§4 · Agent 4 · Audit: counterexample and mechanism · 2026-10-17T23:02:04Z

Summary 4.20 · Verdict of the second audit: the mechanism claims hold, with minor repairs

proved ★ key step

None of the seven mechanism steps fails. Setup 1.1, Theorem 1.2, Proposition 1.6 and Lemma 1.7 hold as stated; Theorem 1.3, Proposition 1.8 and Remark 2.12 hold with minor, repairable defects. The audit confirms the decategorification statement (rows versus columns of one \(R\)-matrix; twisted HH \(=C_K\) with retract classes invisible to the Satake image), and that inside (R) and (C) the remaining obligation is spanning on the uncovered labels.

Summary 4.20.

  1. Hold as stated: Setup 1.1 (Weil \(K\)-groups over \(R\)); Theorem 1.2 (Theorem A: \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\iff\) no zero row of \(M\); \((\mathrm{SG})_n\iff\operatorname{coker}M\otimes_{\mathrm{ev}_n}\overline{\mathbb Q}_\ell=0\); retract-free generation \(\iff\operatorname{coker}M\otimes_\varepsilon\mathbb Z=0\)); Proposition 1.6 (forcing criteria, with the finite-generation proviso it states); Lemma 1.7 (close-field transfer, conditional on (H-int) and (H-Cartan), as stated).
  2. Hold with minor defects: Theorem 1.3 (formality citation mismatched; twisted additivity asserted, not proved; LT needs functoriality; literature in step 7 uncheckable but unused); Proposition 1.8 (item (c) proved only for automorphism-induced autoequivalences); Remark 2.12 (reduction sketch for trivially acting simple factors; opposite convention for Weil classes; one heuristic phrase).
  3. Added here: \(C_K\) is finitely generated over \(\mathcal H_K\) for the unitary twist of every split \(H\) (Audit 4.16). So, by Proposition 1.6(S), its failure of (SG) is witnessed by a Hecke eigenfunctional vanishing at \(\varphi_0\).
  4. Consequence for the first audit: the remark of Lemma 2.2 that depended on Remark 2.12 is now confirmed (Audit 4.19). The scope statement of Summary 4.12 stands: inside (R) and (C), a counterexample must sit on an uncovered label, through cancellation in the \(U\)-projected system (Audit 4.18(e)).
  1. Checks performed. Simples and \(K_0\) of Weil categories (Audit 4.13). The weight argument and Thomason's classification in Theorem A (Audit 4.14). The Koszul computation of twisted HH, the recollement, LT, Morita, cyclicity and Barr–Beck (Audit 4.15). Criteria (S), (d), (b), (c), plus a finite-generation lemma (Audit 4.16). The transfer counts and finiteness of support via \(\tau\) and (H-Cartan) (Audit 4.17). Connectedness and splitness of \(T^\theta\), \(S[2]\subseteq T^\theta\), the KL sign and positivity, and the basis/projection criterion (Audit 4.18). The (R),(C) argument for dominant lifts (Audit 4.19).
  2. Citations verified in local copies: Laumon (1.1.1.1)–(1.1.1.4) and the paragraph before Théorème (1.1.2); BBD Cor. 5.4.4 (via the local reviewed excerpt); Chen–Yi §8.2.1, (41), Thm 93; Zhu Lemma 5.6.1 and (5.6.2); Lurie HA 4.7.3.5 and 7.1.2.1; Ponto–Shulman Prop. 2.4, Prop. 6.2, Ex. 6.3; Sakellaridis Cor. 8.0.3, 8.0.4; verdict.html Theorem 4 / Lemma 4.8. Not local but published and checkable: Thomason 1997 Thm 2.1; Keller 1998; Kuznetsov 2009 (untwisted). Flagged as uncheckable as cited (no result numbers, not local), and unused: AGKRRV, Gaitsgory–Varshavsky, Zhu on categorical traces, Ben-Zvi–Francis–Nadler, Ben-Zvi–Nadler (Theorem 1.3 step 7); Remark 2.12 item 5 is labeled heuristic.
  3. Repairs to make.
    • Theorem 1.3 step 2: replace "as used by Chen–Yi, Thm 93" by the free-cohomology argument: \(\operatorname{Sym}V\to C^*(BL_\mu)\) is a quasi-isomorphism of commutative algebras, Frobenius-equivariant up to one homotopy since Frobenius is scalar on \(H^{2d}\).
    • Theorem 1.3 step 3: prove or cite additivity of HH twisted by an endofunctor preserving both pieces of a semi-orthogonal decomposition.
    • Theorem 1.3 step 1: define LT by functoriality along the Frobenius-equivariant stalk functors.
    • Proposition 1.8(c): restrict the claim to \(\sigma\) induced by automorphisms of \((G,\theta)\).
    • Remark 2.12 item 4: carry out the quotient by trivially acting simple factors.
    • Remark 2.12 item 5: state the convention \(\beta:M\to\mathrm{Fr}^*M\).
  4. What the mechanism says (team question (2)). Under (R), (C), D0–D3, sheaf generation is the support pattern of \(M\) (retracts), while (SG) is the column span of \(\mathrm{ev}_1(M)\). In twisted HH, \(\mathrm{HH}_0(\mathcal C_k,\mathrm{Fr})\cong C_K\) contains every retract class \([\mathrm{IC}_\nu,\mathrm{can}]\), but the image of \(\mathrm{HH}_0(\mathcal H,\mathrm{Fr})\) consists only of the full Satake classes. Outside (R), non-rational Satake objects carry no Weil structure (Audit 4.8), which produces the unitary-twist counterexample (Audit 4.5). Inside (R) and (C) the twisting mechanisms vanish (Audit 4.18), and so does the injective-leading-term mechanism (Audit 4.19). What remains is spanning of \(p_U(T_\lambda\ast\varphi_0)\) on the uncovered labels. That is the open obligation of IId(3); none of the audited steps settles it.

Builds on

  • Audit 4.13 · Audit: Weil Grothendieck groups over \(R\) and the multiplicity matrix \(M\)
  • Audit 4.14 · Audit: Theorem A, rows versus columns of the \(R\)-matrix \(M\)
  • Audit 4.15 · Audit: Theorem T, twisted HH of \(\mathcal C_k\) is \(C_K\), and retract classes
  • Audit 4.16 · Audit: vanishing eigenfunctionals, multiplicity one, extra endomorphisms, row relations
  • Audit 4.17 · Audit: the close-field transfer of a (SG) witness to highly ramified \(p\)-adic fields
  • Audit 4.18 · Audit: under (R), (C), D0–D3 the twisting mechanisms vanish
  • Audit 4.19 · Audit: under (R), (C), injective leading terms force dominant lifts

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