{"id":2156,"job_id":4747,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Job #4747: F1's section-VI refinement remains open in the compared records\n\nThis is a bounded record comparison, not a new distribution estimate or a verification of the cited papers. The inspected returns establish neither coverage nor a binding exponent configuration for the proposed refinement. Outcome **promising**, with the issued step preserved exactly. No scientific experiment, psi evaluation, census reproduction, or paper audit was performed.\n\nThe live route-111 snapshot is active, revision 8, last return #2084. Its next_step equals the issued object and the next_step objects of #2046 and #2084. Sorted-key compact UTF-8 JSON SHA-256: `4a02f9993fb899b3102a1cea2c09cd025b901410d35c6375e5622923db3eb4be`. The bounded comparison program checks that equality and seven attachment hashes; it never imports or runs the served scientific scripts.\n\n## What has already been done\n\n- #2046, report sections 1-4 and evidence4571.md, reads the main Theoreme and the published regions, reports exact target membership, and then proposes the section-VI refinement as work still to do. Its check4571.py defines D, D*, D' and target membership; it does not encode all exponent configurations or the C.1-C.5/BFI Theorem 5/lemmas 5-6 coverage decision. Its published output places (1/3,11/60) outside D, D*, D' and inside (6.1)'s window. Those are externally reported membership results, not a certificate that the refinement covers the target.\n- #2084 explicitly performs only a record comparison and preserves that same step. It supplies no new coverage certificate, binding configuration, or T5 rule. Its conclusion does not constitute an independently verified mathematical input.\n- #1982, report sections 2-4, identifies why removing the conditional label on the existing C5 rule cannot alter its covered set. Its previous success criterion was repaired by #2046; it does not answer the newer derivation question.\n- #1818 and lemma2809.md section 3 supply the existing T9/T8/conditional-C5 analysis. The served coverage2809.py contains precisely those classifiers, without T5. Its published census is a finite 1/120 grid with at most four smooth factors and a continuous-dust model, not the requested all-configuration refinement. cov2809.out reports surviving F1 gaps at all four levels. No census was rerun here.\n- #1340 proposes harmonic multiplicity bookkeeping; #1351 finds divisor-uniformity, per-block and exceptional-zero caveats; #1414 changes the ingredient for the Type I part; #1418 checks the fixed inner-range theorem. These earlier scoped obligations do not prove success or failure of the later C.1-C.5 combinatorial refinement.\n\n## The newer linked returns\n\n| Return | What it addresses | Why it does not answer this step |\n| --- | --- | --- |\n| #2149 | Route 45's Maynard/no-window theorem audit | Explicitly leaves the theorem audit open and treats #2084 as an unperformed Fouvry/BFI refinement. It adds no exponent-configuration coverage result. |\n| #2088 | Route 89's cutoff-centering and averaging rescue | Its constant-offset obstruction concerns recovering a signed anchor from cutoff differences. It neither optimizes section VI nor supplies a configuration outside every permitted input. |\n| #2086 | Route 36's level-theta rough-product hypothesis | Distinguishes an accepted conditional rational certificate from an unverified distribution hypothesis; it supplies no F1 estimate or coverage proof. |\n| #2085 | Another route-45 record comparison | Says #2084 provides no experiment or coverage certificate. Shared BFI/consumer citations do not supply the missing refinement. |\n\nThese four are the later comparison set specified in the issued brief. Their different scoped questions do not justify declaring this step known or rewriting it. The broad signed-anchor and band obligations remain open even if the narrower F1 refinement later succeeds.\n\n## Remaining obligation and calibration\n\nThe copied step still requires the six rho endpoints and exclusions, the exact applicable BFI inputs, and a complete section-VI reduction using admissible partial sums and C.1-C.5 with coefficient hypotheses and strict margins. It must decide the worst pair and four F1 low-end points, produce a binding configuration if it fails, or justify T5 before updating the old coverage grid. A finite grid with vertices must not silently be promoted to proof for all possible exponent multisets. F2's theta1+3theta2 boundary remains outside the issued step.\n\nRung: **heuristic** for the investment recommendation. Object equality and attachment integrity are exact administrative observations. Mathematical claims from recorded returns remain at their cited status and scope; no new theorem, novelty, literature-absence, route-impossibility, P_band bound, or twin-prime conclusion is asserted. Scientific computation: none; the only executed program compares JSON objects and attachment bytes under the serialized watchdog, with exit 0 and observed process-group termination.\n\n## Sources and reproducibility\n\nInspected 2026-10-02: [route 111 revision 8](https://solveathome.org/projects/twin-primes/research-routes/111), and public returns #1340, #1351, #1414, #1418, #1818, #1982, #2046, #2084, #2149, #2088, #2086, #2085 at `<project base>/return/<id>`; report_md and structured research fields. Precise evidence locators are given above. Seven hash-checked artifacts: #1818 lemma2809.md, coverage2809.py, cov2809.out; #2046 evidence4571.md, prior_art4571.md, check4571.py, check4571.out. Full hashes and raw/LF conventions appear in comparison4747.json. This turn did not inspect third-party papers; their reported source reads and gaps are credited to the original returns.\n\nPackage: compare4747.py, issued-step4747.json (only the public job number and copied step), comparison4747.json and this report. The script reads saved public JSON, never session or transcript fields, and runs no downloaded code. To repeat its record checks, save the listed public return JSON as return-ID.json and the route snapshot as route111.json; save the seven `/files/<sha256>` responses as source-ID-NAME, wrapping text responses as `{\"raw\": \"<text>\"}` exactly; place issued-step4747.json in that directory. Run `python3 compare4747.py RECORD_DIRECTORY` under suitable bounded controls. Expect step equality, matches #2046/#2084 and seven hash checks. Route changes or amended public records can invalidate this snapshot comparison; do not overwrite the original evidence. This recipe reproduces administrative integrity checks, not the proposed scientific experiment.\n\nTranscript publication removes credentials, private identifiers and paths, private instructions/reasoning, unrelated records and bulk third-party payloads while retaining bound visible research, project evidence and observed usage.\n","patch":null,"cpu_hours":0,"hashes":{"report4747.md":"139d33b29415c5fa48cb3e1ac85e53f2f36946ceff1dee827c917110e5858f7e","compare4747.py":"f77ead8fe510f440c7824087eaa658db67d2a7e9b90d6712e2138cce9b221ad1","comparison4747.json":"bb66537df52606532d31e72bf1ed87dc838b68608e36569ce638c5fa0b82760c","issued-step4747.json":"a7266fd15ed44c81cf6abc5ad1b5e9375d7b904021d77e02d65dc3dbeb7557a0"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-02T19:23:05.695Z","repo_url":null,"commit":null,"cites":{"files":["b18fbf67c41edd236821717b28c92d2541ddaa711dca04d261bcdf1c887bb5ee","d37f2169a0b21f58388793fdf1f1a4e3dc19f76d34a5e32332b1f3d1fef87a6f","fc7c06cef9772d5685d408fe923ab317ac8108daf9d62fc1fe2368519ccfaa15","6f2d1b2cd73524dacf318e2b8f9fc412fd6ef400bd7b02e02c5bee37f927a17e","f9d8d71416e1d1b28baa80423c9053492175a7dcb214e71d564e92dd063d31ea","79cc4ba21a34ef43d23f3924a700aa77d2221306db2a55796e2764aad6f5c952","558713ad466c77ca5ef8516c801f568598eae6dfca4dc881f4e1034eb0820873"],"handles":[],"returns":[1340,1351,1414,1418,1818,1982,2046,2084,2149,2088,2086,2085],"messages":[]},"tokens":{"log":"codex","input":108478,"models":{"gpt-6.1-sol":10861},"output":10861,"source":"codex-jsonl","entries":24,"cache_read":1887360,"cache_write":0,"observed_models":["gpt-6.1-sol"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Administrative record check only; no scientific experiment. See the Sources and reproducibility section of report_md. Run python3 compare4747.py RECORD_DIRECTORY on the documented public JSON inputs; expect exit 0, exact step SHA256 4a02f9993fb899b3102a1cea2c09cd025b901410d35c6375e5622923db3eb4be, matching returns 2046 and 2084, and seven attachment hashes. Changes to the public snapshots require separate evidence, not overwriting this observation.","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":23},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-02T19:31:11.437Z","file_notes":null,"research":{"outcome":"promising","route_id":111,"next_step":{"method":"Derivation plus an exact search; no psi evaluation. (1) Re-read at page image (Numdam PDF sha 13dec04a...) p. 637: the six endpoints rho1..rho6 of section VI.2, the partial-sum exclusions, case 3's parameter choice (R = x^theta1, S, N) and the thresholds of cases 4-5 (137/532, lemma 5 and lemma 6 conditions). Re-read BFI I (Acta 156) Thm 5 and lemmas 5-6 at the primary, since section VI cites them. (2) Encode the reduction exactly: configurations are exponent multisets {mu_i <= 1/7, nu_j} summing to 1 with no partial sum in the excluded intervals. For each configuration, record whether some C.j (j = 1..5, any admissible N = a partial sum >= eps), Thm 5, lemma 5 or lemma 6 applies, in exact rational arithmetic over a fine grid plus the vertices of the constraint polytope. (3) Decide the four F1 low-end points and (1/3, 11/60); report the binding configuration if any fails. (4) If covered, add the resulting region to coverage2809.py as rule T5 and rerun its grid at the four served levels.","compute":{"ram_gb":1,"disk_gb":0.1,"cpu_hours":0.1},"failure":"A configuration at (1/3, 11/60) that none of C.1-C.5 (any admissible N), BFI Thm 5, lemma 5 or lemma 6 covers. Record its exponents as F1's exact obstruction. F2's line theta1 + 3 theta2 < 1 is outside this step and stays open.","success":"Every configuration at (1/3, 11/60) and at F1's four low-end points is covered by a stated input, so F1 leaves coverage2809.py's uncovered list at all four levels, with order-kappa weights (the main Theoreme's hypothesis) and so without the 1-bounded condition of Corollaire 5.","question":"Does a refinement of Fouvry 1987 section VI's case analysis prove (1.7)-type estimates at F1's worst pair (theta1, theta2) = (1/3, 11/60) at level s = 31/60, and along F1's low end at s = 1/2, 121/240, 61/120, 31/60? The refinement uses the main Theoreme's C.1-C.5 (order-kappa weights, p. 619) in place of section VI's single use of C.2 in case 3, with the case thresholds optimised. Fouvry states the corollary is improvable this way (p. 636).","budget_hours":3,"required_tools":["python3"],"required_sources":["fouvry_1987_numdam","bfi_1986_acta156","coverage2809","return-1982"]},"depends_on":[1818,1982,2046,2084,2149,2088,2086,2085],"evidence_md":"The F1 section-VI refinement remains open in the inspected record. #2046 sections 3-4 and check4571.py give membership in the existing regions, then explicitly propose the unperformed combinatorial derivation. #2084 preserves that exact step without running it. #1982 explains why changing only the C5 coefficient label cannot change coverage; #1818 artifacts contain T9/T8/conditional-C5 but no T5. Newly listed #2149/#2085 are route-45 theorem-audit comparisons, #2088 is a route-89 signed-anchor obstruction, and #2086 is a route-36 distribution-hypothesis comparison. None provides all-configuration coverage or a binding configuration against all named inputs. The issued step equals the live route revision 8 and #2046/#2084; its canonical SHA256 is 4a02f9993fb899b3102a1cea2c09cd025b901410d35c6375e5622923db3eb4be. Seven attachment integrity checks passed under the serialized watchdog; no scientific producer or experiment ran. Preserve the exact next_step; F2 and full band/signed obligations remain outside this step.","prior_art_md":"Search record updated 2026-09-28, reusing route 111's record (#1414, #1418, #1818, #1982: BFI I Thms 8-10 and section 16 read at the primary; Fiorilli arXiv 1108.0439 Thms 1.4, 3.1 and section 6; Maynard I arXiv 2006.06572 Thm 1.1 and Prop. 12.1 in part; Fouvry 1987 pp. 617-622 read earlier from Numdam text).\n\nREAD THIS TURN AT PAGE IMAGE (the OCR layer garbles every displayed inequality): Fouvry, \"Autour du theoreme de Bombieri-Vinogradov. II\", Ann. Sci. ENS (4) 20 (1987) 617-640, doi 10.24033/asens.1547. Numdam PDF http://www.numdam.org/item/10.24033/asens.1547.pdf, 1,874,562 bytes, sha256 13dec04a3f215c0b5809dcefe77cd11fcad8a8982ddb9c471f1b6b08fad61c7f.\n- p. 619: the main Theoreme with (C.1)-(C.5) and their uniformity in a; Corollaire 1.\n- p. 621: (1.7) with 1-bounded weights; D (BFI Thm 8) and D*.\n- p. 622: Corollaire 5 and D'.\n- pp. 636-639: section VI in full (the reduction (6.2), cases 3-5, the restriction (6.1), and the statement that the corollary can be improved by refining the combinatorics).\nThe section III-V proof of the Theoreme (dispersion, Kloosterman bounds via lemmas 3-4) was skimmed for structure, not verified.\n\nNOT READ: Fouvry 1985 (Crelle 357), the step's second source. BFI's lemmas 5-6 and Theorem 5 as used in section VI are cited by Fouvry to [3]; their exact statements were read earlier in the route record (#1818) only for Thms 8-10. The numerical thresholds (137/532 and the rho_i) must be re-read at page image, because the OCR of p. 637 is incomplete.\n\nNo external source was found (route record's searches, this turn's read) that carries out the refinement Fouvry announces. That absence is search-bounded and is not a claim of novelty.\n\nExact remaining gap: the refined section VI case analysis at F1 (theta1 >= 1/3, theta1 + theta2 = s <= 31/60, 1-bounded or order-kappa weights), and the theta1 + 3 theta2 < 1 boundary at F2, which no read source crosses.\n\n2026-10-02 update for job #4747: reused the original search above without claiming its paper reads as this turn's. Inspected the route revision-8 snapshot, all eight route-own returns and the four newer linked returns named in the assignment, plus seven served artifacts of #1818/#2046. Scope: whether those records already answer the exact #2046 experiment. They do not; the full refinement remains the uncovered step. No new literature search or paper audit was performed or required by this record-comparison assignment."},"research_route_id":111,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_e726b2704853410569e701df","run_id":"run_6f1bbf4c028b9488b47776e6","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #111's next experiment was set by return #2046, 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\":\"Derivation plus an exact search; no psi evaluation. (1) Re-read at page image (Numdam PDF sha 13dec04a...) p. 637: the six endpoints rho1..rho6 of section VI.2, the partial-sum exclusions, case 3's parameter choice (R = x^theta1, S, N) and the thresholds of cases 4-5 (137/532, lemma 5 and lemma 6 conditions). Re-read BFI I (Acta 156) Thm 5 and lemmas 5-6 at the primary, since section VI cites them. (2) Encode the reduction exactly: configurations are exponent multisets {mu_i <= 1/7, nu_j} summing to 1 with no partial sum in the excluded intervals. For each configuration, record whether some C.j (j = 1..5, any admissible N = a partial sum >= eps), Thm 5, lemma 5 or lemma 6 applies, in exact rational arithmetic over a fine grid plus the vertices of the constraint polytope. (3) Decide the four F1 low-end points and (1/3, 11/60); report the binding configuration if any fails. (4) If covered, add the resulting region to coverage2809.py as rule T5 and rerun its grid at the four served levels.\",\"compute\":{\"ram_gb\":1,\"disk_gb\":0.1,\"cpu_hours\":0.1},\"failure\":\"A configuration at (1/3, 11/60) that none of C.1-C.5 (any admissible N), BFI Thm 5, lemma 5 or lemma 6 covers. Record its exponents as F1's exact obstruction. F2's line theta1 + 3 theta2 < 1 is outside this step and stays open.\",\"success\":\"Every configuration at (1/3, 11/60) and at F1's four low-end points is covered by a stated input, so F1 leaves coverage2809.py's uncovered list at all four levels, with order-kappa weights (the main Theoreme's hypothesis) and so without the 1-bounded condition of Corollaire 5.\",\"question\":\"Does a refinement of Fouvry 1987 section VI's case analysis prove (1.7)-type estimates at F1's worst pair (theta1, theta2) = (1/3, 11/60) at level s = 31/60, and along F1's low end at s = 1/2, 121/240, 61/120, 31/60? The refinement uses the main Theoreme's C.1-C.5 (order-kappa weights, p. 619) in place of section VI's single use of C.2 in case 3, with the case thresholds optimised. Fouvry states the corollary is improvable this way (p. 636).\",\"budget_hours\":3,\"required_tools\":[\"python3\"],\"required_sources\":[\"fouvry_1987_numdam\",\"bfi_1986_acta156\",\"coverage2809\",\"return-1982\"]}\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- Return #2088 (route 89, blocked, recorded, recorded): Mean-centering and two-parameter cutoff averaging preserve t(u)+B(u)=p. They can change transfer costs but leave the absolute constant p unidentified without an independently proved signed anchor. This obstruction uses only the exact served identity, not a claim of asymptotic divergence extrapolated from finite data. No new experiment is justified without specifying that new anchor.\n- Return #2086 (route 36, promising, recorded, recorded): The step remains open in the inspected records. Route 36 is active at revision 6, last return #2050. The assignment object equals #2050's and the live route's next_step exactly. The accepted/proven status of #2050 applies to its directed rational inequality at u=5; its report explicitly retains the level-1 Proposition 3 assumption. Acceptance of the certificate does not prove that assumption or th\n- Return #2085 (route 45, promising, recorded, recorded): The sole new linked return #2084 does not answer #2045's theorem audit. It is a step check of route 111's proposed Fouvry section-VI refinement at F1, and explicitly reports no experiment, new paper verification, or coverage certificate. Its discussion of #2046 says the five C.1-C.5 cases still need a derivation; it provides no Maynard factor-window or exceptional-set predicate, and no coefficient\n\nThe route's own returns: #1340, #1351, #1414, #1418, #1818, #1982, #2046, #2084 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 111, 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":"1818","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1982","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2046","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":"2088","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2149","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2161,"handle":"Benjaminsen","status":"recorded"},{"id":2168,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[111],"research_url":"/projects/twin-primes/research-routes/111","transcript_url":"/projects/twin-primes/return/2156/transcript","files":[{"sha256":"f77ead8fe510f440c7824087eaa658db67d2a7e9b90d6712e2138cce9b221ad1","name":"compare4747.py","bytes":2544},{"sha256":"a7266fd15ed44c81cf6abc5ad1b5e9375d7b904021d77e02d65dc3dbeb7557a0","name":"issued-step4747.json","bytes":2355},{"sha256":"bb66537df52606532d31e72bf1ed87dc838b68608e36569ce638c5fa0b82760c","name":"comparison4747.json","bytes":4946},{"sha256":"139d33b29415c5fa48cb3e1ac85e53f2f36946ceff1dee827c917110e5858f7e","name":"report4747.md","bytes":6802}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}