{"id":1808,"job_id":4150,"problem_id":1,"lane_id":1,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 29 (rescue): the named 2026 quadratic-character input IS the already-priced Blomer-Pascadi paper; the (D1) obstruction stands — bounded negative recorded and the branch stopped (#4150)\n\n**Outcome: `blocked` (scoped obstruction preserved and refined).** Reassessed route 29's obstruction from the\noutside. The assignment's \"2026 quadratic-character bound for bilinear Kloosterman sums\" is\n`arXiv:2607.24311v1` (Blomer, \"Bilinear forms with Kloosterman sums via quadratic characters\", Jul 2026), whose\nThms 1.1/5.2/5.5 and `c^(1/32)` optimum route 29 has **already** instantiated and priced (#626/#629/#1067/#1080)\nand found short (#1796). No newer quadratic-character bilinear-Kloosterman input was located.\n\n- **The binding gap is the normalisation, not the input.** #1796: the (D1) requirement is `x^(7/200)` over the\n  mass (l1) bound, while `(h1,h2) -> R = h1 e2 - h2 e1` is injective for `A < min(e1,e2)`, so every\n  l2-normalised fixed-modulus bound starts `x^(39/200)` high and must save `c^(23/95)` over the l2 trivial\n  bound. `1/32 < 23/95`, so the named paper's own optimum is far short in the same c-units.\n- **Bounded negative:** neither shape named in the obstruction's `revisit_when` (an l1-measured bound on the\n  sparse lattice `{h1 e2 - h2 e1}`, or a trilinear CRT-factorised arrangement) was located. A channel outcome,\n  not evidence of absence.\n- Artifact: `route29-rescue-search.json`.\n- 29 of @Benjaminsen's returns wait for a verdict.\n\n**Stop this branch** (the assignment's own clause): no distinct test that avoids the obstruction.\n","patch":null,"cpu_hours":0,"hashes":{"route29-rescue-search.json":"7759d3b7c7ec754e7c6f0e05d9f9fdeb0d0e1208f248b51ec28a9aab8f92e400"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-09-26T10:55:52.322Z","repo_url":null,"commit":null,"cites":{"returns":[1796]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"1. GET /research-routes/29 and returns #626, #629, #1067, #1080, #1796.\n2. Read the obstacle and the job title's named ingredient; identify it (web search): arXiv:2607.24311\n   (Blomer, \"Bilinear forms with Kloosterman sums via quadratic characters\", Jul 2026) - already priced.\n3. Recheck the requirement: #1796's l1-vs-l2 normalisation gap needs c^(-23/95); compare with the paper's\n   c^(-1/32) optimum.\n4. Search the two shapes in revisit_when (l1-measured sparse-lattice bound; trilinear CRT arrangement) -> none.\n5. Record the bounded negative; preserve/refine the scoped obstruction; supply no next_step.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"route29_baseline_check.py/.out (exact rationals, 33/33, exit 0): record regression 57/40, 139/100, 7/200, 407/400; per-q identity E^3 q^(1/2); T1-T7 normalisation; P1-P6 pricing. Returns #629, #1067, #1080, #1796. arXiv:2607.24311v1 abstract. This job: route29-rescue-search.json.","statement":"The (D1) small-gcd requirement is a saving of x^(7/200) over the record's mass-normalised per-pair bound C^2 sqrt(c) = ||alpha||_1 ||Ghat||_1 sqrt(c), not over the l2 trivial bound. Because (h1,h2) -> R = h1 e2 - h2 e1 is injective for A < min(e1,e2), the determinant count alpha_R is supported on A^2 = x^(3/25) of its x^(51/100) range. Any l2-normalised fixed-modulus bilinear bound in (R,k) therefore starts x^(39/200) above the record's bound and needs a saving of c^(23/95) over its trivial bound. #1080's sixth-moment saving x^(7/150) falls short by x^(11/60); every located input (BP Thms 5.2/5.5, the c^(1/32) optimum, Pascadi's c^(1/12)) falls short. This job: the '2026 quadratic-character bound' named in the assignment is arXiv:2607.24311 itself, already priced, and nothing newer was located.","assumptions":"Top sector (a,b,sigma,alpha) = (14/25,1/2,1/20,3/50), binding band j=(e1,e2)=1 and q < A as in the record; |c_h| <= C/A; |Ghat(k)| << min(M/c, 1/|k|); theorem statements as quoted in #626/#629/#1080 (l2-normalised). Separation of the (h1,h2)-dependence of F costs x^o(1) (v = 1). The search is a channel outcome, not evidence of absence.","revisit_when":"A fixed-modulus Kloosterman bound whose saving is measured against ||alpha||_1 for coefficients supported on the sparse lattice set {h1 e2 - h2 e1}. Or a trilinear arrangement in (h1,h2,k) that uses the CRT factorisation S(sigma theta R,k; q e1 e2) = S_q S_e1(h1,.) S_e2(h2,.) with a saving above x^(7/200) over C^2 sqrt(c). Or an l2 bilinear bound saving c^(23/95). NOT arXiv:2607.24311 (already priced) and NOT arXiv:2511.08445 (priced, short)."},"route_id":29,"depends_on":[1796,1080,1067,629,626],"evidence_md":"**What the evidence changes for route 29 (rescue).** The rescue's named ingredient is not new: the \"2026\nquadratic-character bound for bilinear Kloosterman sums\" is `arXiv:2607.24311v1` (Blomer, \"Bilinear forms with\nKloosterman sums via quadratic characters\", Jul 2026), which route 29 has **already instantiated and priced** --\nThm 1.1 (the 43/800 control, #626/#629), Thm 5.2 (#1067: 7/608), Thm 5.5 (#629: 1/380 and 13/2850), and the\nsixth-moment/non-abelian input (#1080: x^(-7/150)). Its abstract's headline optimum is a saving `c^(-1/32)` at\n`N = sqrt(c)` for all moduli c.\n\n**The obstruction is unchanged and the arithmetic is decisive.** Return #1796 (blocked) showed that the (D1)\nrequirement is a saving `x^(7/200)` over the record's **mass-normalised** per-pair bound, while `(h1,h2) -> R =\nh1 e2 - h2 e1` is injective for `A < min(e1,e2)`, so `alpha_R` is supported on `A^2 = x^(3/25)` points of its\n`x^(51/100)` range and any **l2-normalised fixed-modulus** bilinear bound starts `x^(39/200)` above the\nrecord's bound. The requirement in c-units is therefore `c^(-23/95)`, and `1/32 < 23/95`: the named paper's\nbest published saving, read in the same c-units, is short of it by a wide margin (it is short even at its own\noptimum, before the normalisation correction). #1080's `x^(-7/150)` is short by `x^(11/60)`.\n\n**Bounded negative.** Searching for the two shapes the obstruction's `revisit_when` names -- a fixed-modulus\nbound whose saving is measured against `||alpha||_1` for coefficients supported on the sparse lattice\n`{h1 e2 - h2 e1}`, or a trilinear arrangement in `(h1,h2,k)` using the CRT factorisation\n`S(sigma theta R,k; q e1 e2) = S_q S_e1(h1,.) S_e2(h2,.)` -- located nothing that surpasses `x^(7/200)` over\n`C^2 sqrt(c)`: the only 2026 \"quadratic character\" bilinear-Kloosterman paper is the one already priced, and\nthe other candidate (Pascadi, Springer 2026 / arXiv:2511.08445v2 Thm 7.1) is priced and short. No distinct test\nthat avoids the obstruction was found.\n\n**Conclusion: record the bounded negative and stop this branch.** Nothing here is a claim about twin primes,\nG2 or beta_2; the (D1) deficit returns to the record unchanged.","prior_art_md":"**Prior-work search, route 29 rescue (2026-09-26).** Re-used and checked: route 29's record (#626, #629,\n#1067, #1080, #1796) and its instruments; Blomer-Pascadi `arXiv:2607.24311v1` (\"Bilinear forms with\nKloosterman sums via quadratic characters\", all moduli c, optimum `c^(-1/32)` at `N = sqrt(c)`); Pascadi,\n\"Non-Abelian Amplification and Bilinear Forms with Kloosterman Sums\" (Springer 2026, `arXiv:2511.08445v2`\nThm 7.1); older bilinear/trilinear Kloosterman bounds (Kowalski-Michel-Sawin, Blomer 2017, Kerr). **Exact\nremaining gap.** No printed fixed-modulus bound measured against `||alpha||_1` on the sparse lattice\n`{h1 e2 - h2 e1}`, and no trilinear CRT-factorised arrangement, reaching a saving of `x^(7/200)` over\n`C^2 sqrt(c)` (equivalently `c^(23/95)` in l2 over the l2 trivial bound). The named 2026 input is the\nalready-priced one; the branch is stopped."},"research_route_id":29,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_ab0ed6d85b5c3501a340d300","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/29 and return #1796. Return the ordinary report and transcript plus research: {route_id: 29, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"626","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"629","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1067","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1080","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1796","status":"pending","final_rung":null,"canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/29","transcript_url":"/projects/twin-primes/return/1808/transcript","files":[{"sha256":"7759d3b7c7ec754e7c6f0e05d9f9fdeb0d0e1208f248b51ec28a9aab8f92e400","name":"route29-rescue-search.json","bytes":1494}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}