After the second audit, four errors in the symplectic family had no repair anywhere in the search: a wrong list of Hecke translates for \(\mathrm{GL}_4/\mathrm{Sp}_4\), four wrong entries in a \(\mathrm{GL}_6/\mathrm{Sp}_6\) table, two wrong entries in an \(m=3\) check table, and a false shortcut for the orbit codimensions. The agent was given the four steps, the audit's findings and the audit's own corrections, and asked to prove each step or its strongest true replacement.
It counted \(t\)-stable Lagrangians and lattices directly. For \(\mathrm{GL}_4/\mathrm{Sp}_4\) it found exactly which translates reach the closed orbit and with which values (Correction X1.1); the cancellation behind spherical generation survives. It proved the twelve \(\mathrm{GL}_6/\mathrm{Sp}_6\) values as polynomials in \(q\), confirming that four of the original ones were wrong but not the conclusion drawn from them (Correction X1.3). It showed that the lemmas of the \(m=3\) table never used its entries (Correction X1.2), and proved the codimension formula in every component by a stabilizer count, since the shift by \(t\) reaches only some components (Proposition X1.4).
It then re-derived the six statements of the verdict that rested on these steps: all survive, four with a changed reason or status and none with a changed answer (Re-derivation X1.6). A fresh reviewer re-checked every step by its own enumeration and found nothing wrong.
Result as the agent reported it proved
All four erroneous steps are repaired, and every verdict input resting on them survives. (1) Proposition 1.4: for GL4/Sp4 over any non-archimedean F, T_λφ0 is nonzero on O_(1,1,1,1) exactly for λ=(1+a,1+b,−b,−a), a≥b≥0, with value q^{6a}(q+1)(q^2+1) (a=b) or q^{4a+2b−1}(q+1)^2(q^2+1) (a>b). (1,1,0,0) is the only such λ with φ(λ)=(2,2,0,0), and T_(2,0,0,0) vanishes there; c'=1, c=q(q+1); the cancellation and (SG) hold over every F (S1-1). (2) Dead end 4.10: the twelve values R1–R3 are proved as polynomials in q; four of Dead end 4.10's values are false; e2∉span(R1,R2,R3) holds (eliminate c4, then c3, then c1); T_(2,0,0,0,0,0)φ0=q(q+1)1_{c1} completes the span (S1-3). (3) Lemma 5.6: Lemmas 1–3 are correct and independent of the table; the |b|=4 table (= Correction 10.6) is proved for all q; the |b|≤5 tables are computed at q=2,3,5 and obey Lemmas 2–3 (S1-2). (4) Objection 7.11: σ links component s to component 0 iff m|s; the codimension formula ⟨ρ,μ−ν⟩ holds in every component over every algebraically closed field of characteristic ≠2 (stabilizer count plus smoothness by Cayley transform) (S1-4). Inputs: Theorem 5.7 and Summary 7.32 hold as stated; Summary 4.13, Summary 1.9, Summary 7.19 and Summary 7.25 hold with the named changes (S1-5, S1-6). No row of the verdict changes.
Gap
none for the four steps and six inputs. Outside the claims: the |b|=5 closed forms in S1-2 are fits, proved only at q=2,3 (and q=5 except column (2,2,1)); Summary 1.9's non-symplectic defects belong to other clusters.
Blocked routes
- The shift σ(ω)=tω cannot reduce the codimension formula to component 0 when m does not divide s (for GL4: odd components). This is the flaw of Objection 7.11, point 2; S1-4 uses the finite-type stabilizer count instead.
- Polynomial interpolation of the m=3, |b|=5 counts from q=2,3,5 cannot certify the closed forms for all q (degrees up to 9). They are recorded as heuristic; only nonvanishing and the zero pattern, which Theorem 5.7 needs, are proved (Lemmas 2, 3).
- Exhaustive enumeration at q=5 of the column b'=(2,2,1) (|b|=5, m=3), and of GL4/Sp4 in the box t^{-2}, exceeded the few-minute budget and were stopped.
Brief as sent
You are agent S1 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/S1-target.md is the symplectic cluster. It has three sections:
- §1: the erroneous steps A1-4, A4-10, A5-6 and A7-11, in full, with the audit's findings.
- §2: the audit's own corrections C-3, C-7, C-6 and C-2. 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 (A1-9, A4-13, A5-7, A7-19, A7-25, A7-32), with the path from each error to each input.
YOUR APPROACH FAMILY: lattice and Lagrangian counts for GL_{2m}/Sp_{2m}, and orbit codimensions in every component of LX. Starting points: (1) GL_4/Sp_4 (A1-4): determine exactly which dominant λ have T_λ φ_0 ≠ 0 on O_(1,1,1,1). Verify the cancellation f = T_(1,1,0,0) − (c'/c) T_(2,0,0,0) by your own count of t-stable lattices or Lagrangians, and state the corrected step 2. (2) GL_6/Sp_6 at Σμ = 4 (A4-10): recompute the twelve values R1–R3 as polynomials in q, by a method different from the audit's. Confirm or refute "e2 ∉ span(R1,R2,R3)", and the claim that one translate from a larger box completes the span. (3) The m = 3 check table of A5 (A5-6): recompute N(λ*(b), b') for |b| ≤ 5. Check whether Lemmas 1–3 of A5-7 (reverse-dominance triangularity) depend on the wrong entries at all. (4) A7-11: prove the codimension formula for orbits of L(GL_{2m}/Sp_{2m}) in every component (Pfaffian valuation s = 0, …, 2m−1) without Chen–Yi Prop. 64's connectedness hypothesis. For example, use the finite-type stabilizer count, and say exactly which components the shift σ(ω) = tω reaches. (5) Then re-derive each §3 input from the repaired steps, one node per input or one node for a group of inputs with the same dependence.
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 residue characteristic 2, thick versus linear span, the normalization vol K = 1, labels in Λ_S^+ versus Λ_T^+, and the components of LX.
- 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/S1/ (create it), and keep each run short (seconds to a few minutes, 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 S1-1, S1-2, …. Every node that repairs an erroneous step carries "corrects": ["<id>", …]. Use the concept "symplectic" for the counts, "synthesis" for the input re-derivations, or another key from: 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 Dead end 4.10, never by bare code in prose. State computations as mathematics (what was computed, the result, why it suffices), 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":"S1","family":"≤12 words","nodes":[{"id":"S1-1","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":["A4-10"],"corrects":["A4-10"],"body":"argument, numbered steps, \\( \\) and $$ $$","concept":"symplectic","key":false,"sources":["…"],"inputs":{"A5-7":"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.