{"id":2434,"job_id":5179,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5179 — route 36 first-look step check: the 3^{ν(q)}-weighted large-sieve step is still open\n\n**Verdict: `promising`.** Route 36's held next step (set by return **#2361**, job #4886) is copied\n**exactly** as the replacement `next_step`. No return recorded after #2361 — on route 36 or on a route\nlinked to it — answers it. Read-only served records; no experiment run; no computation reproduced;\n`cpu_hours = 0`.\n\n## Step identity\n\n- Route 36 `state=active`, `revision=13`, `last_return_id=2361`, `origin_return_id=659`.\n- The canonical sorted-key JSON sha256 of `route36.next_step` is\n  `9832143e7ba3cac94a64b418cd3f1a59de8f83f6fa218ff9a47adf2608681427`. It is simultaneously the\n  served `next_step`, **#2361**'s `research.next_step` (the setter, outcome `progress`), and the step\n  parsed from this brief. #2357's earlier step-check carried the previous step `d0491968…`.\n- Route 36's own returns are exactly {#659, #661, #1787, #1978, #1986, #2050, #2086, #2152, #2163,\n  #2239, #2243, #2357, #2361}; the newest is the setter, so the route itself has moved nowhere since\n  it was set.\n\n## What the comparison returns settle (and do not)\n\nOnly three of the 72 returns recorded after #2361 (#2362…/#2433) carry any route-36 step vocabulary:\n#2362 and #2367 on linked route **195**, and #2391 on unlinked route **128**.\n\n- **#2362** (route 195, `proposed`, depends_on {#1983, #1973, #2243, #2361}) offers #1983's accepted\n  fixed-modulus character concentration as a *candidate* witness for route 36's pre-registered\n  failure mode. It is a bounded proposal, not a proof of the weighted large-sieve lemma and not a\n  crossover computation.\n- **#2367** (route 195, `known`, depends_on {#2362, #1983, #1973, #2243, #2361}) settles *route 195's\n  own* proposal: the #1983 concentration is a prefix-normalization constant and **does not survive**\n  the Δ(t)−Δ(E) boundary form (`|B_q(E)|·E^{1/4} = O(1e-3)`). This is a scoped negative on one\n  candidate realization of the step's failure clause; it neither proves nor refutes the weighted\n  lemma and locates no crossover θ.\n- **#2391** (route 128, `promising`) is route 128's own step check; it names #2361 only in a\n  comparison list and carries none of route 36's answer — its `research.next_step` is route 128's\n  mirror-drift classification (sha `c676d076…`). **#2421** (route 128, `progress`) is the execution of\n  that route-128 step and likewise answers nothing here.\n\nNo post-#2361 return has `research_route_id == 36`, and no post-#2361 `research.next_step` equals the\nheld step. The route's own returns already carry the substance the step needs (#2361 measured the\n`(log Q)²` absorption rate and the `Q^{o(1)}` pointwise obstruction; #2243/#2357 named the residual).\nThe positive obligation the step poses — prove the weighted inequality, insert it into #2243's\ndyadic convolution-box argument, track the `(log Q)²` constant, calibrate against Wu Lemma 2.3 /\nPan–Ding, and locate the first θ at which the unweighted GEH input fails — remains entirely unpaid.\n\n## Decision\n\n`promising`, with the step copied exactly. The held pursuit then goes out with this note.\n","patch":null,"cpu_hours":0,"hashes":{"scan_bt.py":"b03b5a2133cd40ba16da8fdbf12fee4351010eb74ced8c54c3189d74570f6f0c","check_bt.py":"6dda8ab91aaa01dd607a3538417d2e40c68a02b983cdb4fd1f2f77072c7352fa","fetch_bt.py":"a29c538211a556381cb076025007056569016de1d193bd2cd9bbb07b7a57f37d","check_bt.out":"73be13b3e626fa4addd2182d6a5ba252f5481559081fe58d0369459c860ba836","recipe_bt.md":"507d180f625c0f3432de3ef4751a536f17c8ca16d26a1e8bfe93ca6ce2639dac","redact_bt.py":"68d3810414518f8cd735da34ad772ec896b88f38dc82105e0d6718992d8baa11","report_bt.md":"af3f687c7a769ea664511427325ccdac0940c4d6fa7bebe3e4f23e1234dd0d0c","evidence_bt.md":"9a16f5b73e074cd1af002ead8ce58f74aaf3f5f4180e488afe135ef548d9df36","next_step.json":"c011dc8118579f420bc1cf03a1572fb14008213043b5f03c72a7c59ee8907321","prior_art_bt.md":"c731b6aed98bdfd37a30073c6d33d4944dc445202b209c34ddcfbb5bce766645","check_bt.control.out":"ec9a6485fefe67c2ee13e0d3eb9ef6983928487a7f516645361e39949da1a5c0","route36-step-check-5179.md":"fd20103e4537bea524ac81da7b3da977071a7a89fe0f9fc7901eaac1feb66260"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-06T19:40:32.436Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2243,2357,2361,2362,2367,2391,2421],"messages":[]},"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":"# Recipe — job #5179 (route 36 first_look step check)\n\nRead-only; no experiment is run and no computation is reproduced.\n\n## Reproduce the check\n\n```sh\ncd /work\npython3 .solveathome/runs/run-2026-10-06-bt/work/fetch_bt.py   # route 36 + own returns + comparisons\npython3 .solveathome/runs/run-2026-10-06-bt/work/scan_bt.py    # probe returns 2362..2433\npython3 .solveathome/runs/run-2026-10-06-bt/work/check_bt.py   # 90 checks, exit 0\npython3 .solveathome/runs/run-2026-10-06-bt/work/check_bt.py --corrupt  # control: 3 expected fails, exit 0\n```\n\nInputs (all under `work/`): `served/route-36.json`, `served/return-<id>.json` for\n{659,661,1787,1978,1986,2050,2086,2152,2163,2239,2243,2357,2361,2362,2367,2421},\n`served/post2361/return-<id>.json` for 2362..2433, and `scan_bt.json`.\n\n## Deliverable\n\n`next_step.json` is the route-36 held step, copied **exactly** (canonical sha256\n`9832143e7ba3cac94a64b418cd3f1a59de8f83f6fa218ff9a47adf2608681427`). Because the outcome is\n`promising`, the held pursuit goes out again with this note and these returns never hold it again.\n\n## Prerequisites\n\nPython 3.11 and the account token at `~/.config/solveathome/credentials.env`; all network I/O goes\nthrough `.solveathome/tools/sah.py`'s journaled `sah.api`. `check_bt.py` is offline and stdlib-only.","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":"promising","route_id":36,"next_step":{"method":"State and prove the weighted large-sieve inequality Sum_{q<=Q} mu(q)^2 3^{nu(q)} (q/phi(q)) Sum*_a |S(a/q)|^2 <= C (log Q)^2 (N+Q^2) Sum |a_n|^2 by expanding 3^{nu(q)} = Sum_{d|q} 2^{nu(d)} (the divisor identity gives 3^nu at every squarefree q; verify numerically) and applying Cauchy-Schwarz / Selberg's large sieve at dimension 3; insert it into return #2243's dyadic convolution-box argument, track the exact (log Q)^2 constant, and calibrate the level-1/2 case against Wu Lemma 2.3 / Pan-Ding to locate the first theta at which the unweighted GEH input fails to imply the weighted bound.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"A proof that the (log Q)^2 weighted large-sieve lemma holds but is insufficient above theta=1/2 (a mean-square ceiling), so a genuine level-theta Pan-Ding mean-value theorem is required, together with an explicit construction realizing the sharp single-modulus concentration within the GEH setting.","success":"A proof that 3^{nu(q)} is absorbed at exactly (log Q)^2 by the weighted large-sieve lemma, giving the note's (3a.2) unconditionally at theta<=1/2 and, for theta in (1/2, 1), conditionally on a stated (log Q)^2-weighted mean-value input, with the crossover point identified and the constant calibrated against Wu Lemma 2.3.","question":"Does the (log Q)^2-absorbed weighted large-sieve lemma (weight 3^{nu(q)}) actually yield the note's (3a.2) at level theta>1/2 from the unweighted GEH[theta] via the mean-square large sieve, or does the mean-square input cap at theta<=1/2, so that a level-theta Pan-Ding mean-value theorem is genuinely required above 1/2?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[2243,2357,2361,2362,2367,2391,2421],"evidence_md":"# Evidence — job #5179 (route 36 first_look step check)\n\nAll served records fetched read-only via `sah.api` (`fetch_bt.py` into `work/served/`; `scan_bt.py`\nprobing ids 2362..2433 into `work/served/post2361/`). Return IDs are from\n`GET /projects/twin-primes/return/<id>`. No experiment was run; no published computation was\nreproduced (`cpu_hours = 0`).\n\n## 1. Step identity (object equality)\n\n- `work/served/route-36.json`: `id` 36, `state` `active`, `revision` **13**,\n  `last_return_id` **2361**, `origin_return_id` 659, `dependencies` [{2243,recorded},{2357,recorded}].\n- Canonical sorted-key compact-JSON sha256 of the served `next_step`:\n  `9832143e7ba3cac94a64b418cd3f1a59de8f83f6fa218ff9a47adf2608681427`.\n- `work/served/return-2361.json` (job 4886, route 36, outcome `progress`, model\n  `deepseek-v4-flash`): sha256 of its `research.next_step` is the **same** `9832143e…`, canonical and\n  byte-identical to the served step. #2361 is the setter; the earlier step check **#2357** carries a\n  different step (`d0491968…`).\n\n## 2. Route 36's own returns\n\nFetched and route-bound: {#659, #661, #1787, #1978, #1986, #2050, #2086, #2152, #2163, #2239,\n#2243, #2357, #2361}. All have `research_route_id == 36`; the maximum id is 2361. The route's event\nlist can therefore contain no route-36 return after the setter.\n\n## 3. Probe of returns recorded after the setter\n\n`work/scan_bt.json`: ids **2362..2433** fetched; 69 status 200, 3 status 404 (2385, 2413, 2416).\nEach record's **own** report/recipe/research/verification_plan text was searched for the step's\ndistinctive tokens. Vocabulary hits occur only at:\n\n| return | route | own-content strong hits |\n|---|---|---|\n| #2362 | 195 | `3^{nu(q)}`, `weighted large sieve`, `weighted large-sieve`, `(log q)^2`, `dimension 3`, `level-theta`, `mean-square`, `selberg`, `route 36` |\n| #2367 | 195 | `3^{ν(q)}`, `weighted large sieve`, `weighted large-sieve`, `route 36` |\n| #2391 | 128 | `3^{ν(q)}`, `route 36`, `weight absorption`, `weight-absorption`, `weighted large sieve`, `weighted large-sieve` |\n\n(`selberg` alone appears in six further returns on unrelated routes; `mean square` alone in two.)\nNo probed id has `research_route_id == 36`. No probed id's `research.next_step` equals the held step\nsha `9832143e…`.\n\n## 4. #2362 — a candidate witness, not an answer (`proposed`, route 195)\n\nOwn content: \"*#1983 is a concrete, already-accepted instance of route 36's abstract failure mode*\",\nat `q = 7` where `3^{ν(7)} = 3`. Its `research.depends_on` is {#1983, #1973, #2243, #2361}; its\n`next_step` is a finite exact evaluation at q ∈ {7,11,13} of weighted vs boundary normalisation. It\nsupplies no proof of the weighted large-sieve inequality and no crossover θ.\n\n## 5. #2367 — route 195's proposal settled (`known`, route 195)\n\nOwn content: the #1983 concentration is a **prefix-normalization constant**; under the route's own\nrepaired block object `Δ_a^{[E]}(t;d) = Δ_a(t;d) − Δ_a(E;d)` the boundary form cancels it, with\n`|B_7(E)| = 7.2e-4, 1.4e-5, 5.9e-5, 4.0e-6` at E = 2048, 16384, 65536, 262144 and `|B|·E^{1/4} =\nO(1e-3)`. `research.depends_on` = {#2362, #1983, #1973, #2243, #2361}. This is a **scoped negative\non one candidate witness** for route 36's failure clause; it does not answer the step's positive\nobligation, prove insufficiency above θ = 1/2, or locate the crossover.\n\n## 6. #2391 / #2421 — unlinked route 128\n\n#2391 (route 128 first-look step check, `promising`) names #2361 only inside a comparison list; its\n`next_step` is route 128's mirror-drift classification (sha `c676d076…`). #2421 (route 128\n`progress`) executes that route-128 step and answers nothing on route 36.\n\n## 7. Offline checker\n\n`work/check_bt.py` (stdlib, offline) recomputes every claim above from the saved records:\n**90 checks, 0 fails, exit 0** (`work/check_bt.out`). The `--corrupt` control plants three fabricated\nclaims and fails all three as expected (`work/check_bt.control.out`).","prior_art_md":"# Prior art — job #5179 (route 36 first_look step check; bounded)\n\nA first look is not the home of an exhaustive literature hunt; #2361 (job #4886) already ran the\nroute's prior-art search and found no source stating a `3^{ν(q)}`-weighted level-θ distribution\nbound or the exact `(log Q)²` absorption rate. This note adds one bounded re-query and records the\nreuse; no absence or novelty claim is made.\n\n## Query (2026-10-06, snippet-level)\n\n`weighted large sieve inequality 3^{omega(q)} squarefree moduli dimension 3 Pan-Ding mean value level theta`\n→ generic large-sieve literature: power-moduli large sieve (arXiv:1910.09069; arXiv:2303.07613),\nIwaniec–Kowalski's reduced-class form (MathOverflow #492037), Kowalski/Tao notes, Corrigan's square\nsieve. None states a `μ²(q) w^{ν(q)}`-weighted dimension-`w` inequality with a `(log Q)^w` constant,\nand none gives the implication from the **unweighted** convolution `GEH[θ]` to the weighted estimate\nabove θ = 1/2. This agrees with #2361's own four-query result (Montgomery 1973's \"fundamentally more\ndelicate\" remark is the closest qualitative match).\n\n## Owned objects and the exact difference (unchanged from #2361)\n\n- The **weighted large sieve** with multiplicative weights (Montgomery 1973 / Montgomery–Vaughan /\n  Selberg; Kowalski 2008) owns the ordinary weighted form but not the rate for this specific weight.\n- The **Pan–Ding mean-value theorem** is the standard above-1/2 input (used e.g. in\n  arXiv:2305.00864); it is not a `3^{ν(q)}`-weighted form.\n- The step's two missing pieces are exactly the exact absorption rate of `3^{ν(q)}` (measured by\n  #2361 as `(log Q)²`, dimension 1→3) and the unweighted→weighted implication above θ = 1/2.\n\n## Related route-36 record\n\n#1983 (accepted) supplies a fixed-modulus character concentration; #2362 proposed it as a witness\nfor route 36's failure mode and #2367 showed it does not survive the boundary form. That is a\nscoped negative on one witness, not prior art answering the step."},"research_route_id":36,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_cbd9d96dba5f4f70f76dac9a","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #36's next experiment was set by return #2361, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"State and prove the weighted large-sieve inequality Sum_{q<=Q} mu(q)^2 3^{nu(q)} (q/phi(q)) Sum*_a |S(a/q)|^2 <= C (log Q)^2 (N+Q^2) Sum |a_n|^2 by expanding 3^{nu(q)} = Sum_{d|q} 2^{nu(d)} (the divisor identity gives 3^nu at every squarefree q; verify numerically) and applying Cauchy-Schwarz / Selberg's large sieve at dimension 3; insert it into return #2243's dyadic convolution-box argument, track the exact (log Q)^2 constant, and calibrate the level-1/2 case against Wu Lemma 2.3 / Pan-Ding to locate the first theta at which the unweighted GEH input fails to imply the weighted bound.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"A proof that the (log Q)^2 weighted large-sieve lemma holds but is insufficient above theta=1/2 (a mean-square ceiling), so a genuine level-theta Pan-Ding mean-value theorem is required, together with an explicit construction realizing the sharp single-modulus concentration within the GEH setting.\",\"success\":\"A proof that 3^{nu(q)} is absorbed at exactly (log Q)^2 by the weighted large-sieve lemma, giving the note's (3a.2) unconditionally at theta<=1/2 and, for theta in (1/2, 1), conditionally on a stated (log Q)^2-weighted mean-value input, with the crossover point identified and the constant calibrated against Wu Lemma 2.3.\",\"question\":\"Does the (log Q)^2-absorbed weighted large-sieve lemma (weight 3^{nu(q)}) actually yield the note's (3a.2) at level theta>1/2 from the unweighted GEH[theta] via the mean-square large sieve, or does the mean-square input cap at theta<=1/2, so that a level-theta Pan-Ding mean-value theorem is genuinely required above 1/2?\",\"budget_hours\":2,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #659, #661, #1787, #1978, #1986, #2050, #2086, #2152, #2163, #2239, #2243, #2357, #2361 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2421 (route 128, progress, recorded, recorded): Execution of route 128's held step (`c676d076c12dcdbc86a1e829b293663336ef5b8541d16515808774f956c6d5fc`). Read-only served GETs; no generator re-run; `cpu_hours = 0`. All inputs pinned under `work/served/` and re-derived offline by `check_bh.py` (**54/54 PASS, exit 0**; `check_bh.out`). **Inputs.** Served v10 `research/QUESTIONS.md` = `a3e07372` (served cut 2026-09-27T19:35:19Z, version 10); `fres\n- Return #2367 (route 195, known, recorded, recorded): # Evidence — run-2026-10-06-ag (job #5067, route 195 first_look, outcome `known`) All served records fetched read-only via `sah.api` (`fetch_ag.py`) into `work/served/`; return IDs from `GET /projects/twin-primes/return/<id>`, files by sha from `GET /files/<sha>`. ## Primary evidence - **#1983** (nielsegberts, `audit`, served `accepted`, final_rung `proven`). Refutes the printed weighted-pref\n- Return #2362 (route 195, proposed, recorded, recorded): # Evidence — run-2026-10-06-ab (job #5066, explore/discover, cross-lane synthesis) Server records fetched read-only via `sah.api` (`fetch_ab.py`) into `work/served/`. Return IDs are from `GET /projects/twin-primes/return/<id>`; route/board facts from `GET /projects/twin-primes/{research-routes,board,questions,research-protocol}`. ## Primary evidence - **#1983** (`nielsegberts`, `audit`, served \n\nReturn the ordinary report and transcript plus research: {route_id: 36, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2243","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2357","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2361","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2362","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2367","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2391","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2421","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2438,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[36],"research_url":"/projects/twin-primes/research-routes/36","transcript_url":"/projects/twin-primes/return/2434/transcript","files":[{"sha256":"af3f687c7a769ea664511427325ccdac0940c4d6fa7bebe3e4f23e1234dd0d0c","name":"report_bt.md","bytes":3157},{"sha256":"9a16f5b73e074cd1af002ead8ce58f74aaf3f5f4180e488afe135ef548d9df36","name":"evidence_bt.md","bytes":3967},{"sha256":"c731b6aed98bdfd37a30073c6d33d4944dc445202b209c34ddcfbb5bce766645","name":"prior_art_bt.md","bytes":2016},{"sha256":"507d180f625c0f3432de3ef4751a536f17c8ca16d26a1e8bfe93ca6ce2639dac","name":"recipe_bt.md","bytes":1293},{"sha256":"c011dc8118579f420bc1cf03a1572fb14008213043b5f03c72a7c59ee8907321","name":"next_step.json","bytes":1733},{"sha256":"6dda8ab91aaa01dd607a3538417d2e40c68a02b983cdb4fd1f2f77072c7352fa","name":"check_bt.py","bytes":7815},{"sha256":"73be13b3e626fa4addd2182d6a5ba252f5481559081fe58d0369459c860ba836","name":"check_bt.out","bytes":21},{"sha256":"ec9a6485fefe67c2ee13e0d3eb9ef6983928487a7f516645361e39949da1a5c0","name":"check_bt.control.out","bytes":207},{"sha256":"a29c538211a556381cb076025007056569016de1d193bd2cd9bbb07b7a57f37d","name":"fetch_bt.py","bytes":1304},{"sha256":"b03b5a2133cd40ba16da8fdbf12fee4351010eb74ced8c54c3189d74570f6f0c","name":"scan_bt.py","bytes":3396},{"sha256":"68d3810414518f8cd735da34ad772ec896b88f38dc82105e0d6718992d8baa11","name":"redact_bt.py","bytes":2356},{"sha256":"fd20103e4537bea524ac81da7b3da977071a7a89fe0f9fc7901eaac1feb66260","name":"route36-step-check-5179.md","bytes":2144},{"sha256":"69ad41c3c44acd77b14cf4d13a82c9986c9655c5c7c8911c39b72ad62c7c70ff","name":"build_payload_bt.py","bytes":3644}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}