The agent audited, by recomputation and without the other agents' opinions, the eleven steps behind the non-split counterexample and then the seven steps behind the mechanism.
For the counterexample it rechecked the \(\mathbb F_q\)-structures of the unitary and Galois twists and proved the one clause asserted without proof, an \(\mathbb F_q\)-rational \(\theta\)-stable Borel pair (Audit 4.1). It recomputed the convolution over \(k\), the classification of \(K\)-orbits and the leading-term lemma, and re-derived the theorem for a torus, \(\mathrm{SL}_2\), \(\mathrm{PGL}_2\) and \(\mathrm{GL}_n\) (Audit 4.5). An independent coset enumeration reproduced every explicit Hecke count for \(q=3\) and \(5\) (Computation 4.6). None of the eleven steps failed; six needed one-line repairs.
For the mechanism it confirmed Theorem A (rows against columns of one matrix) and Theorem T (the twisted Hochschild homology of \(\mathcal C_k\) is \(C_K\), and the Satake image consists of whole classes). It also found three gaps of rigour in the latter: the formality argument must come from free even cohomology with scalar Frobenius, additivity along the recollement is asserted rather than proved, and the trace map must be defined by functoriality (Audit 4.15). It proved that \(C_K\) is a finitely generated Hecke module for every unitary twist, so the counterexample's failure is witnessed by a Hecke eigenfunctional vanishing at \(\varphi_0\) (Audit 4.16). Finally it confirmed that inside (R) and (C) the injective-leading-term criterion never produces a failure (Audit 4.19): the remaining obligation there is a spanning condition on the labels without dominant lifts.
Result as the agent reported it proved
Two audits, no failures found. (1) Counterexample claims (Lemma 2.2..A2-11, Counterexample 1.5): all eleven steps survive independent re-derivation (five as stated, six with minor repaired defects). For every split connected reductive H ≠ 1 over F_q (q odd), the unitary twist X_u = Res_{F_{q^2}/F_q} H / (Res H)^{ϑσ} (hermitian matrices under Res GL_n; hermitian SL_2 of determinant 1, one G(F)-orbit; the norm torus) has C_k^Sat = C_k over k (V_{(c,0)} * IC_0 ≅ IC_c, connected stabilizers) and fails (SG) over F_q((t)) (rational Hecke labels (a,a) have leading orbit 2a, so 1_{O_c} is not generated for dominant c ∉ 2Λ_H); (SG)n fails for odd n, holds for even n; it satisfies (C), D0, D1, D3, D2 (purity for Frobenius-stable λ), has an F_q-rational θ-stable Borel pair, and violates (R) only through non-split T, T_0; exact recomputation (q = 3, 5) confirms all Hecke counts. (2) Mechanism claims (Setup 1.1, Theorem 1.2, Theorem 1.3, Proposition 1.6, Lemma 1.7, Proposition 1.8, Remark 2.12): all seven hold (four as stated, three with minor repairable defects): under (R), (C), D0–D3, C_k^Sat = C_k ⟺ no row of the R-matrix M is zero, (SG)_n ⟺ coker(M) ⊗{ev_n} Q̄ℓ = 0, retract-free generation ⟺ coker(M) ⊗_ε Z = 0 (Thomason); twisted HH(C_k, Fr) ≅ C_K in degree 0 with all retract classes [IC_ν, can], while the Satake image consists of full classes; inside (R) and (C), X(F_n) = G(F_n)x_0, top entries m{φ(λ),λ} ≠ 0, the injective-leading-term criterion never fails, and the remaining obligation is spanning of p_U(T_λ * φ_0) on uncovered labels. Added: C_K is finitely generated over H_K for every unitary twist, so criterion (S) applies there.
Gap
None for IId(1). Open: IId(3), a counterexample or proof inside (R) and (C), which reduces to the spanning condition on uncovered labels (Proposition 1.8(e)); not settled by any audited claim. Repairs to make: Lang–Steinberg Borel pair (Setup 2.4/A2-6); purity wording (Setup 2.4/A2-5); one-line fix in Computation 2.9 step 3; φ1φ2 = 1 caveat (Counterexample 1.5 step 6); C→F̄_q transfer in Dead end 2.11(1); Theorem 1.3: correct formality argument, proof or citation of twisted additivity along recollements, LT via functoriality; Proposition 1.8(c) scope; Remark 2.12 item 4 reduction and item 5 convention.
Blocked routes
- A counterexample inside (R) by the unitary-twist mechanism: impossible, since (R) gives a split maximal torus, so σ = 1 on Λ_T and every Satake object V_λ is Frobenius-stable; under (R) and (C) the injective-leading-term criterion cannot fail either (Audit 4.19). A counterexample there needs a lemma producing non-spanning of p_U(T_λ * φ_0) on uncovered labels, e.g. cancellation of Frobenius traces on multiplicity spaces with all rows geometrically nonzero.
- Transfer of a p-adic failure (Sakellaridis-type) to F_q((t)) through the close-field lemma Lemma 1.7: blocked unless the failure and (H-Cartan) are proved at p-adic places with residue field F_q and unbounded ramification; Sakellaridis's Theorem 8.0.2 needs (A1)–(A5), with (A3) known only at almost all places of a global field.
- Detecting failure inside (R) and (C) by a Hecke eigenfunctional vanishing at φ_0 (criterion (S) of Proposition 1.6): the converse direction needs finite generation of C_K over H_K in characteristic p, proved here only for unitary twists (Audit 4.16).
Brief as sent
Audit, independently and by recomputation, the eleven steps A2-2 to A2-11 and A1-5 (verbatim in scripts/R/A4-packet.md): F_q-structures and hypotheses; the categorical side over k (group case, IC_0, thick generation); the function side over F_q((t)) itself (K-orbits, Hecke basis, support bound, top value, leading-term lemma), tested on G_m, GL_2, SL_2 for small q; G(F)-orbits; (SG)_n by parity of n; the rows of m^(1) for unitary SL_2 and the GL_2 column; the Galois twist; the torus criterion; hidden hypotheses. Return check/objection nodes naming the audited step and the outcome. Second task (2026-10-17T22:44Z, same agent continued): audit A1-1, A1-2, A1-3, A1-6, A1-7, A1-8 and A2-12 (verbatim in scripts/R/A4-packet2.md): the R-matrix theorem, twisted Hochschild homology and the trace map, the forcing criteria, close-field transfer, the reduction under (R)+(C).