§4 · Agent 4 · Audit: counterexample and mechanism · 2026-10-17T22:15:32Z

Summary 4.12 · Verdict of the audit: the unitary twist is a valid counterexample; the repairs are minor

proved ★ key step

None of the eleven audited steps fails. Five hold as stated, six hold with minor, repaired defects. The unitary twist of the group case (hermitian \(\mathrm{SL}_2\), hermitian \(\mathrm{GL}_n\), the norm torus) has \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) and fails (SG) over \(\mathbb F_q(\!(t)\!)\), proved over \(\mathbb F_q(\!(t)\!)\) itself. It lies outside (R), so IId(3) is untouched.

Summary 4.12.

  1. Hold as stated: Counterexample 2.3, Theorem 2.6, Computation 2.7, Lemma 2.8, Proposition 2.10.
  2. Hold with a minor defect, repaired here: Lemma 2.2 (its first remark depends on the unaudited Remark 2.12); Setup 2.4 (rational \(\theta\)-stable Borel pair asserted, not proved; D2 unqualified); Lemma 2.5 ("pure" needs a Frobenius-stable \(\lambda\)); Computation 2.9 (single-vector equivalence in step 3); Dead end 2.11 (a \(\mathbb C\)-to-\(\overline{\mathbb F}_q\) transfer in a side remark); Counterexample 1.5 (the Weil-summand sentence of step 6 needs \(\phi_1\phi_2=1\); theorem and example are correct).
  3. Established: for every split connected reductive \(H\ne1\) over \(\mathbb F_q\), \(q\) odd, the symmetric space \(X_u=\operatorname{Res}_{\mathbb F_{q^2}/\mathbb F_q}H\big/(\operatorname{Res}H)^{\vartheta\sigma}\) satisfies \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\), and (SG) fails over \(F=\mathbb F_q(\!(t)\!)\): \(\mathbf 1_{O_c}\notin\mathcal H_K\ast\varphi_0\) for every dominant \(c\notin2\Lambda_H\). \((\mathrm{SG})_n\) fails for odd \(n\) and holds for even \(n\). The example satisfies (C), D0, D1, D3, D2 (purity for Frobenius-stable \(\lambda\)), and every clause of (R) except "\(T\) split" and "\(T_0\) split". For \(H=\mathrm{SL}_2\), \(X(F)\) is a single \(G(F)\)-orbit, so the failure is not an orbit-count artefact. Mechanism: the rational Hecke labels are \((\Lambda_T^+)^\sigma=\{(a,a)\}\), whose leading orbits \(\varphi(a,a)=2a\) miss the Frobenius-stable labels \(c\notin2\Lambda_H\), while sheaf generation uses the non-rational \(V_{(c,0)}\).
  1. Checks performed (Audit 4.1–Audit 4.11): the \(\mathbb F_q\)-structures and the hypothesis profile; the convolution identification over \(k\); the \(K\)-orbit classification (proof via Lang and connected \(J_c\), plus a finite truncated check); the leading-term lemma; the main theorem for every split \(H\ne1\), including a torus, \(\mathrm{SL}_2\), \(\mathrm{PGL}_2\), \(\mathrm{GL}_n\); exact recomputation of all hermitian Hecke counts for \(q=3,5\) and a volume identity; the \(G(F)\)-orbit statements; the trace-side statements and the explicit \(m^{(1)}\) columns; the Galois twist; the \(\mathrm{GL}_2\) non-split-torus dead end; the torus criterion.
  2. Repairs to make in the write-up.
    • Setup 2.4(6), Theorem 2.6(3): add the Lang–Steinberg construction of an \(\mathbb F_q\)-rational \(\theta_u\)-stable Borel pair \(B_0=\mathrm{Fr}\,B_2\times B_2\), \(B_2\) stable under \(\vartheta\circ\mathrm{Fr}\) (Audit 4.1, item 5).
    • Setup 2.4(6), Lemma 2.5(2): read "pure" for \(\sigma\lambda=\lambda\) over \(\mathbb F_q\), or over \(\mathbb F_{q^2}\).
    • Computation 2.9 step 3: add \(\operatorname{tr}\mathrm{IC}_{\alpha^\vee}=\mathbf 1_{O_{\alpha^\vee}}+\varphi_0\), so that \(e_1\notin\) span \(\iff\mathbf 1_{O_{\alpha^\vee}}\notin\mathcal H_K\ast\varphi_0\).
    • Counterexample 1.5 step 6: "the canonical \(\mathrm{IC}_1\) is a Weil summand" only when \(\phi_1\phi_2=1\).
    • Dead end 2.11(1): either drop "\(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\)" for \(\mathrm{GL}_2/T_E\) or justify it over \(\overline{\mathbb F}_q\) (Proposition III with Chen–Yi's D0–D2); the dead end does not depend on it.
    • Lemma 2.2: its first remark ("never yields a failure under (R) and (C)") stands or falls with Remark 2.12, which was not in this audit.
  3. No hidden transfer in the main line. The function-level failure is proved over \(\mathbb F_q(\!(t)\!)\) by Cartan decompositions, Lang's theorem and an elementary leading-term count; the Hironaka remark of Theorem 2.6 is labeled heuristic and unused; no Sakellaridis theorem, no characteristic-\(0\) input, no unproved geometric Langlands statement is used. The categorical side is proved over \(k\) directly (connected stabilizers and \(V_{(c,0)}\ast\mathrm{IC}_0=\mathrm{IC}_c\)).
  4. Scope. Under (R), \(G\) has a split maximal torus, so \(\sigma\) is trivial on \(\Lambda_T\), every \(V_\lambda\) is rational, and the mechanism above is absent. A counterexample inside (R) and (C) would need either cancellation of Frobenius traces on multiplicity spaces (rows nonzero geometrically, columns not spanning) or non-injective leading terms. That is the remaining obligation of IId(3), not addressed by the audited claims.

Builds on

  • Audit 4.1 · Audit: the \(\mathbb F_q\)-structures of the unitary and Galois twists
  • Audit 4.2 · Audit: \(V_{(c,0)}\ast\mathrm{IC}_0\cong\mathrm{IC}_c\); sheaves generate for all forms
  • Audit 4.3 · Audit: \(K\)-orbits on \(X(F)\) for the twists, one per Frobenius-stable orbit label
  • Audit 4.4 · Audit: the orbit obstruction and the injective leading-term lemma
  • Audit 4.5 · Audit: the unitary twist fails (SG) over \(\mathbb F_q(\!(t)\!)\) for split \(H\ne1\)
  • Computation 4.6 · Computation: hermitian Hecke counts \(T_\lambda\ast\varphi_0\) for \(q=3,5\)
  • Audit 4.7 · Audit: \(G(F)\)-orbits of the hermitian spaces, and failure on the base orbit
  • Audit 4.8 · Audit: Weil structures, rational columns and \(m^{(1)}\) for unitary twists
  • Audit 4.9 · Audit: \(\mathrm{GL}_n(E)/\mathrm{GL}_n(F)\) satisfies (SG) over \(\mathbb F_q(\!(t)\!)\)
  • Audit 4.10 · Audit: split \(\mathrm{GL}_2\) over a non-split torus gives no counterexample
  • Audit 4.11 · Audit: the torus criterion and the norm-torus counterexample

Built on by

  • Remark 6.2 · Registry after the split agent and the first audit: audit the split candidate and the mechanism
  • Theorem 6.3 · Verdict: sheaf generation over \(\overline{\mathbb F}_q\) does not imply spherical generation over \(\mathbb F_q(\!(t)\!)\)

Sources