The agent audited the two general claims, the Iwahori summand criterion in two independent versions and the functions ⇒ sheaves bridge in two versions, step by step against the checklist. The gluing of perverse hearts along the levels and the thickness of the constituent subcategories hold over \(\overline{\mathbb F}_q\) and over \(\mathbb C\) ( (Audit 7.1)). So do the generators and the criterion ( (Audit 7.2)); the only repairs are pro-unipotent level groups and counting local systems at every level.
Its main objection is about the decomposition theorem ( (Counterexample 7.3)). A non-equivariant splitting does not suffice: for \(\mathbb G_m\) acting on a point, the cone of a generator of \(H^2(B\mathbb G_m)\) has simple perverse cohomology but generates nothing. The splitting must come from the stack (equivariant) decomposition theorem, which applies to the finite-type models. It also showed by counterexample that full faithfulness on derived categories must be assumed of the transitions, not deduced ( (Audit 7.4)).
For the bridge, the Weil heart, the Lang bases and the trace of convolution hold ( (Audit 7.6)). So does the implication from (IG) to every simple object being a summand ( (Audit 7.7)), as does the variant with disconnected stabilizers ( (Audit 7.8)). Nothing in the bridge reaches \(\mathbb C\). Its verdict ( (Summary 7.11)) is the form of Propositions I and II used on the verdict page.
Result as the agent reported it proved
Every audited step survives, some after minor repairs. (I_I) holds over F_q-bar and over C (Betti) under (L1)-(L3), (I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).), (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.)/(I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.') and G_w in C_I: Thick(M_I) <= C_I^Iwa <= D_{M_I^+} unconditionally, and under (DT_I) C_I^Iwa = Thick(M_I) and C_I^Iwa = C_I iff M_I = all simple objects. (DT_I) holds over F_q-bar via a proper finite-type model and Sun Thm 1.2 (verified for A^n and the group); over C it is the cited Bernstein-Lunts theorem. (II_I) holds over F_q-bar with F = F_q((t)) under (L1), (L2), (D1)-(D3): (IG) implies M_I = Irr P_I, hence C_I^Iwa = C_I, plus pointwise and disconnected-stabilizer forms ((IG)n for all n). Converses hold only as stated (non-zero rows; (WT) sufficient; for the group, IC_omega is not in M_I, so Delta_e is not in C_I^Iwa). Repairs: equivariant splitting (Sun 3.12, not BBD 5.4.5) in Proposition 5.7 and over C in Proposition 5.4; D^b full faithfulness in Proposition 5.4 (L); local systems at every level in Theorem 2.4(e); shift [-2d] in Lemma 2.3(c); (Lin) over every F{q^r} in Proposition 5.7; congruence choice of I_{N,M} for the group; three citation fixes.
Gap
(II_I) for Betti sheaves over C (needs uniform-in-w spreading out); shared cited inputs: Laszlo-Olsson = Bernstein-Lunts equivariant category, equivariant decomposition theorem over C; the claims' own verifications for A^n and the group (Proposition 2.6, Proposition 2.8, Theorem 2.9, Corollary 3.8, Proposition 3.9) were not in the audited file, and classical-affine.html Theorem 3 was not audited.
Blocked routes
Brief as sent
Audit target: The file /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/b7fae8cb-6986-4266-9b6a-db15dbad022f/scratchpad/solve/A7/claims.md contains claimed statements with the proofs exactly as their authors wrote them. Audit every claimed step in it against the adversarial checklist below and against the exact meaning of each term. Do not trust the authors' status labels; check each proof yourself, step by step, and recompute what can be recomputed. The file gives two independently written versions of Claim I (the Iwahori summand criterion) and two of Claim II (functions ⇒ sheaves at Iwahori level), plus the converse statements. Check in particular: the gluing of perverse hearts along the levels and the shift conventions; that the constituent subcategory is thick; the generators of C_I^{Iwa}; the decomposition DT_I and the finite-type model of the convolution; the identification of C_I-functions with traces at a finite level (Lang, Frobenius-stability, connected stabilizers); linear independence of traces of simple objects; the treatment of non-Frobenius-stable constituents and of multiplicity spaces; what is proved over C versus over \overline{F}_q.
For each claimed step return one or more nodes of kind check or objection whose parents list the claimed step ids audited, and whose statement begins with the outcome: holds · holds with a minor repair (give the repair) · needs hypothesis H (name H and where it is used) · fails at step k (with a counterexample or the exact missing lemma). Where a step fails, say whether the claim survives through another version of it in the file or through a repair you prove. End with a verdict node: which claims survive, with exactly which hypotheses, over which fields.