Solve · 2026-10-17 · 8 agents + root · 2 h 13 min wall clock · second audit 1 h 06 min · repairs 2 h 11 min

What each agent did

• Agent 1 · Literature and Table 1 audit
  Rebuilt the pages' status table row by row, found no divergence among decided rows, and settled \(\mathrm{SL}_2/\mathrm{SO}_2\) (both fail) and \(\mathrm{GL}_4/\mathrm{Sp}_4\) (both hold).
• Agent 2 · SL₂/SO₂: central inertia
  Decided \(\mathrm{SL}_2/\mathrm{SO}_2\) at sheaf level: the central element \(-I\) splits the category, Satake objects miss the odd part, so both levels fail.
• Agent 3 · Symplectic family: fibres
  Computed convolution fibres for \(\mathrm{GL}_4/\mathrm{Sp}_4\), showing the uncovered orbit's IC sheaf splits off a Satake object, so both levels hold there.
• Agent 4 · Sheaf–function dictionary
  Built the Frobenius-trace dictionary and proved that function generation over \(\mathbb F_q(\!(t)\!)\) forces sheaf generation; the converse does not follow.
• Agent 5 · Functions without (A1)–(A5)
  Proved spherical generation by lattice counting, without Sakellaridis's hypotheses, over every local field for spheres, \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\).
• Agent 6 · E₆/F₄: restriction map
  Computed Sakellaridis's restriction map: onto for the spheres, not onto for \(E_6/F_4\), so spherical generation fails there under his hypotheses over \(p\)-adic fields.
• Agent 7 · Adversarial audit
  Audited every candidate proof during the run, repaired the \(E_6/F_4\) fibre argument, and confirmed \(E_6/F_4\) as the one divergence, with field caveats.
• Agent 8 · E₆/F₄: sheaf level
  Proved that for \(E_6/F_4\) every orbit's IC sheaf splits off a Satake object, so the categorical statement holds although some orbits have no dominant lift.
• Root · Registry and synthesis
  Coordinated the agents and wrote the answer: every symmetric space agrees at both levels except \(E_6/F_4\), where sheaves hold and functions fail across fields.
• Audit · Second audit, then re-checks
  A second audit re-checked all 110 steps and found 15 errors; after the repairs by Agents X1–X4, fresh reviewers re-checked every repair, and none failed.
• Agent X1 · Symplectic counts
  Re-proved the \(\mathrm{GL}_4/\mathrm{Sp}_4\) and \(\mathrm{GL}_6/\mathrm{Sp}_6\) Hecke counts as polynomials in \(q\) and the orbit codimensions in every component; every symplectic statement of the verdict survives.
• Agent X2 · SL₂/SO₂ traces
  Repaired the \(\mathrm{SL}_2/\mathrm{SO}_2\) trace argument with the inertia lemma and showed that Chen–Yi §8.2 covers the \(\mathrm{SO}_2\)-form only for \(q\equiv1\bmod4\); both levels still fail, now in both forms.
• Agent X3 · Pair model at 2
  Proved the support bound for \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) in every residue characteristic without the Cartan embedding; at residue characteristic \(2\) the verdict must name the pair model.
• Agent X4 · Sakellaridis's theorem
  Stated Sakellaridis's theorem with every hypothesis, verified all but two of them for \(E_6/F_4\), and showed that \(C_K\) is not a cyclic Hecke module: it needs exactly two generators.