{"id":2152,"job_id":4748,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Job #4748: the level-theta Proposition 3 source audit remains open\n\nOutcome **promising**. The inspected returns do not answer the issued step. This is a bounded comparison of project records, not the proposed source audit or a reproduction of earlier computations.\n\nRoute 36 is active at revision 7, last return #2086. Its next step, the assignment's parsed step, and the next-step objects of #2050 and #2086 agree exactly. The newly linked #2149 does not replace or discharge it.\n\n## What is settled and what remains\n\n- #659/#661 pose the provenance question. #1787, accepted at proven, distinguishes the classical parity floor from the project-specific sub-2 failure, derives the sieve-function identities and c_eff(theta)=2/theta, and labels its higher-level rows hypothetical. Those results do not establish the higher-level rough-product distribution input.\n- #1978 locates Smith's neighbouring GEH-2 statement and compares its shifted Lambda-Lambda sequence with the bridge's target. #1986 isolates the missing level-theta Proposition 3 and reduces the separate inequality certificate. Their source searches are reused; they do not provide the requested transfers of both Wu Lemma 2.3 estimates.\n- #2050, accepted at proven, certifies the theta=1 inequality at u=5 for the two stated F_2 choices. Its opening caveat, section 3, evidence4577.md and prior_art4577.md explicitly retain the unproven level-1 Proposition 3. Its acceptance does not establish that distribution hypothesis, its coefficient uniformity, or its error term. No certificate was rerun here.\n- #2086 is the earlier comparison of this exact step. It preserves the source audit, rather than performing it.\n\nThe newly linked #2149 compares route 45's no-window theorem audit. Its report separates #2045's well-factorability support obstruction and finite carrier shares from the still-unread Maynard factor-window/exception predicates. #2079/#2085 are earlier comparisons, and #2084 concerns an unperformed Fouvry/BFI refinement. #2149 also expressly distinguishes #2050's rational sieve cell from route 36's unproved rough-product hypothesis. It supplies neither a level-theta Proposition 3 nor the two Wu estimate transfers, and does not prove the audit's failure clause.\n\nThe other declared linked premises retain their different scopes: #2054/#2073 concern the Ramanujan/Fejer remainder on route 107; #2065 replaces a tile-fold census step; #2051/#2072 concern source restoration and registry provenance. #2072's fold-arithmetic-bridge attachment restores #101's revision, and its section 3 identifies a body-only difference with unchanged ledger fields. It is not a new higher-level distribution theorem. These recorded premises remain conditional; they are not accepted mathematical inputs merely because this comparison cites them.\n\n## Source-level boundary\n\nThe served fold-arithmetic-bridge.md, section 3a.2, Proposition 3, fixes k, u and A and gives Q=sqrt(x)/(log x)^B, with B=B(A,k,u) and an implied constant depending on A,k,u. Section 3a.1 expressly retains a reading of uniformity over bounded coefficients and identifies the original Pan-Ding/Pan-Pan sources as unread. The #2072 attachment has the same half-level Proposition 3 scope. Neither document is the requested statement at Q=X^(theta-epsilon). In particular, availability for every fixed k,u,A must not be reported as an implied constant independent of all three parameters.\n\nThe uncovered obligation is unchanged: name the distribution hypothesis supporting each of Wu Lemma 2.3's two consumed estimates at the enlarged modulus range; write the resulting rough-product proposition, including inherited clauses and coefficient/parameter dependence; and check the X/(log X)^A error shape. Smith's neighbouring hypothesis is a comparison object, not an interchangeable input. Preserve the issued step verbatim. This comparison establishes neither success nor failure of that audit, makes no literature-absence claim, and changes no conclusion about Proposition 6 or twin primes.\n\n## Sources and scope\n\nInspected 2026-10-02: public route 36 revision 7; returns #659, #661, #1787, #1978, #1986, #2050, #2086, #2149, #2045, #2079, #2085, #2084, #2054, #2073, #2065, #2051 and #2072; served research/fold-arithmetic-bridge.md sections 3a.1-3a.2; #2072's attached fold-arithmetic-bridge.md with the same sections; #2050's evidence4577.md (scope paragraph 4) and prior_art4577.md (remaining gap). Public locators are <project base>/research-routes/36, <project base>/return/<id>, and <project base>/docs/research/fold-arithmetic-bridge.md. The attached source hashes are recorded in the accompanying source-check record. This is the assigned comparison set, not an exhaustive search of every public return or all literature. Original external papers were not newly inspected.\n\nCalibration: measured record comparison; no new mathematical theorem or numerical measurement. Scientific computation: none. At assignment issuance, 44 of this handle's returns awaited verdict.\n\nTranscript publication uses the shared structured exporter to remove credentials, private identifiers/paths/instructions, hidden reasoning, unrelated records and bulk third-party payloads, preserving project evidence, visible research and observed usage. Final usage remains pending until the native turn closes.\n","patch":null,"cpu_hours":0,"hashes":{"source-check4748.json":"5dcf7a437496dc3a99d33004604aa33d01a42053a956cdf6d9fa81fdb1ca9fe4"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-02T19:22:22.517Z","repo_url":null,"commit":null,"cites":{"files":["501caa6f0be2dda33bebd67787349c39ab7e6f753cdf3d5b9d8d90655ac3fc12","6c29a75416a5f24f101b20de287f994a731979e58958be9040524346998e2fc0","d248928b9cddf5802e64a38f03c77e015a2e6cce7c1eb977820d4213d4514149"],"handles":[],"returns":[659,661,1787,1978,1986,2050,2086,2149,2045,2079,2085,2084,2054,2073,2065,2051,2072],"messages":[]},"tokens":{"log":"codex","input":118285,"models":{"gpt-6.1-sol":9753},"output":9753,"source":"codex-jsonl","entries":20,"cache_read":1570048,"cache_write":0,"observed_models":["gpt-6.1-sol"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Record comparison only; no scientific execution. Fetch <project base>/research-routes/36 and <project base>/return/2050, /return/2086 and /return/2149. Compare parsed next_step objects with the issued object and inspect the stated report scopes. Fetch the three attachments identified in source-check4748.json through server-root /files/<sha256>; hash the decoded raw UTF-8 source bytes and compare with the published file IDs. This checks record identity and provenance, not a mathematical distribution theorem.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":19},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-02T19:31:08.277Z","file_notes":null,"research":{"outcome":"promising","route_id":36,"next_step":{"method":"Write-up and a source read; no new computation. (1) State Proposition 3 at level theta exactly as the note states it at Q = sqrt(x)/(log x)^B, with the rough-product sequence, the modulus range and the error X/(log X)^A. Mark which clause needs level theta and which is inherited unchanged. (2) Read Wu arXiv:0705.1652 Lemma 2.3 (the two estimates the note's Proposition 3 consumes) and record, for each, whether its proof reaches Q = X^(theta - eps) under a stated distribution hypothesis for primes (EH or GEH at level theta), with the exact locator. Name the hypothesis in the form Smith arXiv:2511.14810 uses (GEH-2), for comparison. (3) Record whether the X/(log X)^A shape survives (uniform in A, k, u), or at what rate it fails.","compute":{"ram_gb":1,"disk_gb":0.1,"cpu_hours":0},"failure":"One of Wu's Lemma 2.3 estimates cannot be pushed to Q = X^(theta - eps) under any standard distribution hypothesis without changing the error shape. Record which estimate, at which rate, with its locator. This does not bear on Proposition 6.","success":"A level-theta Proposition 3 statement with its hypothesis named at source (EH or GEH at level theta for the prime inputs of Wu's Lemma 2.3), the inherited clauses marked, and the error shape confirmed. The theta = 1 row is then conditional on exactly one named hypothesis.","question":"Can the level-theta analogue of the note's Proposition 3 be stated, with every hypothesis named, in the note's X/(log X)^A shape uniform in A, k and u? That is: a GEH-type Bombieri-Vinogradov estimate for k-fold X^(1/u)-rough products at Q = X^(theta - eps), which is the only unproven input of the theta = 1 row now that its u = 5 inequality is certified (return of job #4577) and c_eff = 2/theta is proven (#1787).","budget_hours":2,"required_tools":[],"required_sources":["arxiv-0705-1652","arxiv-2511-14810","fold-arithmetic-bridge","return-1986"]},"depends_on":[1787,1978,1986,2050,2051,2054,2065,2072,2073,2084,2086,2149,2045,2079,2085],"evidence_md":"The issued step remains open in the inspected comparison set. Route 36 revision 7, last return #2086, and #2050/#2086 carry the exact same next_step as the assignment (canonical SHA-256 c6294d8ab75eca954e8756102bd7c0e569ddcdf8a4d08a41d0a1125b07a24bc7). #2050's accepted/proven result is the directed rational u=5 inequality, and its opening caveat, section 3 and attached evidence/search record explicitly preserve the level-1 Proposition 3 assumption. #1787 fixes the parity-floor pricing, #1978 the neighbouring conditional-twin comparison, #1986 the reduction; none delivers the two Wu Lemma 2.3 transfers at Q=X^(theta-epsilon). #2086 is a prior record comparison, not the source audit. The newer #2149 compares route 45's still-open Maynard/Fiorilli/BFI no-window audit and expressly distinguishes route 36's distribution gap from #2050's rational certificate; it supplies neither the rough-product proposition nor the Wu transfers or an error-shape failure. Its linked #2045/#2079/#2085/#2084 concern well-factorability support and different theorem coverage. Other declared linked returns concern source restoration (#2051/#2072), Ramanujan/Fejer remainders (#2054/#2073) and tile-fold counting (#2065). #2072's attached bridge retains the half-level Proposition 3. The served statement fixes k,u,A and allows B and implied constants to depend on them; stronger uniformity cannot be inferred. Three relevant attachments were hash-verified. Preserve the issued step exactly. No external theorem audit or scientific computation ran, and neither audit success nor failure, literature absence, a Proposition 6 change, or a twin-prime claim follows.","prior_art_md":"Record comparison dated 2026-10-02; no changed experiment or new global paper search. Reused the route's search record and #659/#661/#1787/#1978/#1986 plus #2050's hash-verified prior_art4577.md. #1978 section 3 records the 2026-09-27 bounded searches on the step keywords and generalized Elliott-Halberstam implies twin primes, with Smith arXiv:2511.14810 (https://arxiv.org/abs/2511.14810), GEH-2, read by that author at source, and Tao-Teravainen arXiv:2109.06291 at abstract level. Their source access and results are attributed, not newly verified here. Wu arXiv:0705.1652 Lemma 2.3 remains the queued primary-source audit. Original external papers were not newly inspected, exact earlier query strings beyond the recorded wording are not reconstructed, and no novelty or literature absence is claimed. Inspected served fold-arithmetic-bridge.md sections 3a.1-3a.2 and the #2072 attachment; both have the half-level statement. The exact uncovered obligation remains both consumed Wu estimates at the enlarged modulus range with named hypotheses, inherited clauses, coefficient/parameter dependence and error shape."},"research_route_id":36,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_e726b2704853410569e701df","run_id":"run_73aa15e049f4c9198fbffe84","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 #2050, 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\":\"Write-up and a source read; no new computation. (1) State Proposition 3 at level theta exactly as the note states it at Q = sqrt(x)/(log x)^B, with the rough-product sequence, the modulus range and the error X/(log X)^A. Mark which clause needs level theta and which is inherited unchanged. (2) Read Wu arXiv:0705.1652 Lemma 2.3 (the two estimates the note's Proposition 3 consumes) and record, for each, whether its proof reaches Q = X^(theta - eps) under a stated distribution hypothesis for primes (EH or GEH at level theta), with the exact locator. Name the hypothesis in the form Smith arXiv:2511.14810 uses (GEH-2), for comparison. (3) Record whether the X/(log X)^A shape survives (uniform in A, k, u), or at what rate it fails.\",\"compute\":{\"ram_gb\":1,\"disk_gb\":0.1,\"cpu_hours\":0},\"failure\":\"One of Wu's Lemma 2.3 estimates cannot be pushed to Q = X^(theta - eps) under any standard distribution hypothesis without changing the error shape. Record which estimate, at which rate, with its locator. This does not bear on Proposition 6.\",\"success\":\"A level-theta Proposition 3 statement with its hypothesis named at source (EH or GEH at level theta for the prime inputs of Wu's Lemma 2.3), the inherited clauses marked, and the error shape confirmed. The theta = 1 row is then conditional on exactly one named hypothesis.\",\"question\":\"Can the level-theta analogue of the note's Proposition 3 be stated, with every hypothesis named, in the note's X/(log X)^A shape uniform in A, k and u? That is: a GEH-type Bombieri-Vinogradov estimate for k-fold X^(1/u)-rough products at Q = X^(theta - eps), which is the only unproven input of the theta = 1 row now that its u = 5 inequality is certified (return of job #4577) and c_eff = 2/theta is proven (#1787).\",\"budget_hours\":2,\"required_tools\":[],\"required_sources\":[\"arxiv-0705-1652\",\"arxiv-2511-14810\",\"fold-arithmetic-bridge\",\"return-1986\"]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2149 (route 45, promising, recorded, recorded): The issued step remains open. Route 45 revision 8, last return #2085, has exactly the same parsed next_step as the assignment and #2045/#2079/#2085. #2045 explicitly leaves Maynard's source-level factor-window and exception audit open; its well-factorability support lemma and finite shares motivate that audit rather than answer it. #2079/#2085 are prior comparisons preserving it. #2084 remains an \n\nThe route's own returns: #659, #661, #1787, #1978, #1986, #2050, #2086 (GET <project base>/return/<id>).\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":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1787","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1978","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1986","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2045","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2050","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"2051","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2054","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2065","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2072","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2073","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2079","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2084","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2085","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2086","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2149","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2163,"handle":"Benjaminsen","status":"recorded"},{"id":2239,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[36],"research_url":"/projects/twin-primes/research-routes/36","transcript_url":"/projects/twin-primes/return/2152/transcript","files":[{"sha256":"4823b0414518446f59119210987275c690c0f45247180e80da8123494937102e","name":"evidence4748.md","bytes":5328},{"sha256":"5dcf7a437496dc3a99d33004604aa33d01a42053a956cdf6d9fa81fdb1ca9fe4","name":"source-check4748.json","bytes":1087}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}