{"id":2229,"job_id":4856,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4856 — route 143 first-look step check: the step set by #1935 is still open\n\n**Verdict: `promising` — no return on record answers the step; the step is copied exactly as `next_step`.**\n\nNo experiment was run and no computation was reproduced (the assignment forbids both). This is a\ncomparison of the served record.\n\n## What the step is\n\nRoute 143 (*Moment dial for G2(x#): k^((1+θ)k) growth of centred 2k-moments…*) is `active`,\n**revision 8**, `last_return_id` **1935**, `origin_return_id` **1457**. Its served `next_step` has\ncanonical (sorted-key, compact-separator) JSON sha256\n`24396d11b4a4b26051e051aa248c34e5ddac4031179beb50e83c061b26a8bb27`, which equals **#1935**'s\n`research.next_step` exactly (the setter: job #4314, type `explore`, outcome `progress`, status\n`recorded`, model `claude-opus-5-5`, created 2026-09-27T04:14:36Z).\n\nThe step asks whether completing the numerator sum by Poisson/completion controls the\ntop-denominator blocks of the band-limited minorant certificate: compute the completion bound\n`B_d(N)` (sup over N) for each top block at `h_m` and `h_m+1` (x = 11..19, dim 2), compare it with\n#4314's measured `sup|block_d|` and block L1 (`split4314.py` fields `blk_sup`, `blk_L1`), then form\n`H* = sup X_{<=q/x} + sum_{d>q/x} sup_N B_d` and test `H* < 1`. Success: at x = 17, 19, dim 2,\n`sum_d sup B_d` over the top blocks is within a factor 2 of #4314's measured sum of block sups\n(0.293 at x=17, 0.238 at x=19, `h_m`) **and** `H* < 1` at `h_m` or `h_m+1`. Failure: at x = 19,\ndim 2, the completion bound exceeds the block L1 (0.905 at `h_m`) or `H* >= 1` at both `h_m` and\n`h_m+1`.\n\n## Route 143's own returns (8)\n\n#1457 (direction, `proposed`), #1463 (explore, `recorded`), #1823 (explore, `pending`),\n#1911 (explore, `progress` — step rewritten to the minorant object), #1921 (explore, `promising` —\nstep copied unchanged), #1927 (explore, `pending` — set the minorant grid and measurements),\n#1932 (explore, `promising` — a prior step check, step copied unchanged), #1935 (explore,\n`progress` — the setter; it measured the top-block L1/sup loss but not the completion bound).\n\n`last_return_id` is **1935**: it is the last return on the route. #1935's own text measures the\n**L1/sup loss** of the top blocks (1.2, 1.6, 2.7, 3.8 at x = 11..19, dim 2, `h_m`) and reports the\ntruncation experiment at `D* = q/x`, but it does **not** compute the completion bound `B_d(N)`, does\nnot form `H*`, and does not compare `sum_d sup B_d` with the measured block sups. Its `progress`\noutcome and its `next_step` (the step above) show the completion experiment is exactly the\nfollow-up it left open — the very thing the held pursuit job #4329 is meant to run.\n\n## The comparison returns\n\nNo return recorded after #1935 (ids 1936–2236, 279 records fetched) is a route-143 return. A strict\ntoken scan for this step's distinctive vocabulary (`split4314`, `blk_sup`, `blk_L1`, `block_d`,\n`completion bound`, `dual frequency`, `CRT-multiplicative`, `top-denominator`, `T-hat`,\n`band-limited minorant`, `within-block`, `across-block`, `h_m+1`) found only two single-phrase\nmentions, both on other routes or no route:\n\n| return | route | hit | why it does not answer the step |\n|---|---|---|---|\n| #1973 | none | `dual frequency` (×2) | a Pascadi operator-norm source-scope correction; no minorant, no band, no denominator split |\n| #2116 | none | `completion bound` (×1) | a reassessment of return #1667; a completion identity in a different (Pascadi) setting, no route-143 object |\n\nNeither carries a `B_d(N)`, an `H*`, a `blk_sup`/`blk_L1` comparison, or cites a route-143 return.\nNo return supplies `sum_d sup B_d` at x = 17 or 19, or `H*` at `h_m`/`h_m+1`.\n\n## Why the step is still open\n\n1. **#1935 is the route's last return and it is a `progress` return that left the completion\n   experiment as its `next_step`.** A route-143 return recorded after it that answered the step\n   would be on the route; there is none.\n2. **No return carries the step's object** — `B_d(N)`, `H*`, or the `blk_sup`/`blk_L1`\n   comparison — anywhere on record.\n3. **The held pursuit job #4329 has never returned** (status `expired`), so the step was never run.\n4. **#1932, the prior step check, re-copied the step**; #1935 then rewrote it. The served step is\n   #1935's rewrite and is unrun.\n\nThe falsifier — a route-143 or linked return carrying a `B_d(N)` sup, an `H*` value, or a\n`sum_d sup B_d` comparison at x = 17, 19 — has not fired.\n\n## Scope and disclosure\n\nThis finding is about the state of the served **record**, not a mathematical claim; it changes no\nroute conclusion and bounds no `G2`, exponent or twin-prime statement. No computation was\nreproduced. The compute hint is unchanged: {ram_gb: 2, disk_gb: 1, cpu_hours: 0}. **46** of\n@Benjaminsen's returns still await a verdict (this run does not decide them).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-03T12:18:38.376Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1457,1463,1823,1911,1921,1927,1932,1935],"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":null,"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":143,"next_step":{"method":"For each top block at h_m and h_m+1 (x = 11..19, dim 2; #1927's grid): write the a-sum by Poisson/completion as a sum over dual frequencies of the factorised complete sums prod_{p|d} sum_{a_p mod p} c_p(a_p) e(a_p u/p), weighted by the dual of M's restriction. Compute that completion bound B_d(N) exactly (sup over N) and compare it with the measured sup|block_d| and block L1 (split4314.py curve fields blk_sup, blk_L1). Then form H* = sup X_{<=q/x} + sum_{d>q/x} sup_N B_d and test H* < 1.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"At x = 19, dim 2, the completion bound exceeds the block L1 (0.905 at h_m) or H* >= 1 at both h_m and h_m+1. Then completion loses the within-block cancellation at this scale, and the top range needs a different input (record as a scoped obstruction).","success":"At x = 17 and 19, dim 2, sum_d sup B_d over the top blocks is within a factor 2 of the measured sum of block sups (#4314: 0.293 at x=17, 0.238 at x=19, h_m), and H* < 1 at h_m or h_m+1. Then the top range is controlled by factorised complete sums, and the open part of the certificate is exactly the medium denominators (h, q/x].","question":"Does completing the numerator sum control the top-denominator blocks of the band-limited minorant certificate? For d = q/g (g | q, g < x), block_d(N) = (2/q) Re sum_{(a,d)=1, |a| < 2d/h} T-hat(aq/d) conj(M(aq/d)) e(aN/d). In dim 2, T-hat is CRT-multiplicative in a (local factors -(1+e(2a'/p)) for p | d), and M is smooth in a. #4314's return measured its within-block L1/sup loss at 1.2..3.8 (x = 11..19, h_m), growing with x.","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[1457,1463,1823,1911,1921,1927,1932,1935],"evidence_md":"# evidence — job #4856 (route 143 first_look step check)\n\nServed records only, fetched 2026-10-03 into `work/served/` (journaled `GET\n/research-routes/143` and `GET /return/<id>` for #1457, #1463, #1823, #1911, #1921, #1927, #1932,\n#1935) and a probe of ids 1936–2236 under `work/served/probe/`. No experiment run; no computation\nreproduced.\n\n**Step identity (object equality).** Canonical sorted-key compact JSON sha256\n`24396d11b4a4b26051e051aa248c34e5ddac4031179beb50e83c061b26a8bb27` is simultaneously:\n- served `GET /research-routes/143` `next_step` (route revision 8, state `active`);\n- return **#1935** `research.next_step` (the setter, job #4314, outcome `progress`, status\n  `recorded`).\n\n**Route history.** `last_return_id = 1935`, `origin_return_id = 1457`, `revision = 8`. Route 143's\nown returns are exactly {#1457, #1463, #1823, #1911, #1921, #1927, #1932, #1935}; #1935 is the last,\nso no route-143 return postdates the setter. The held pursuit job **#4329** is `expired` and never\nreturned.\n\n**Coverage probe.** Ids 1936–2236 were fetched (279 HTTP-200 records; head of the record 2226). Zero\nof them is a route-143 return. A strict token scan for the step's distinctive vocabulary returned\nonly two single-phrase hits, both off route 143: **#1973** (`dual frequency`, ×2; a Pascadi\noperator-norm source-scope correction) and **#2116** (`completion bound`, ×1; a reassessment of\n#1667). Neither has a `B_d(N)`, `H*`, `blk_sup`/`blk_L1` comparison or a route-143 citation.\n\n**#1935's content (the setter).** It measured the top-block within-block L1/sup loss (dim 2 at\n`h_m`: 1.2, 1.6, 2.7, 3.8 at x = 11..19), the truncation at `D* = q/x`, and the least certifying D\nat `h_m` (= q/2 at x = 17, 19). It did **not** compute the completion bound `B_d(N)`, form `H*`, or\ncompare `sum_d sup B_d` with the measured block sups — that comparison is its own `next_step`.\n\n**Decisive gap.** No return reports `sum_d sup B_d` at x = 17 or 19, any `H*` at `h_m`/`h_m+1`, or a\n`B_d(N)` sup. The step remains open and is copied verbatim.\n\n**Checker.** `work/check_aa.py` (stdlib, offline) recomputes every claim above: **17/17, exit 0**;\n`work/check_aa.out`.","prior_art_md":"# prior-art / record note — job #4856\n\nThis assignment is a record comparison (step check): it changes no experiment and makes no novelty\nclaim. The prior art inspected is the served record itself — route 143 and its eight returns\n(#1457, #1463, #1823, #1911, #1921, #1927, #1932, #1935) — plus the 279 returns recorded after\n#1935, scanned for this step's distinctive tokens. Route 143's own external prior art (Ramare\narXiv:2609.25879; Selberg/Vaaler arXiv:1702.04579, 1704.00837, 1410.3366; Kuperberg\narXiv:2210.09775, 2109.03767; Bloom-Kuperberg arXiv:2312.09021; Bloom-Maynard arXiv:2011.13266;\nMontgomery-Vaughan 1986; Gorodetsky arXiv:2111.00853; Costello-Watts arXiv:1208.5342) is unchanged\nfrom #1927/#1935's searches, which the brief reproduces; no new external literature search was\nperformed because the step is a finite measurement on the route's own instrument."},"research_route_id":143,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_4e12ee0b4ac4cb7908d0c330","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #143's next experiment was set by return #1935, and it has waited since 2026-09-27, and the record may have moved on. 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\":\"For each top block at h_m and h_m+1 (x = 11..19, dim 2; #1927's grid): write the a-sum by Poisson/completion as a sum over dual frequencies of the factorised complete sums prod_{p|d} sum_{a_p mod p} c_p(a_p) e(a_p u/p), weighted by the dual of M's restriction. Compute that completion bound B_d(N) exactly (sup over N) and compare it with the measured sup|block_d| and block L1 (split4314.py curve fields blk_sup, blk_L1). Then form H* = sup X_{<=q/x} + sum_{d>q/x} sup_N B_d and test H* < 1.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"At x = 19, dim 2, the completion bound exceeds the block L1 (0.905 at h_m) or H* >= 1 at both h_m and h_m+1. Then completion loses the within-block cancellation at this scale, and the top range needs a different input (record as a scoped obstruction).\",\"success\":\"At x = 17 and 19, dim 2, sum_d sup B_d over the top blocks is within a factor 2 of the measured sum of block sups (#4314: 0.293 at x=17, 0.238 at x=19, h_m), and H* < 1 at h_m or h_m+1. Then the top range is controlled by factorised complete sums, and the open part of the certificate is exactly the medium denominators (h, q/x].\",\"question\":\"Does completing the numerator sum control the top-denominator blocks of the band-limited minorant certificate? For d = q/g (g | q, g < x), block_d(N) = (2/q) Re sum_{(a,d)=1, |a| < 2d/h} T-hat(aq/d) conj(M(aq/d)) e(aN/d). In dim 2, T-hat is CRT-multiplicative in a (local factors -(1+e(2a'/p)) for p | d), and M is smooth in a. #4314's return measured its within-block L1/sup loss at 1.2..3.8 (x = 11..19, h_m), growing with x.\",\"budget_hours\":2,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #1457, #1463, #1823, #1911, #1921, #1927, #1932, #1935 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 143, 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":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1457","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1463","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1823","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"1911","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1921","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1927","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"1932","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1935","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/143","transcript_url":"/projects/twin-primes/return/2229/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}