§5 · Agent 5 · Audit: split candidate · 2026-10-17T23:25:08Z

Summary 5.12 · Verdict: (SG) fails, proved; \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) modulo (O1)–(O3)

open ★ key step

All function-level claims survive. Every geometric and combinatorial sheaf-level input re-derives and is confirmed by independent computation. 'Sakellaridis's \(X\) is a split counterexample to IId' is proved modulo one named input: Chen–Yi's \(\ell\)-adic framework for this non-symmetric \(X\) in odd characteristic, i.e. obligations (O1)–(O3) of Objection 5.10. It is not established unconditionally.

Summary 5.12.

Claims that hold.

Repairs needed.

Status of "Sakellaridis's \(X\) is a split counterexample to IId": proved modulo a named input. The function half is proved. The categorical half, \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) over \(\overline{\mathbb F}_q\), follows from the verified geometry once Chen–Yi's formalism (Thm 36 via Remark 37, a dimension theory, App. A, Prop. 70) is available for this \(X\). Concretely, that means (O1) [SW22, §2.2 and Lemma 2.3.10] in odd characteristic, (O2) the dimension theory \(\delta=3d_1+3d_2+e\), and (O3) the verbatim transfer.

Hypotheses satisfied by the example (analogues of (R), (C), D0–D3).

Evidence produced by this audit.

  1. A hand re-derivation of every step. Two statements are false as written, A3-3(d) at \(a=-b^{\pm1}\) and A3-10's reason; neither is used in the function-level proof.
  2. A finite-field Mackey test of A3-4, which matched in 40/40 cases (Computation 5.4).
  3. Brute-force lattice counts over \(\mathbb F_q(\!(t)\!)\) for \(q=3,5,7\), with exact agreement with the position formula and exact mass identities (Computation 5.7).
  4. Freudenthal-based branching, which equals the leading coefficients on 5247 pairs (Audit 5.8).
  5. Smoothness of the generic \(B\)-stabilizer for \(p\le11\) (Objection 5.10).

Mechanism (for question (2)). In the conditional picture, \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) because \(c\) is the branching matrix of a subgroup (every row is nonzero). (SG) fails because fusion makes the restriction of characters non-surjective (the columns do not span). The trace of Frobenius sees \(V_\lambda\ast\mathrm{IC}_0\) only with \(e=\mathrm{id}\); the retracts that thick closure adds are exactly what is lost.

Builds on

  • Audit 5.1 · Audit of A3-1: (SG) forces \(\dim\operatorname{Hom}_{H(F)}(\pi,\mathbb C)\le 1\)
  • Audit 5.2 · Audit of A3-3: the model of X, its stabilizer, the dual group and fusion
  • Audit 5.3 · Audit of A3-4: the Mackey filtration and the two SL₂-quotients of I(χ)
  • Computation 5.4 · Finite-field test: Hom from Ind of Sp₄(F₁₁) to 1⊠σ_c over SL₂(F₁₁)²
  • Audit 5.5 · Audit of A3-5: two periods on \(\pi\boxtimes\pi\), so \(\mathcal H_K\ast\varphi_0\ne C_K\)
  • Audit 5.6 · Audit of A3-6: orbits \((d_1,d_2,e)\), connected stabilizers, \(\delta=3d_1+3d_2+e\)
  • Computation 5.7 · Brute-force counts of \((T_\lambda\ast\varphi_0)(x_\nu)\) over \(\mathbb F_q(\!(t)\!)\)
  • Audit 5.8 · Audit of A3-7, A3-8: position formula, semismallness, dual branching
  • Audit 5.9 · Audit of A3-9, A3-2(c): given D1–D2, \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\)
  • Objection 5.10 · Objection to A3-10: placidity needs no τ (Chen–Yi Remark 37); the gap is elsewhere
  • Audit 5.11 · Audit of A3-11, A3-2: no n-dependence; retracts reach the trace, Hecke does not

Built on by

  • Theorem 6.3 · Verdict: sheaf generation over \(\overline{\mathbb F}_q\) does not imply spherical generation over \(\mathbb F_q(\!(t)\!)\)

Sources