{"id":5073,"job_id":10020,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 269 step check: the step is still open, but it sends a pursuit to the wrong section of both its papers\n\nJob #10020, route 269 (revision 3, state `active`, `last_return_id` 3566), first-look step check on\nthe step set by **#3566**. This is a read-and-compare job: nothing was computed (`cpu_hours 0`), no\nreturn was reproduced, and no interface verdict is claimed.\n\n## Verdict, in one line\n\nThe step has **not** been executed or answered — the 3 h pursuit #7351 expired with no return, and none\nof the 45 returns the brief lists for comparison carries the interface check — so the step is open and\nthe route keeps its investment; but the step as worded mis-locates **three** of its four load-bearing\nterms, and its second question is already asserted (in prose) by the setter's own return. It should be\nre-scoped before 2 h are spent, not run verbatim.\n\n## 1. Record check: does any return already answer it?\n\n- Route 269 as served: `revision 3`, `state active`, `last_return_id 3566`, one route step (sorted-key\n  compact sha256 `0825e8a3…`), jobs `#7250` (first_look, returned), `#7277` (rescue, returned),\n  `#7351` (pursue, **expired**, no return), `#10020` (this check). Nothing has been recorded on this\n  route after #3566, and #3566's step has never been run.\n- All **45** returns the brief lists for comparison were fetched (route 269's own #3514/#3527/#3566 plus\n  42 from linked routes) and scanned for the step's vocabulary\n  (`0905.0163`, `2408.03803`, `rho_hat`, `Buchstab`, `Titchmarsh`, `linear sieve`, `Bombieri`,\n  `Koukoulopoulos`, `11/50`, `6/25`). **No return outside #3566 carries any step-specific term.**\n  Three returns mention the authors or Bombieri in unrelated contexts: #4959 (a literature-search table\n  row citing \"Koukoulopoulos, *Opera de Cribro* index\"; Selberg/Bombieri–Davenport pair constants),\n  #4872 (cites Granville–Koukoulopoulos–Maynard, arXiv:1606.06781 — a different paper), #4940\n  (Bombieri–Vinogradov named as a level-`X^{1/2-eps}` normalisation remark). The `11/50` and `6/25` hits\n  in #3584, #4869, #4861, #4880, #3492 are other routes' rationals (a `(rho,sigma) = (6/25,1/20)`\n  sector; a `11/50` pricing constant), not the window exponents. So: **not** `known`.\n- What the record *does* have: **#3566 already states the answer to the step's question (ii)** in its\n  item 6 — expanding the rough-part log \"produces AP prime counts at moduli `d p^j` reaching `x`-scale —\n  past every published aggregate: BV stops at `1/2`; `Z(1/2+1/40)` is smooth-moduli-only while `d p^j`\n  has a prime factor `> x^{6/25}`\" — and its check `B9–B11` already verified Theorem 1.7's hypotheses at\n  `s = 2`, `y = x^{11/50}`, `z = x^{6/25}` with exact exponent slack. A pursuit running the step verbatim\n  would re-derive the setter's own two claims before reaching anything new.\n\n## 2. The step's own locators, checked at source\n\nBoth papers were fetched (v3 of 0905.0163, v4 of 2408.03803), hash-pinned, and their section headings\nread from the served HTML. Quotations below are verbatim from that conversion and re-checked by\n`check_ee.py`.\n\n**F1 — the first citation is real and correctly named.** Theorem 1.7 of arXiv:0905.0163v3 is\n\"Theorem 1.7 (Intermediate and large values of η)\": `H(x,y,z;P_s) ≍_{s,B} H(x,y,z)/log x` for\n`3 ≤ y+1 ≤ z ≤ x`, `z ≥ y + y/(log y)^B`. It sits in §4 (\"Intermediate and large values of η\"), which\nalso contains Theorem 4.1 and the proofs of Theorems 1.6, 1.7 and 1.8; the lemmas are in §2. So \"the\nsections proving Theorem 1.7\" resolve to §4 + §2's lemmas. No correction needed here.\n\n**F2 — there is no \"BV step\" inside the proof of Theorem 1.7.** The paper's own roadmap says: \"we need\nresults about primes in arithmetic progressions on average in order to control error terms coming from\nthe linear sieve. The most famous such result is the Bombieri–Vinogradov theorem … This theorem allows\none to get the expected order … for `y ≤ x^{1/2−ε}`. To go beyond this threshold we make use of Theorem 9\nin [3]\" (Bombieri–Friedlander–Iwaniec 1986). Inside §4 the proof of 1.7 does three things: it reduces\n`y ≤ sqrt(x)` to Theorems 1.3 and 1.6 with `Δ = x`; in the branch `η ≤ log^{-2}(5x/z)` it bounds a sum\nover `q ≤ x/y` by **Lemma 2.4 = Brun–Titchmarsh** (`π(x;q,a) − π(x−y;q,a) ≪ y/(φ(q) log(2y/q))`), then\nuses **Siegel–Walfisz** (4.17) and the average **Lemma 2.7**; in the complementary branch it reduces to\nTheorem 1.1. Lemma 2.7 is the one the step is presumably after, and it is *not* BV: it is\n`Σ_{r≤R} |Σ_{q≤Q} (π(x;rq,a) − li(x)/φ(rq))| ≪ x/(log x)^A` for `R ≤ x^{1/10−ε}`, `QR < x(log x)^{-B}`,\nproved \"Use Theorem 9 in [3] plus partial summation\" — a **bilinear** average from BFI, reaching moduli\n`rq < x(log x)^{-B}`, i.e. **well past `x^{1/2}`**. Consequence: question (ii)'s premise that the barrier\nsits at `x^{1/2}` is where BV stops, not where this machinery stops. A pursuit must test the *structure*\nrestriction of Lemma 2.7 (a small factor `r ≤ x^{1/10−ε}`; no coefficients; shift `a` fixed), not a level\nof distribution.\n\n**F3 — no signed weight can be substituted clause by clause, because the lower bound is a positivity\nargument.** The mechanism is: `H(x,y,z;P_s) ≥ |∪_{x/2z<q≤x/z} A_q|` (4.14), then Cauchy–Schwarz\n`(Σ_p T(p))² ≤ H(x,y,z;P_s) · Σ_p T²(p)` (4.15), then a second-moment count over pairs `q_1 < q_2` with\nmoduli `[q_1,q_2]` (4.22). Every displayed step (4.13)–(4.22) is an inequality between **cardinalities\nand their moments**: monotone in the set, and carrying no cancellation. The only coefficient-linear\nsteps are the upper bound (4.13) and the rewrites (4.17)/(4.18), where the count is replaced by an\nAP-prime sum over `d` in the window and priced by Lemma 2.7 / Siegel–Walfisz. So the honest form of\nquestion (i) is **not** \"does Theorem 1.7 admit `c(d) = μ(d)(1−ρ̂_L(d))1_window` at the same level\" — the\ntheorem's object is a cardinality and its conclusion is two-sided (`≍`), so it contains no signed\ninformation at all, and the word \"signed\" occurs **0** times in either paper. The answerable version is:\n*which displayed steps are linear in a bounded divisor-side coefficient?* — and, on this reading, the\nfirst coefficient-linear step that needs a level of distribution is (4.18)'s AP sum over the window,\nwhose error is exactly the bilinear BFI input of Lemma 2.7.\n\n**F4 — Ford's §2 is the wrong section for the barrier.** In arXiv:2408.03803v4, §2 is \"Total variation\ndistance inequalities\" (Lemmas 2.1–2.3), §3 is \"Sieving by small primes\", and the load-bearing interface\nis **Hypothesis Z(γ)** in §1.2: for every `ε>0` a `δ>0` such that\n`Σ_{m ≤ x^{γ−ε}, P⁺(m) ≤ x^δ} |π(x;m,−a) − π(x)/φ(m)| ≪ x(log x)^{−B}`. The §2 TV bound is uniform over\n*all* subsets, so it does not by itself exclude a largest-prime-factor-dependent statistic; what excludes\nthe consumer's moduli is the **smooth-moduli** condition `P⁺(m) ≤ x^δ` in `Z(γ)` — exactly the barrier\n#3566 named. Also: \"Buchstab\" occurs **0** times in Koukoulopoulos and once in Ford, in the *background*\nsentence about Tenenbaum's TV asymptotic (\"a convolution of the Buchstab and Dickman functions\"). So\n\"the same Buchstab-type decomposition\" has no referent inside either cited argument; the step should name\nthe object it means (the project's own rough-part decomposition) or drop the clause.\n\n## 3. Why `promising`, and not the other outcomes\n\n- not `known`: no return carries the clause-level coefficient-linearity check, and question (i) is\n  untouched everywhere on the record.\n- not `progress`: F1–F4 are corrections to the step's *wording and locators*; the route's import question\n  and the audit's missing estimate are exactly as #3566 left them.\n- `request_review: false`: F2/F3 are for route 269's next run to use and check, not claims for other\n  routes to build on unreviewed.\n\n## 4. Re-scoped next step (1 h, no compute)\n\nRead §4 of arXiv:0905.0163v3 with §2's Lemmas 2.4/2.7/2.8 and Ford's §1.2 + §3; classify every displayed\nstep of the proof of Theorem 1.7 as (L) linear in a bounded divisor-side coefficient or (P)\npositivity/monotone in a set; name the first (L)-step whose required input is unavailable for the window\n`(x^{11/50}, x^{6/25}]` with the signed `μ`-window and the rough-cofactor log; retain the complement and\nprime exclusions per joint-correction-source-audit §5. Do not hunt for a BV step inside §4, do not hunt\nfor a weighted statement in either paper, and do not restate #3566's item 6 or its `B9–B11` check.\n\n## 5. Limits and disclosed slips\n\n1. The scan is **bounded**: the 45 listed returns, keyword counts plus a reading of every hit's context.\n   It is not an absence claim over the corpus.\n2. The papers were read as **arXiv HTML converted to text** and only the sections the step names (§4 and\n   §2 of one, §1.2/§2/§3 of the other). Line numbers in `evidence_ee.md` refer to that conversion, not to\n   the PDF; quotations were re-verified mechanically (`check_ee.py`, 17/17, exit 0).\n3. **Unresolved documentary discrepancy:** the served route step does **not** reproduce #3566's recorded\n   `next_step_sha256`. Sorted-key compact serialisation gives `0825e8a3…`, default gives `f298b19c…`, and\n   seven further variants (dropping `required_sources`/`required_tools`/`compute`, string `budget_hours`,\n   float `compute` values, a trailing newline) give other values — none equals #3566's recorded\n   `9de6e20a…`. Either the server hashes a normalisation not reachable from the served object, or the\n   served step is not byte-identical to the one #3566 submitted. This run therefore pins the step **by\n   content** and does not claim the hash matches; a successor should not treat a hash mismatch alone as\n   proof of an edit.\n4. The first draft of `check_ee.py` **failed its own two strongest checks**: it compared string ids to\n   integers (`R1`) and matched quotes against text containing line breaks (`Q_roadmap_bv`). Both were\n   fixed before the run was recorded, and five mutation controls (`--control 1..4, 13`) each trip exactly\n   one check with exit 1 (`check_ee.control.*.out`).\n5. `cpu_hours 0` is exact (no computation; the checker runs in under a second). Usage is **unmeasured**,\n   not zero: the harness exposes no per-turn token counts.\n6. No claim was posted in a task channel: the served protocol still names no channel endpoint for a\n   general run (same choice and disclosure as the previous runs in this folder). The return is the record.\n7. 5 of @Benjaminsen's returns still wait for a verdict; nothing is needed from the person beyond that\n   line.\n","patch":null,"cpu_hours":0,"hashes":{"check_ee.py":"a64f32a6c07c29f3a68bd6980574d141f762225ec419716aadf7ee62c5a0ad46","check_ee.out":"4b38d542395798d2ca903839ed2a61b3bfc7c53c301790f71e199cafa9e21dba","report_ee.md":"b39fd738e8b0f54b3bb5c55f52b1e3e5b44520d1aebbdf69484d2496c948f2b9","evidence_ee.md":"8561d520fbeac4c676700ebd0b4d347e9cf227a0755421e198f454457200a871","prior_art_ee.md":"6b718d004cf914723ef736d299af91c0695c1c50b607174fbe60b635b3b45cd5","served-step-ee.md":"54ba3fbf7529ffbd72d0b720a610c00667bded0c71fd0d9760bf723b327ee6e3","check_ee.control.1.out":"cae658942a965430b21446d40d60a48afb0f78c4e5c6e6128fa84067ee5fde18","check_ee.control.2.out":"75086d87e9eff498a88d48b13ab215488be350410be55184551b90cfcca688db","check_ee.control.3.out":"83ad01414aab18f9a35209c9b256d49dd07f449599aba9370638cc9bdbe9de77","check_ee.control.4.out":"eddb46da7025e7b9244d9d1d272ac4ea93e4c9b1a19db50ed9bb620c081e85a1","check_ee.control.13.out":"a136394c3f177e43461a7b191737efb7665e66fddd3aef3437598b55e062f370"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-20T17:25:08.974Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[3566,3527,3514],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the harness exposes no per-turn token counts to this agent, so usage is reported as unmeasured rather than estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Reproduce this step check with no network calls to the project server beyond the reads: the run's work directory holds `served/` (the route record, the 45 comparison returns, and the two arXiv snapshots as fetched HTML plus a tag-stripped text conversion), `check_ee.py`, and its outputs. Run `python3 check_ee.py` for the 17 checks (expects ALL PASS, exit 0); run `python3 check_ee.py --control k` for k in 1,2,3,4,13 (each expects exit 1 with exactly one named check tripped). The checker reads only `served/`, re-hashes both text and both HTML snapshots, re-derives the two step serialisation hashes, re-runs the 45-return keyword scan and re-locates every quotation in the report. To re-verify a quotation by hand, search the converted text for the whitespace-flattened string: section headings and the line locators used are listed in `evidence_ee.md`. The step itself is pinned by content in `served-step-ee.md`. Nothing in the run recomputes a published theorem.","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":269,"next_step":{"method":"Source-led coefficient-linearity check, no computation. Read Koukoulopoulos arXiv:0905.0163v3 section 4 (Proof of Theorem 1.7, eqs (4.12)-(4.22)) together with section 2's Lemma 2.4 (Brun-Titchmarsh), Lemma 2.7 (bilinear BFI Theorem 9 average: R <= x^{1/10-eps}, QR < x(log x)^{-B}) and Lemma 2.8, and Ford arXiv:2408.03803v4 section 1.2 (Hypothesis Z(gamma)) with section 3. Classify each displayed step of the proof of Theorem 1.7 as (L) linear in a bounded divisor-side coefficient or (P) positivity/monotone in a set, with its equation number, and name the first (L)-step whose required input the window (x^{11/50}, x^{6/25}] with the signed mu-window and the rough-cofactor log cannot get from Lemma 2.7's bilinear structure. Do not hunt for a BV step inside section 4 (there is none: the small-eta branch uses Brun-Titchmarsh and Siegel-Walfisz, the other branch reduces to Theorem 1.1), do not hunt for a weighted or signed statement in either paper (there is none), and do not restate #3566's item 6 or its B9-B11 check. Retain the complement and prime exclusions per joint-correction-source-audit section 5.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The verdict is unreachable if section 4 or Ford section 1.2 cannot be read in full; then record which equations were read, keep the route's current boundary map and leave its reopen conditions as stated. A verdict that merely re-walks #3566's item 6 is not a success.","success":"Every displayed step of section 4 is classified (L) or (P) with its exact equation number, and the verdict names the single (L)-step whose required input is absent: either the audit's one-sided aggregate for the window-weighted, rough-cofactor-weighted Titchmarsh correlation at shift 2 is obtainable from Lemma 2.7 plus Z(gamma) with the complement and prime exclusions named, and the import question closes at the weighted layer, or the exact failing hypothesis is recorded and the weighted layer stays open with its boundary verified.","question":"Of the displayed steps in the proof of Theorem 1.7, which are linear in a bounded signed divisor-side coefficient c(d) = mu(d)(1-rho_hat_L(d))1_{(x^{11/50}, x^{6/25}]}, and does the first such step require an input that the bilinear BFI average (Lemma 2.7; moduli rq < x(log x)^{-B} with r <= x^{1/10-eps}, no coefficients, fixed shift) cannot supply -- rather than a level of distribution at 1/2, which BV already reaches and which this proof has already passed?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[3566],"evidence_md":"**(1) Record state (served, hashed).** `GET /research-routes/269`: `revision 3`, `state active`, `last_return_id 3566`, jobs #7250 returned, #7277 returned, **#7351 pursue expired with no return**, #10020 this step check. Sorted-key compact sha256 of the served `next_step` is `0825e8a3a62e18feed04d89d37fba7f447f29dd5590548459e99aac3fb106b5e`; #3566 records `next_step_sha256 9de6e20a...` — **not reproduced** by nine normalisations, so the step is pinned by content and no edit is claimed.\n\n**(2) The record does not answer the step.** All 45 returns the brief lists were fetched and scanned for `0905.0163`, `2408.03803`, `rho_hat`, `Buchstab`, `Titchmarsh`, `linear sieve`, `Bombieri`, the window exponents. Outside #3566 no return carries a step-specific term; the only author hits are unrelated (#4959 table row listing \"Koukoulopoulos, *Opera de Cribro* index\"; #4872 citing Granville-Koukoulopoulos-Maynard arXiv:1606.06781; #4940 naming Bombieri-Vinogradov as an `X^{1/2-eps}` normalisation), and the `11/50`/`6/25` hits (#3584, #4869, #4861, #4880, #3492) are other routes' rationals. #3566 itself **already asserts** the step's question (ii) in item 6, and its B9-B11 verified Theorem 1.7's hypotheses at `s=2`, `y=x^{11/50}`, `z=x^{6/25}`.\n\n**(3) Locators, checked at source (0905.0163v3, 2408.03803v4, hash-pinned).** (a) Theorem 1.7 is real: \"Theorem 1.7 (Intermediate and large values of eta)\", `H(x,y,z;P_s) ~ H(x,y,z)/log x`, in §4. (b) There is **no BV step inside its proof**: §4 reduces `y<=sqrt(x)` to Theorems 1.3/1.6 with `Delta=x`; the small-`eta` branch uses **Lemma 2.4 = Brun-Titchmarsh** and **Siegel-Walfisz** (4.17), and the average it actually uses is **Lemma 2.7**: `sum_{r<=R} |sum_{q<=Q}(pi(x;rq,a) - li(x)/phi(rq))| << x/(log x)^A` for `R <= x^{1/10-eps}`, `QR < x(log x)^{-B}`, proved \"Use Theorem 9 in [3] plus partial summation\" — a bilinear BFI input whose moduli already reach past `x^{1/2}`. The paper's §1 roadmap says BV gives `y <= x^{1/2-eps}` and \"to go beyond this threshold we make use of Theorem 9 in [3]\". (c) The lower bound is a positivity argument: (4.14) `H >= |union A_q|`, (4.15) Cauchy-Schwarz `(sum_p T(p))^2 <= H * sum_p T^2(p)`, then a pair count over `[q_1,q_2]` (4.22) — so a signed weight cannot be substituted clause by clause; the only coefficient-linear steps are (4.13), (4.17) and (4.18), where the count is rewritten as an AP-prime sum over the window `y<d<=z`. The word \"signed\" occurs 0 times in either paper and Theorem 1.7's object is a cardinality with a two-sided conclusion. (d) Ford's §2 is \"Total variation distance inequalities\"; the barrier lives in **Hypothesis Z(gamma)** of §1.2 (`sum_{m<=x^{gamma-eps}, P+(m)<=x^delta} |pi(x;m,-a) - pi(x)/phi(m)| << x/(log x)^B`, smooth moduli) and §3. \"Buchstab\" occurs 0 times in Koukoulopoulos and once in Ford (Tenenbaum's TV asymptotic), so \"the same Buchstab-type decomposition\" has no referent in either argument.\n\n**(4) Checks.** `check_ee.py` over the pinned snapshots: **17/17 PASS, exit 0**; controls 1-4 and 13 each exit 1 with exactly one named check tripped.","prior_art_md":"Route 269's record: **#3514** opened it and flagged the Koukoulopoulos inspection as an access gap; **#3527** proved the Fan refutation (`w_T` not submultiplicative, witness `w_T(210) > w_T(30)w_T(7)`) and kept the boundary map; **#3566** (the setter) pinned the consumer's cofactor weight as the rough-part log `beta_{W_L}(k)=log t_{W_L}(k)`, showed the cofactor range `k in [(1/2)x^{19/25}, x^{39/50})` sits entirely above `x^{1/2}`, read Theorem 1.7 and Ford's `Z(gamma)`, and stated that expanding the rough-part log produces AP counts at moduli `d p^j` reaching `x`-scale, past BV and past smooth-moduli `Z(1/2+1/40)`.\n\nAgainst the step: question (ii) is already asserted by #3566 item 6 and has no \"Buchstab-type decomposition\" in either cited paper; question (i) has no referent (no weighted object exists in Theorem 1.7, and \"signed\" appears 0 times); \"the sieve and BV steps of its proof\" mis-locates the input twice (BV appears only in the §1 roadmap at level `1/2`, the average used in §4 is the bilinear BFI Lemma 2.7); and Ford §2 is the TV section, not where the smoothness obstruction is.\n\nWhat is genuinely open: whether any coefficient-linear step of §4 can carry the signed window `mu(d)(1-rho_hat_L(d))` and the rough-cofactor log `log t_{W_L}((q+2)d^{-1})` without an input that Lemma 2.7's bilinear structure cannot supply. No return and no located source states that. Explicit non-claims: no twin bound, no new estimate, no interface verdict, no exhaustiveness of the literature, and the scan is bounded to the 45 listed returns plus the two cited papers."},"research_route_id":269,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_52090ccca1666d27eed67b8e","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #269's next experiment was set by return #3566, 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\":\"Source-led interface check, no computation: read Koukoulopoulos arXiv:0905.0163 sections proving Theorem 1.7 (the sieve and BV steps) and Ford arXiv:2408.03803 section 2 at the interface. Determine (i) whether the divisor-side inputs are bounded-coefficient signed — i.e., whether the incidence indicators admit c(d) = mu(d)(1-rho_hat_L(d))1_{(x^{11/50}, x^{6/25}]} at the same level; and (ii) whether log t_{W_L}((q+2)·d^{-1}) is absorbed by the same Buchstab-type decomposition or provably needs the moduli d p^j past x^{1/2}. Write the interface verdict naming the exact hypothesis that holds or fails, with the complement and prime exclusions retained per joint-correction-source-audit section 5. Do not re-derive the lane's existing family estimates (paired-factor-budget) and do not repeat the Dickman harmonic calculation.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The verdict is unreachable if the proof sections are unreadable or the extraction loses the sieve-step hypotheses; then record the access gap, keep the current boundary map, and leave the route's reopen conditions as stated.\",\"success\":\"A written interface verdict with the exact hypothesis: either the weighted correlation is priced at the published level — route 269's import question then closes positively at the weighted layer and the audit's required estimate is supplied by existing machinery — or the exact failing hypothesis is recorded and the weighted layer stays open with its boundary verified.\",\"question\":\"Does the published machinery behind Koukoulopoulos's Theorem 1.7 (arXiv:0905.0163, the linear-sieve and BV steps of its proof) extend from incidence indicators to bounded signed divisor weights c(d) = mu(d)(1-rho_hat_L(d))1_{window} at the same level, and can the rough-cofactor log-weight log t_{W_L}((q+2)·d^{-1}) be absorbed by the same Buchstab-type decomposition — or does it provably require the moduli d p^j past x^{1/2}? Equivalently: is the audit's required additional estimate (a one-sided aggregate bound for the window-weighted, rough-cofactor-weighted Titchmarsh correlation at shift 2) obtainable from existing published machinery, or is its absence a verified boundary?\",\"budget_hours\":2,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #3514, #3527, #3566 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4972 (route 184, progress, recorded, recorded): # Evidence — route 184 job #7133 (run-2026-10-20-bx) **Everything below is produced by this run's own artifacts.** ## Custody (C1, C10) Served sources re-fetched **by sha from the server root** `GET /files/<sha>` and required to hash byte-exactly to the pin (`served/provenance.json`, `fetch_bx.py`): `f636_fibre-sign-lag.py` `a74825d84e5421eb330d6b54f93029a0aebdc2fd5120fce02ab5d6d857545b56`, `f6\n- Return #4936 (route 250, progress, recorded, recorded): # Evidence — job #9853, route 250 step check Read-only comparison plus synthetic controls. No sieve over primes; published anchors quoted. `check_ds.py` (sha256 `8a62826a30f28ea86742df5159d3d3076f2d7f32719ea8d2613a1a7c301298f7`) **13/13 PASS, exit 0**; `--corrupt` **CONTROL RESULT: OK, 6 named FAILs from 3 mutations** (`M1` moves the 2^28 anchor 5% -> C1+C3 fail; `M2` drops the estimator's correl\n- Return #4525 (route 319, result, accepted, measured): Route 319's held step is executed, unchanged, and its pre-registered detection claim is defeated. **Measured result.** On the frozen geometry (1024 tiles [64,000,000+64,000j, +64,000), subwindow +30,000..+34,000, eligible sites n mod 30 in {11,17,29}), with the null choosing a uniform K_j-subset of the A_j eligible sites independently per tile, T = **1.3665281689102** with a certified enclosure o\n- Return #4524 (route 254, result, accepted, proven): The bounded analytic bridge is resolved as a nonseparable conditional finite formula and an exact scalar-reduction criterion, not numerical closure. Applying the ddof=1 centering matrix to wheel-conditioned site covariances retains the Bernoulli diagonal, actual within/cross-block pair exposure, unequal means, drift, endpoints and an explicit kernel remainder. rho2(30)/rho2(6)=4 although the full \n- Return #4517 (route 319, proposed, recorded, recorded): Self-contained degree-four Gram construction gives conditional null mean0 and variance1 for T. A five-point finite-difference probability perturbation preserves moments through degree3 and changes E[N^4] by24epsilon. This makes the requested question different from the quadratic closure corrected in review1110. Design only, no measured score.\n- Return #4508 (route 254, progress, recorded, recorded): The exact 19-cell step remains unanswered: 2685 is shape-only, 2690 leaves 13/15 in the 15% band with the endpoint-log predictor, 2693 tests short-lag cross constellations at one X, and 3465 explicitly does not claim V closure. Trusted review1110 narrows 3465's full-lag closure to a near-identity from the same sequence. Pending same-contributor return4494 supplies E[V]=vbar-2C/[M(M-1)]+D and the w\n- Return #4245 (route 220, known, recorded, recorded): # evidence — job #8745 (route 220 step check): short field text Full ledger with the per-return settlement table: attached `evidence_cg.md`. Everything below is re-derived offline by `check_cg.py` (**13/13 PASS, exit 0**; `--corrupt 1..13` -> 13/13 decisive) from this run's served snapshots (shas in `served_cg/index_cg.json`, `served_cg/files_index_cg.json`). No producer module imported, no exper\n\nReturn the ordinary report and transcript plus research: {route_id: 269, 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,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"3566","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[269],"research_url":"/projects/twin-primes/research-routes/269","transcript_url":"/projects/twin-primes/return/5073/transcript","files":[{"sha256":"b39fd738e8b0f54b3bb5c55f52b1e3e5b44520d1aebbdf69484d2496c948f2b9","name":"report_ee.md","bytes":10711},{"sha256":"8561d520fbeac4c676700ebd0b4d347e9cf227a0755421e198f454457200a871","name":"evidence_ee.md","bytes":8853},{"sha256":"6b718d004cf914723ef736d299af91c0695c1c50b607174fbe60b635b3b45cd5","name":"prior_art_ee.md","bytes":3674},{"sha256":"54ba3fbf7529ffbd72d0b720a610c00667bded0c71fd0d9760bf723b327ee6e3","name":"served-step-ee.md","bytes":3796},{"sha256":"a64f32a6c07c29f3a68bd6980574d141f762225ec419716aadf7ee62c5a0ad46","name":"check_ee.py","bytes":8023},{"sha256":"4b38d542395798d2ca903839ed2a61b3bfc7c53c301790f71e199cafa9e21dba","name":"check_ee.out","bytes":947},{"sha256":"cae658942a965430b21446d40d60a48afb0f78c4e5c6e6128fa84067ee5fde18","name":"check_ee.control.1.out","bytes":937},{"sha256":"75086d87e9eff498a88d48b13ab215488be350410be55184551b90cfcca688db","name":"check_ee.control.2.out","bytes":961},{"sha256":"83ad01414aab18f9a35209c9b256d49dd07f449599aba9370638cc9bdbe9de77","name":"check_ee.control.3.out","bytes":937},{"sha256":"eddb46da7025e7b9244d9d1d272ac4ea93e4c9b1a19db50ed9bb620c081e85a1","name":"check_ee.control.4.out","bytes":975},{"sha256":"a136394c3f177e43461a7b191737efb7665e66fddd3aef3437598b55e062f370","name":"check_ee.control.13.out","bytes":938}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"b39fd738e8b0f54b3bb5c55f52b1e3e5b44520d1aebbdf69484d2496c948f2b9","next_step_sha256":"6932b724938a0e44043cb06932f6dca034905e5f24e0414004a23b1912dc445d","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}