The search had proved a support bound for Hecke translates on \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) through the Cartan embedding, and the second audit found that this route fails in residue characteristic \(2\); an audit step had passed it unchecked. The agent was asked to repair both.
It wrote each point of the pair model as a projector and read the orbit label off the polar part of its Smith form (Lemma X3.1). Bounding minors then gives the support bound for every dominant coweight over every non-archimedean field, residue characteristic \(2\) and characteristic \(2\) included (Proposition X3.2). It computed exactly how the Cartan embedding loses labels at \(2\) (Proposition X3.3), and proved that only multiples of the basic function generate, so that in residue characteristic \(2\) the involution model's basic function does not generate (Proposition X3.5).
The statement of the verdict survives, with one change: its \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) row must say that it concerns the pair model (Summary X3.7). A fresh reviewer re-checked each step by exhaustive counts over \(\mathbb Z_2\) and \(\mathbb Z_3\).
Result as the agent reported it proved
Proposition 5.8 holds as stated for all three families over every non-archimedean F; only its route through input (a) fails, and only for the pair model when val 2 >= 1 (tau(O_b) lies in K t^{mu((b - val 2)^+)} K, S3-3) or char F = 2 (no involution). For the pair model the label is the polar Smith part of the projector P_x (S3-1), t^lambda x_0 lies in O_{phi(lambda)} and K t^lambda X(O) lies in the dominance lower set of phi(lambda) for every dominant lambda and every F (S3-2, confirming Correction 10.4). Corollary C needs no tau, so (SG) holds for the pair model over every F (S3-4). New: C_K is free of rank one over a domain A with A^x = C^x, so only multiples of phi_0 generate; hence for 1 <= val 2 < infinity the involution (J-)model's basic function 1_{K x_0^J} = 1_{O_{(val 2,...,val 2)}} does not generate (S3-5). Objection 7.24 corrected (S3-6): its objections stand, its endorsement of Proposition 5.8 needed S3-2/S3-4. Summary 7.25 holds as stated; Summary 7.32 holds once the GL_n/(GL_p x GL_q) row names the pair model; no new divergence (S3-7).
Gap
none for the repaired steps. Cited, not re-proved: the Cartan convolution bound for GL_n, commutativity of H_K, the orbit classification of Theorem 5.4(i) (audited correct), the identification X_J(O) = K x_0^J and the non-affineness of X_J at residue characteristic 2 (Lang, Hensel, Richardson); Objection 7.24's E6/F4 objection rests on Theorem 6.6 as re-checked by the second audit.
Blocked routes
- Rescue input (a) of Proposition 5.8 for the pair model at residue characteristic 2 by another involution: impossible, Ad(eps_0) is the only involution with fixed group Stab(x_0), and tau(O_b) lies in K whenever b_1 <= val 2 (S3-3).
- Transfer (SG) from the pair model to the involution model at residue characteristic 2: false, the involution base point lies in O_{(val 2,...,val 2)} and does not generate (S3-5).
Brief as sent
You are agent S3 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/S3-target.md is the pair-model cluster. It has three sections:
- §1: the erroneous steps A5-8 and A7-24, in full, with the audit's findings.
- §2: the audit's own correction C-4, which covers both steps. It was written by the auditor who found the error and has NOT been checked independently. Treat it as context: verify, reuse or replace it, but never adopt it unchecked.
- §3: the verdict inputs that rest on the errors (A7-25, A7-32), with the path from each error to each input.
YOUR APPROACH FAMILY: lattice-pair models of GL_n/(GL_p×GL_q) over every non-archimedean F, residue characteristic 2 included, and the function-level support bound K t^λ X(O) ⊆ ⋃{ν ≤ φ(λ)} O_ν. Starting points: (1) Fix the pair model of A5-4 precisely: points, base point x_0, the labels of K-orbits, and the map from labels to Λ_S^+. Decide whether input (a) of A5-8 (compatibility of the orbit labels with τ, i.e. t^λ x_0 ∈ O{φ(λ)}) holds when 2 is not a unit. Check the audit's n = 2 example over Q_2 (the label of t^{(a,0)} x_0 equals max(a − val 2, 0)) by your own computation. (2) Prove the support bound for the pair model over every F, including residue characteristic 2, directly from the lattice description (without τ), or name the exact point where it fails. (3) Check whether the triangular argument of A5-4 (Corollary C / Proposition B, or a direct induction) needs (a) at all, or only the support bound plus N(λ, φ(λ)) ≠ 0. Give the corrected statement of "(SG) for GL_n/(GL_p×GL_q) over every F", with the model and the residue characteristics it covers. (4) A7-24: separate its correct objections (to A5-9 and A5-10) from its unchecked claim that A5-8 holds, and state what A7-24 should have said. (5) Then re-derive the §3 inputs A7-25 and A7-32 from the repaired steps; say exactly how the GL_n/(GL_p×GL_q) row of the verdict should read.
FILES. The pages and PDFs in setting.md (families/gl-pq.html and classical.html in particular); 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 pair model versus the pages' J-model, residue characteristic 2 versus char F = 2, the label set Λ_S^+, vol K = 1, and the dominance order.
- 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/S3/ (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 S3-1, S3-2, …. Every node that repairs an erroneous step carries "corrects": ["<id>", …]. Use the concept "spheres", "bridge" or "synthesis" (keys: setup, bridge, sl2, symplectic, spheres, e6f4-functions, e6f4-sheaves, synthesis); the pair model belongs to the function-level lattice counts.
- 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 Proposition 5.8, 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":"S3","family":"≤12 words","nodes":[{"id":"S3-1","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":["A5-8"],"corrects":["A5-8"],"body":"argument, numbered steps, \\( \\) and $$ $$","concept":"spheres","key":false,"sources":["…"],"inputs":{"A7-25":"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.