The agent audited the \(\mathrm{GL}_2/T\) claim by recomputing it independently. It took about ten thousand random geodesics over \(\mathbb F_p(t)\), applied explicit coset representatives as Möbius maps, and confirmed the orbit list, the closures and the connected stabilizers ( (Audit 9.1)) and the Hecke action ( (Computation 9.2)). It recounted the fibres by Hermite normal forms and checked the convolution rules ( (Audit 9.3), (Audit 9.4)).
The sheaf failure at the two closed orbits holds ( (Audit 9.6)), as does the function computation of codimension two over every local field ( (Audit 9.7)). So does the positive-certificate lemma: purity makes every multiplicity a Laurent polynomial with nonnegative coefficients, so positivity on every simple object excludes cancellation ( (Audit 9.8)). Its main objection is that the strict level presentation 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)., with \(I\) acting on finite quotients, is assumed for \(N\ge1\), not checked; the sheaf half is conditional on it, exactly like the pages' Proposition IV.1 ( (Objection 9.5)).
It also found a real gap in the general closed-orbit test: positivity does not give a nonzero specialization at \(v=-q^{-1/2}\), so only the sheaf half of that test is proved ( (Objection 9.10)). Its verdict ( (Summary 9.12)) is the form of Proposition IV on the verdict page.
Result as the agent reported it partial
Claim IV (X = GL_2/T) survives the audit. Functions, over every non-archimedean F with no further hypothesis: H_I*φ_0 = C_K + C_{K'} has codimension 2, and the quotient is ε_- ⊕ ε_+, detected by λ (Steinberg) and ψ (twisted Steinberg). Sheaves, over F̄q (p odd) and over C, assuming the level presentation 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).: M_I = {IC{i,n}}, and C_I^Iwa = Thick(For C_K ∪ For C_{K'}) is a proper subcategory of C_I, missing exactly k_{A_+} and k_{A_-}. A second proof over F̄_q uses the positive certificate 2ψ, and the positive-certificate lemma Lemma 6.7 is sound under its hypotheses (plus W ≠ 1). So the row is not a divergence. Corollary 6.10's closed-orbit test proves its sheaf half; its function half needs a nonzero specialization of ψ at v = -q^{-1/2}.
Gap
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). for GL_2/T in the strict form required by the setting (Y_N = lim Y_{N,M}, with I acting through finite quotients) is not proved for N ≥ 1, nor over C. Chen–Yi give only an I_0-placid presentation over F̄_p. Theorem 6.12(4) wrongly lists 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). as checked.
Blocked routes
- Strict level presentation for Y_N (N>=1) of GL_2/T: needs smoothness of the truncation maps pi_{j+1}(Y_N) -> pi_j(Y_N) for j >= 2N+1, a Greenberg/Newton lemma with jumping gradient order e <= N, or an I-equivariant finite-quotient refinement of Chen–Yi Proposition 39
- Function-level half of the closed-orbit test Corollary 6.10: needs a lemma that a sign-type psi, nonnegative on closed orbits, has a nonzero specialization at v = -q^{-1/2}; false for general Laurent values (v+2v^2 at q=4)
Brief as sent
Audit target: The file /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/b7fae8cb-6986-4266-9b6a-db15dbad022f/scratchpad/solve/A9/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 Claim IV for X = GL_2/T (= pairs of distinct points of P^1; oriented geodesics of the Bruhat–Tits tree). The author's computation scripts are in /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/b7fae8cb-6986-4266-9b6a-db15dbad022f/scratchpad/solve/A6/ (lv_module.py, lv_checks.py, n*.py); you may read and rerun them with python3 -I, but write your own independent checks in your folder. Check in particular: the list of I-orbits on LX, their closures, dimensions and stabilizers (connected? which local systems?); the Hecke action on orbit classes; the IC sheaves of the level sets Y_n and their stalks; the convolution IC_{s_0} ∗ IC(Y_n); the positivity certificate lemma (a Hecke functional nonnegative on IC classes and killing the generators excludes summands — is the positivity really proved for every simple object, with which normalization, and does purity exclude cancellation?); the twisted Steinberg functional; the function-level computation H_I·φ_0 = C_K + C_K' of codimension 2 over every F; whether Chen–Yi's presentation (I1) is needed and checked.
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.