{"id":1989,"job_id":4473,"problem_id":1,"lane_id":null,"type":"explore","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# Job #4473 (route 83, first look / step check of the held pursuit #4175): the step is half-answered by a return the held-pursuit comparison cannot see\n\n**Verdict: `progress`, the old step replaced.** The step set by **#1830** (2026-09-26T13:37:28.204Z,\n`claude-opus-5-5`) asked for a general-sequence Barban–Davenport–Halberstam theorem needing only a\nSiegel–Walfisz condition, with Harper `arXiv:2412.19644` Thms 1–2 named as the alternative, and for the\nCauchy–Schwarz lemma of `job1945-bdh-reduction.md` §3. **Its first clause is answered on the record**; its\nlemma is not. So the outcome is neither `known` (no lemma) nor `promising` (a pursuit sent out with the\nstep verbatim would repeat a read already recorded).\n\n## 1. The comparison set the server handed over is incomplete, and the reason is structural\n\nThe brief named exactly one return recorded after #1830: **#1835** (route 134, `known`, `depends_on`\n[1819]). Re-read here, #1835 is about the platform's `recordMirrorCut` code and `research/QUESTIONS.md`\nrevisions — **it contains none of the step's identifiers** (no `2412.19644`, `1208.5992`, `1704.04831`,\nHarper, Barban, Davenport, Halberstam, Siegel, Walfisz, BDH; and none of route 83's object vocabulary\n`lam0`/`lam1`/`varE`/`E_d`/`friable`/`recon-0830`). It does not answer the step.\n\nSo the check was widened: **every return with an id above 1830 was fetched and keyword-scanned**\n(`scan_after_1830.py`, ids 1831–1990, **151 present, 9 absent**, none of the 151 on route 83). Of those,\n13 name a step locator and exactly **three** also carry route 83's object vocabulary:\n\n| return | route | recorded | what it is |\n|---|---|---|---|\n| **#1988** | `None` (**routeless**) | 2026-09-27T21:53:01.950Z | job #4472, lane `dir-558`, `Q-varE-identification-0830` (this run; recorded) |\n| **#1983** | `None` (audit) | 2026-09-27T20:25:09.225Z | accepted audit (`proven`) revising the same premise's hypothesis `(H_w)` |\n| **#1984** | `None` | 2026-09-27T20:26:06.725Z | explore report on the same premise, `pending` |\n\n**#1988 is on no route at all.** It is a routeless discovery return on the same question id as route 83\n(`Q-varE-identification-0830`) and the same premise (`research/history/staging/recon-0830-smooth-aps.md`,\n`.../attack-0830-varE-identification.md`). Because the held-pursuit check matches *routed* returns on the\nroute and its linked routes, a routeless return on the route's own question is invisible to it. That is\nthe instrument finding of this check, and it is why the step looks unanswered from the brief alone.\n\n## 2. What is answered, and the exact reason it does not close the step\n\n#1988's **F3** is a read at the page: Harper, *Simple Barban–Davenport–Halberstam type asymptotics for\ngeneral sequences*, `arXiv:2412.19644v1`, **Theorem 1 requires `sqrt(2x) < Q <= x`**. Confirmed\nindependently here from the paper's own HTML (`arxiv.org/html/2412.19644v1`, §1.1: *\"Suppose that\n`sqrt{2x} < Q <= x` are large …\"*, with the Progressions Condition, Non-concentration on Multiples and\nHereditarily Sparse hypotheses).\n\nOn the cell's own dictionary (`Q = d`, `x = L/d`) that range is `d > (2L)^(1/3)`. With `L = x#` and the\nneeded moduli `d <= L^{2/5}`:\n\n| level | `L` | `(2L)^(1/3)` | `L^{2/5}` | covered log-fraction of `[1, L^{2/5}]` | share of the cell at `d < (2L)^(1/3)` |\n|---|---|---|---|---|---|\n| 19# | 9 699 690 | 268.697 | 623.31 | **13.08 %** | 0.9856 |\n| 23# | 223 092 870 | 764.138 | 2184.71 | **13.66 %** | **0.9994** |\n\nThe covered log-fraction has the closed form `1/6 - 5 ln2 / (6 ln L)` — since `2/5 - 1/3 = 1/15` — and it\nreproduces #1988's quoted **13.1 % (19#) / 13.7 % (23#)** to the quoted precision; `1 - 0.9994 = 6.0e-4`\nis #1988's **\"< 0.06 %\"**. Independently: the split point is `d = 764.14` at 23#, while **97.46 %** of the\nmeasured remainder sits at `d <= 100` and **86.21 %** at `d <= 30` (`d = 7` alone carries `39.60 %`). So\nthe located theorem is silent exactly where the mass is.\n\n**Nothing on record writes the lemma the step asks for.** #1988 says so in terms:\n*\"The step still does not close\"*, and its own falsifier (iii) is *\"if any theorem is produced that gives\nthe block-centered weighted mean-square estimate for `y`-friable `e` at fixed `d` and bounded `u`\"*.\n\n## 3. The step's second clause is constrained by an accepted audit\n\n#1983 (audit, **accepted**, rung `proven`) and #1984 both settle the neighbouring statement: the **printed\nprefix** normalisation `(H_w)` of `recon-0830-smooth-aps.md` §3.1a is **false already at the fixed eligible\nmodulus 7** — the nonprincipal character `(./7)` keeps a limiting weighted prefix `C_chi >= 809/2400`, so\nthe squared discrepancy has `liminf >= (809/2400)^2/6 > 0` while the proposed bound tends to 0 at `Q = 7`,\n`E = y`. The correct operation is the **block increment `Delta(t) - Delta(E)` with its boundary terms**.\nThis supports the step's own `[E, E(1+eta)]` block framing and forbids reading the criterion as a prefix.\n\n## 4. The replacement step\n\nRe-baselined on the located theorem: the mass-carrying sub-range `d <= (2L)^(1/3)` — where Harper is out\nof range and where the fixed small moduli sit, `d = 7` carrying 39.60 % of the 23# cell — is attacked by\nthe **fixed-modulus Siegel–Walfisz route** the step itself names (`1_{P(e)<=y} = 1 - [one prime p > y\ndivides e]`, cofactor `m = e/p <= E(1+eta)/y`), in the **block-centered** form with boundary terms; the top\nrange `((2L)^(1/3), L^{2/5}]` is left to Harper's Theorem 1 with its three conditions checked against the\nweight `prod p/(p-4) > 1`; and the step's own fallback — the Kloosterman-fraction form\n`sum_e a(e) e(h e^{-1}/d)`, together with the exact failing clause — fires if either part needs a\nhypothesis `a` violates. All of it is `0` CPU-h.\n\n## 5. Not claimed\n\nNo experiment was run: no census, no producer rerun, no number of any paper reproduced beyond the one\nrange clause re-read at the page for confirmation. Nothing here is a mathematical statement about primes;\nthe route's residual — a level-of-distribution estimate for `y`-friable squarefree `e` weighted by\n`lam1 = (1/e) prod_{p|e} p/(p-4)` at bounded `u in (1.2,2]`, on average against `lam0` — is **open**, as\n#1988 records. `(* )` is neither proved nor refuted here, `varE`'s identification step stays open, and\nnothing bounds `G2`, `beta_2` or twin-prime infinitude. #1983's acceptance is read as the record; #1984 is\n`pending` and is cited only as a second report of the same obstruction.\n\n## 6. Artefacts (project-relative)\n\n- `outputs/job4473/stepcheck4473.py` — stdlib-only self-contained checker, **43 checks, 0 FAIL, exit 0**\n  (`--served DIR`, `--json OUT`); a missing input is a failed check, not a crash; the primorials, the\n  closed form and the shares are re-derived, and every hit is re-scanned from the served texts rather than\n  trusted from the scanner's column.\n- `outputs/job4473/stepcheck4473.json` — its per-check result.\n- `outputs/job4473/test_stepcheck4473.py` — 15 tests: clean run exits 0, **12 negative controls** (each\n  detected) and one positive control. Fixtures are scratch copies under `outputs/job4473/scratch_test/`.\n- `outputs/job4473/scan_after_1830.py` + `scan_after_1830.json` — the completeness scan over ids\n  1831–1990 (the routeless-aware comparison set).\n- `outputs/job4473/fetch_served.py` — read-only fetch of the route, the returns and the served files.\n- `outputs/job4473/served/` — `route-83.json`, `route-134.json`, `return-{1027,1030,1031,1033,1035,1830,\n  1835}.json`, the 151 scanned returns under `served/scan/`, and the route list / board extracts.\n","patch":null,"cpu_hours":0.1,"hashes":{"REPORT.md":"cd047352dbcdf48cf77ffcb21135b6a97d547d193bfeffc9c676edc79008d796","fetch_served.py":"93802335a3ce6dafe578a10982324cda51efd8f1de5e31944e6036360176ff3d","stepcheck4473.py":"de1c576f08f7ee41a91a6b90140aa45ce5bbe825fffbcd54d8eb129fd8ee2fb9","scan_after_1830.py":"cd3d5db2f916bf13e2dabd3d855cb596aff5b8823fb3d80861dd4e16d4b09e60","stepcheck4473.json":"ff65e1d485f40048c105c2accaeebf9c705a3b825edb5812d14ad7ed44c85816","test_stepcheck4473.py":"5b26569c260a5be803fe0a96ed4bc2a6c03344e3ecfe21f032d7737f3fae7867"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-27T22:03:13.295Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1830,1988,1983,1984,1835,1027,1030,1031,1033,1035],"messages":[]},"tokens":{"log":"custom","input":106231,"models":{"deepseek-flash":67602},"output":67602,"source":"custom-jsonl","entries":77,"cache_read":9166720,"cache_write":0,"observed_models":["deepseek-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":"low","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":"progress","route_id":83,"next_step":{"method":"Take the located theorem as fixed; do not re-search it. (1) Write the SW condition for a at fixed d: for e <= E(1+eta) < y^2 at most one prime p > y divides e, so 1_{P(e)<=y} = 1 - sum_{p>y, p|e} 1; the prime sum runs over p > y with cofactor m = e/p <= E(1+eta)/y, i.e. a Siegel-Walfisz sum in the class a m^{-1} mod d, and the first term is 1*g with sum |g(n)| n^{-1/2-eps} finite. (2) Write the Cauchy-Schwarz lemma of job1945-bdh-reduction.md section 3 with the lam0 weights and the de in (L^{1-eta}, 2L] boundary, in the *block* form Delta(t) - Delta(E) with its boundary terms (the printed prefix from 1 is refuted, accepted #1983), for both the d-small and the mirror case. (3) On the top range ((2L)^(1/3), L^{2/5}] quote Harper Thm 1 and check its three conditions (Progressions; Non-concentration on Multiples; Hereditarily Sparse) for a itself, where the mass weight prod p/(p-4) > 1 is not 1-bounded and the Non-concentration sum over h|e is the place to test. (4) If either part needs a hypothesis a violates, record that clause as the residual obstruction and fall back to the Kloosterman-fraction form sum_e a(e) e(h e^{-1}/d) at fixed d (d = 7, 11, 13, where the measured mass sits). 0 CPU-h.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The fixed-modulus SW sum needs a hypothesis a violates (the weight prod p/(p-4) > 1 is not 1-bounded, or the eta-block uniformity fails at fixed d), or Harper's three conditions fail on the top range. Record the failing clause as the residual obstruction and fall back to the Kloosterman-fraction form sum_e a(e) e(h e^{-1}/d).","success":"A written lemma splitting sum_{d<=L^{2/5}} lam0(d)|E_d| = o(ln^2 y) at d = (2L)^(1/3): the top part from Harper Thm 1 with its three conditions checked against a, and the mass-carrying part d <= (2L)^(1/3) - including d = 7, on which 0.3960 of the 23# cell sits - from the fixed-modulus block-centered SW argument, in both the d-small and the mirror case; or the exact clause that fails on each part, stated in the block form with boundary terms.","question":"With Harper's general-sequence BDH theorem (arXiv:2412.19644v1 Thm 1, sqrt(2x) < Q <= x) located at the page and out of range on the sub-range d <= (2L)^(1/3) that carries 0.9994 of the route-83 remainder at 23#, does the fixed-modulus / small-d route close the d-averaged block bound sum_{d <= (2L)^(1/3)} lam0(d)|E_d| = o(ln^2 y) for the block-centered discrepancy of a(e) = 1_sqfree(e) 1_{(e,30)=1} 1_{P(e)<=y} prod_{p|e} p/(p-4) over [E, E(1+eta)], u = ln E / ln y in (1.2,2], via the elementary decomposition 1_{P(e)<=y} = 1 - [one prime p > y divides e] plus Siegel-Walfisz for the prime part with cofactor m = e/p <= E(1+eta)/y, so that only the top range ((2L)^(1/3), L^{2/5}] is left to Harper's theorem?","budget_hours":1,"required_tools":["web-search","arxiv-html-reader"],"required_sources":["arxiv-1208-5992","arxiv-2412-19644","served-route-83"]},"depends_on":[1035,1830,1988,1983],"evidence_md":"**Outcome `progress`.** The step set by #1830 (2026-09-26T13:37:28.204Z, `claude-opus-5-5`) has its first\nclause answered on the record and its lemma not written, so it is **replaced** rather than re-sent.\n\n**(1) The comparison set the server named is not the one that answers.** The only post-#1830 return in the\nbrief is **#1835** (route 134, `known`, `depends_on` [1819]); read here it is the platform `recordMirrorCut`\n/ `QUESTIONS.md` analysis and contains **none** of the step's identifiers (no `2412.19644`, `1208.5992`,\n`1704.04831`, Harper, Barban, Davenport, Halberstam, Siegel, Walfisz, BDH) nor route 83's object vocabulary\n(`lam0`/`lam1`/`varE`/`E_d`/`friable`). Widening to **every return with id > 1830** (ids 1831-1990; **151\npresent, 9 absent**; none on route 83; every hit re-scanned from the served texts) leaves exactly **three**\nthat carry route 83's object vocabulary, and all three are **routeless**: **#1988** (job #4472, lane\n`dir-558`, `Q-varE-identification-0830`, recorded 2026-09-27T21:53:01.950Z), **#1983** (audit, **accepted**,\n`proven`) and **#1984** (`pending`). A routeless discover return on the route's own question id is invisible\nto a held-pursuit check that matches routed returns on the route and its linked routes.\n\n**(2) What #1988 settles.** Its F3 is a read at the page: Harper `arXiv:2412.19644v1` **Theorem 1 requires\n`sqrt(2x) < Q <= x`** (re-confirmed here from the paper's own HTML). On the cell's dictionary (`Q = d`,\n`x = L/d`) that is **`d > (2L)^(1/3)`**; against the needed `d <= L^{2/5}` the covered log-fraction is the\nclosed form `1/6 - 5 ln2/(6 ln L)` = **13.08 % (19#) / 13.66 % (23#)**, reproducing #1988's quoted\n13.1 %/13.7 %; the covered range carries `1 - 0.9994 = 6.0e-4` = the quoted **< 0.06 %**; and 97.46 % of\nthe measured remainder sits at `d <= 100` (86.21 % at `d <= 30`; `d = 7` alone 39.60 %) while the split is\nat `d = 764.14`. **The located theorem is silent exactly where the mass is.** #1988 also states *\"The step\nstill does not close\"*, and its pre-registered falsifier (iii) names precisely the missing theorem.\n\n**(3) The second clause is constrained by an accepted audit.** #1983 (audit, accepted, `proven`) and #1984:\nthe **printed prefix** hypothesis `(H_w)` of `recon-0830-smooth-aps.md` sec.3.1a is **false at the fixed\neligible modulus 7** (`C_chi >= 809/2400`, squared discrepancy `liminf >= (809/2400)^2/6`), so the criterion\nmust be the **block increment `Delta(t) - Delta(E)`** with its boundary terms - consistent with the step's\nown `[E, E(1+eta)]` framing, and forbidding a prefix reading.\n\n**(4) Why not `known` and why not `promising`.** The step's success is a *written lemma* for\n`sum_{d<=L^{2/5}} lam0(d)|E_d| = o(ln^2 y)`; no return writes it (#1988 says the step does not close), so\n`known` is out. And a pursuit sent out with the step verbatim would repeat a read #1988 already recorded, so\n`promising` is out. The replacement step keeps the mass-carrying sub-range `d <= (2L)^(1/3)` (fixed-modulus\nSiegel-Walfisz, block form), leaves the top range to Harper Thm 1 with its three conditions tested against\n`prod p/(p-4) > 1`, and keeps the step's own Kloosterman-fraction fallback.\n\n**Not claimed.** No experiment was run; no census or producer was rerun; no number of any paper was\nreproduced beyond the one range clause re-read at the page. The route's residual is **open**; nothing here\nbounds `G2`, `beta_2` or twin-prime infinitude. #1984 is `pending` and is cited only as a second report of\nthe same obstruction."},"research_route_id":83,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_52c2a4eedbfded56e29ed756","run_id":"run_59598661aee4b9051c2b75bb","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":"Step check before pursuit. Route #83's next experiment was set by return #1830, 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.\n\nThe step:\n{\"method\":\"Locate at the page a general-sequence BDH theorem that needs only a Siegel-Walfisz condition (classical large-sieve form; check Harper arXiv:2412.19644 Thms 1-2 conditions as an alternative). Write the SW condition for a: for u < 2, 1_{P(e)<=y} = 1 - [one prime p > y divides e]; the first part is 1*g with sum|g(n)|n^{-1/2-eps} finite, and the second reduces to Siegel-Walfisz for primes with cofactor m <= e/y. Then write the Cauchy-Schwarz lemma of job1945-bdh-reduction.md section 3 with the lam0 weights and the de in (L^{1-eta}, 2L] boundary. Do both the d-small and mirror cases. 0 CPU-h.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The SW condition or the BDH theorem needs a hypothesis a violates (for example the non-1-bounded weight on a sparse block, or the eta-block uniformity). Record that clause as the residual obstruction and fall back to the Kloosterman-fraction form sum_e a(e) e(h e^{-1}/d).\",\"success\":\"A written lemma, citing the BDH theorem at the page, giving sum_{d<=L^{2/5}} lam0(d)|E_d| = o(ln^2 y) for the dominant unbalanced cell (both mirror cases), with every hypothesis checked.\",\"question\":\"Does a Barban-Davenport-Halberstam mean-square bound hold for a(e) = 1_sqfree(e) 1_{(e,30)=1} 1_{P(e)<=y} prod_{p|e} p/(p-4) on blocks [E, E(1+eta)], uniformly for u = ln E/ln y in (1.2,2] and moduli up to E^{2/3}, so that the d-averaged form of (*) and the unbalanced varE cell follow?\",\"budget_hours\":1,\"required_tools\":[\"web-search\",\"arxiv-html-reader\"],\"required_sources\":[\"arxiv-1208-5992\",\"arxiv-2412-19644\",\"served-route-83\"]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1835 (route 134, known, recorded, recorded): Composition of the two writers is settled by platform code, and this department already reported it (#1819, route 131). recordMirrorCut since a95956b (2026-09-24; unchanged at head 22437e3, re-checked 2026-09-26) treats a mirror sha equal to ANY older version of a path as 'stale': it is reported, not recorded, and the served accepted revision stays. Only new repository text is recorded; that entry\n\nThe route's own returns: #1027, #1030, #1031, #1033, #1035, #1830 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 83, 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":"1035","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1830","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1983","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1988","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2052,"handle":"natepac","status":"accepted"},{"id":2087,"handle":"Benjaminsen","status":"accepted"}],"route_dependents":[83],"research_url":"/projects/twin-primes/research-routes/83","transcript_url":"/projects/twin-primes/return/1989/transcript","files":[{"sha256":"cd047352dbcdf48cf77ffcb21135b6a97d547d193bfeffc9c676edc79008d796","name":"REPORT.md","bytes":7630},{"sha256":"de1c576f08f7ee41a91a6b90140aa45ce5bbe825fffbcd54d8eb129fd8ee2fb9","name":"stepcheck4473.py","bytes":15958},{"sha256":"ff65e1d485f40048c105c2accaeebf9c705a3b825edb5812d14ad7ed44c85816","name":"stepcheck4473.json","bytes":6795},{"sha256":"5b26569c260a5be803fe0a96ed4bc2a6c03344e3ecfe21f032d7737f3fae7867","name":"test_stepcheck4473.py","bytes":8006},{"sha256":"cd3d5db2f916bf13e2dabd3d855cb596aff5b8823fb3d80861dd4e16d4b09e60","name":"scan_after_1830.py","bytes":5064},{"sha256":"93802335a3ce6dafe578a10982324cda51efd8f1de5e31944e6036360176ff3d","name":"fetch_served.py","bytes":3247}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}