Two errors about \(\mathrm{SL}_2/\mathrm{SO}_2\) had no repair in the search: a circular step in the proof that the sign sheaves are never reached by Satake objects, and a false remark that Chen–Yi's §8.2 applies to the \(\mathrm{SO}_2\)-model. The agent was asked to prove each step or its strongest true replacement and to re-derive the four verdict statements built on them.
It identified the square-class automorphism as a translation by an element of the arc group over \(\overline{\mathbb F}_q\) (Lemma X2.1), and proved the missing invariance directly: the central element \(-I\) acts on each simple equivariant IC sheaf by the scalar \(\chi(-1)\) (Lemma X2.2). With this the original theorem goes through, without Chen–Yi's §8.2 (Theorem (repair of A4-11) X2.4). It then read §8.2 exactly: its hypotheses hold for the \(\mathrm{SO}_2\)-form only when \(q\equiv1\bmod4\), which is also exactly when that form is isomorphic to the split one (Correction (repair of A7-16) X2.5). As a bonus, function-level generation fails for the \(\mathrm{SO}_2\)-form for every odd \(q\) (Proposition X2.6).
No conclusion of the search rested on the false remark, and all four verdict statements stand as stated (Summary X2.8). A fresh reviewer re-checked every step, including the slice computations and the Hecke counts in both forms.
Result as the agent reported it proved
Theorem 4.11 holds after two named changes (S2-4): sigma must be the PGL2(O_n)-action A -> u^{-1} s A s^T (conjugation by s only on the split form), and the missing sigma-invariance is the Weil inertia lemma S2-2: -I acts on every stalk of IC_{chi,nu} by chi(-1), so tr_n IC_{chi,nu} o sigma_n = chi(-1) tr_n IC_{chi,nu} and sigma_n^*(IC,Phi) = (IC, chi(-1)Phi); in particular no sign local system occurs in any cohomology sheaf of IC_{triv,mu} and IC_{sgn,mu} vanishes on L^+X (explicit for mu = alpha^vee, 2alpha^vee in S2-3). This confirms Correction 10.8, in both F_q-forms, with a second PGL2-descent proof for chi = triv; the repaired theorem uses neither Chen-Yi 8.2 nor Thm. 81. Audit 7.16's remark is false: Chen-Yi 8.2 needs a theta-stable Borel pair (B_0,T_0) over F_q with T_0 split; the SO2-form satisfies this iff q = 1 mod 4, exactly when it is F_q-isomorphic to the split form (S2-5, confirming Correction 10.5); its conclusion (Lang's count in both forms) is true for an independent reason, and no conclusion of the run rests on the remark (S2-7). New: (SG) also fails for the non-split SO2-form for every odd q and every F (for q = 3 mod 4, X(F) has two SL2(F)-orbits by parity of the minimal valuation), closing the gap of Summary 2.9 (S2-6). The verdict inputs Summary 4.13, Summary 7.19, Summary 7.25 and Summary 7.32 all hold as stated (Summary 7.32's 'every q' refers to the split form, as Table 1 defines the row).
Gap
None beyond the run's standing inputs D0-D2The standing inputs of the sheaf–function dictionary over F̄_q, p ≠ 2: D0 Chen–Yi's orbit geometry; D1 traces of Frobenius and the perverse t-structure; D2 decomposition of Satake objects into shifted IC sheaves; D3 Lang's theorem; D4 unitriangularity of Satake traces; D5 comparison of F̄_q with C. (l-adic equivariant Weil formalism on placid quotients, decomposition theorem, purity) for the l-adic statement; residue characteristic 2 for the non-split SO2-form is not treated (outside the symmetric-space setting p != 2).
Brief as sent
You are agent S2 in a bounded team of three that is salvaging a finished, audited search on one mathematical question. Work independently; you will not see the other agents.
SETTING. First read /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/f0f18b89-aa0d-4ce6-93d2-78c269eb7d7d/scratchpad/salvage/setting.md. It holds the problem, the setting, the completion criterion, the references with absolute paths, the run's record and the problem's failure checklist, and every agent gets the same text. To read references: sed -e 's/<[^>]*>/ /g' FILE | tr -s ' \n' for HTML; pdftotext -layout FILE - for PDFs.
QUESTION (team-level). Eight agents searched the question and a root wrote a verdict: every row agrees at both levels except E6/F4. An independent second audit then graded every step and found errors that no step of the search repaired. The team's question is: for each erroneous step, prove it as stated, or prove the strongest true replacement; then re-derive every input of the verdict that rests on it, and say whether that input holds as stated, holds with a named change, or fails.
YOUR TARGET. /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/f0f18b89-aa0d-4ce6-93d2-78c269eb7d7d/scratchpad/salvage/S2-target.md is the SL_2/SO_2 cluster. It has three sections:
- §1: the erroneous steps A4-11 and A7-16, in full, with the audit's findings.
- §2: the audit's own corrections C-8 and C-5. Each was written by the auditor who found the error and has NOT been checked independently. Treat them as context: verify, reuse or replace each one, but never adopt one unchecked.
- §3: the verdict inputs that rest on the errors (A4-13, A7-19, A7-25, A7-32), with the path from each error to each input.
YOUR APPROACH FAMILY: Weil structures and Frobenius traces of equivariant IC sheaves on L(SL_2/SO_2) under the square-class automorphism, and the hypotheses of Chen–Yi §8.2 for the SO_2-model. Starting points: (1) A4-11, step 2: prove or refute σ-invariance of tr_n IC_{triv,μ} without circularity. Decide whether σ is induced by an element of L^+SL_2(k̄), or of a larger group normalizing the stabilizers. Then check whether σ*IC_{χ,ν} ≅ IC_{χ,ν} as Weil sheaves, and compute the scalar by which σ acts on each trace function (the correction claims χ(−1)). (2) Determine whether the cohomology sheaves of IC_{triv,μ} along lower orbits can contain sign local systems; at least compute the small cases μ = α^∨ and 2α^∨ explicitly. Give a proof that does not reduce to the claim being proved. (3) A7-16: read Chen–Yi §8.2 (chen-yi-v4.pdf) and state exactly its hypotheses on (G, B_0, T_0, θ) over F_q. Decide over which q the SO_2-model of SL_2/SO_2 (θ(g) = (g^T)^{-1}) satisfies them, and whether the diagonal (split) model is isomorphic to it over F_q. (4) Decide whether any conclusion of the run about SL_2/SO_2 (the sheaf failure over C via central inertia, A2-4; the function failure for every q) needs the false remark. If one does, give a replacement route, for example via the split model over F_q or by working over C directly. (5) Then re-derive each §3 input from the repaired steps.
FILES. The pages and PDFs in setting.md; the run's data.json, nodes/, returns/ and scripts/ (read-only; scripts/C/ holds the second audit's scripts).
RULES.
- Return concrete lemmas, computations or counterexamples, each with a proof or an explicit gap; no status reports. Label every claim: proved (here, or cited with an exact result number) · heuristic · gap. An analogue at one level does not certify an inference at the other.
- Check the exact meaning of every term you use: the field (C, F̄_q, F_q((t)) or p-adic), q odd versus q ≡ 1 mod 4, split versus quasi-split models, the Weil structure versus the underlying sheaf, thick versus linear span, and labels in Λ_S^+.
- Do not search the web for a solution of this question; ordinary background may be looked up. Prefer the local sources.
- Short Python checks for finite computations: run them with
python3 -I, keep scripts in /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/f0f18b89-aa0d-4ce6-93d2-78c269eb7d7d/scratchpad/salvage/S2/ (create it), and keep each run short and single-threaded. Do not modify anything under /Users/miltonlin/Projects/Trees. - Timestamps: at the start of every node, run
date -u +%Y-%m-%dT%H:%M:%SZ and record the result. - Work in nodes and produce 4–10 of them. Node ids are S2-1, S2-2, …. Every node that repairs an erroneous step carries "corrects": ["<id>", …]. Use the concept "sl2", "bridge" or "synthesis" (keys: setup, bridge, sl2, symplectic, spheres, e6f4-functions, e6f4-sheaves, synthesis).
- Write every node as a stage of a mathematics paper: a plain-words name, a one-sentence summary, the precise statement, and the argument as numbered steps. Cite nodes as Theorem 4.11, never by bare code in prose. State computations as mathematics, not as program output.
- FINAL ANSWER: one JSON object inside a ```json fence, with nothing outside the fence. Escape JSON strings properly (double every backslash, escape quotes and newlines). Schema:
{"agent":"S2","family":"≤12 words","nodes":[{"id":"S2-1","t":"<date -u>","kind":"idea|lemma|computation|check|objection|blocked|verdict","status":"proved|refuted|blocked|open|heuristic|superseded","label":"Lemma|Proposition|Computation|Correction|Audit|Summary|…","name":"≤44 chars, plain text","title":"≤90 chars, inline TeX allowed","summary":"1–2 sentences","statement":"Markdown+TeX","parents":["A4-11"],"corrects":["A4-11"],"body":"argument, numbered steps, \\( \\) and $$ $$","concept":"sl2","key":false,"sources":["…"],"inputs":{"A4-13":"holds as stated | holds with change: … | fails"}}],"result":{"status":"proved|partial|refuted|blocked","statement":"…","gap":"…or none"},"blocked_routes":[],"scripts":["paths"]} "inputs" is optional and goes only on the nodes that re-derive §3 inputs. "parents" must name the run's steps (or your own earlier nodes) that each node uses.