic B-stabilizer is verified for p = 3,5,7,11, while SW22's characteristic hypotheses are removed, per Chen–Yi's footnote 6, by Bao–Song [BS25], a paper on symmetric spaces whose reach to this X is unverified. (O2) δ(ν) = 3d_1+3d_2+e satisfies Chen–Yi's dimension-theory compatibility (8); Chen–Yi's own Prop. 41 uses τ. (O3) Chen–Yi App. A.5–A.7 and Prop. 70 (with Sun Thm 1.2) then yield H2, PH, D1, D2 for X verbatim. Until then C_k is not defined on the pages for this non-symmetric X.
Blocked routes
- Placidity via the open embedding X ⊂ IG(2, V_1⊕V_2): fails, because the induced topology on LX is not the ind-topology (Dead end 3.10 step 2, correct).
- Dimension theory via Chen–Yi Lemma 40 / Prop. 41: needs τ: X → G with τ^{-1}(L^+G) = L^+X; for X replace it by δ from relative Haar volumes (Audit 5.6) plus a check of Chen–Yi (8).
- Placidity of L^+G-orbit closures from the closed embedding X ⊂ Sp_8/(Sp_6×Sp_2): needs an argument for the smaller group L^+G ⊊ L^+Sp_8; not immediate.
- Deducing the categorical statement over F̄_q from a Betti statement over C (a D5-type comparison): gives only p outside an unspecified finite set.
Brief as sent
Audit, independently and by recomputation, A3-1 to A3-11 (verbatim in scripts/R/A5-packet.md): the multiplicity criterion; the model of X and the fusion; the Mackey analysis of I(χ)|_{Sp_2×Sp_2} (orbits, Ext formula, lifting, genericity, characteristic-freeness); the two H(F)-periods on π⊠π and their independence; the orbit labels, stabilizers, dimensions, closures and counts on LX; semismallness, relevance and the branching multiplicities; whether placidity is the only gap on the sheaf side. Return check/objection nodes and a verdict on the status of the split counterexample.