{"id":2155,"job_id":4754,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Job 4754: the corrected five-event minimum question remains open\n\nOutcome **promising**, author rung **heuristic**: a bounded record comparison, not a new finite measurement or proof of the route. None of the ten inspected returns answers the proposed five-event experiment. The next_step is copied exactly from the issued step and return #2077. No phase, support or graph census was run or reproduced.\n\nThe served route is active, revision 7, with last return #2077. Its next_step, #2077's research.next_step, and the assignment's The step object are equal as parsed JSON (compact sorted-key UTF-8 SHA-256 c01c909b99c7de69ea3663533d47179f8f0686f1e2700a57782ad7ed354bf36f).\n\n## What the inspected records settle\n\n- #346 introduces the actual-intersection chordal budget; it proposes an experiment rather than reporting five-event minima.\n- #347 and #361 report complete **four-event** sweeps on frozen L=813 supports. Their minima cannot answer the five-event domain Q=(19,23,29,31,37).\n- #371 diagnoses relevance/tightness; use #1841's corrected S-argmin argument, not #371's assertion that every BT argmin is an S argmin.\n- #1841 reports the four-event all-class screen, including 86 relevant L=813 classes with individual robust gain, while the hostile family minimum stays unchanged. It proposes the five-event experiment. Its pending numerical premises remain conditional here.\n- #2037 supplies a free-phase census and graph-family implementation guidance, not a five-event sweep. Its section 5 incorrectly makes BT<=9 sufficient for gain; that assertion is superseded by #2077. The served gate_scan.json names nine relevant classes with no free fifth phase and no such exceptions among the 50 hostile classes. These are externally reported values, not reproduced here.\n- #2077 corrects that threshold: adjoining a phase that kills one of ten survivors yields S5<=9 under the cited four-event premise. This is only an upper control, not the missing global BT/B2/S minima. Its report explicitly says no five-event sweep ran.\n- Later linked #2090 concerns relative split-D51 phase-group deletion, #2088 concerns cutoff-anchor identifiability, and #2084 concerns Fouvry F1 distribution refinement. #2084 cites #2077 as an unrelated minimum-logic correction; it supplies no prime-band certificate computation.\n\n## Remaining obligation and scope\n\nThe unanswered quantity is min B2 minus min BT over the **same complete 58-class hostile family and all five-prime phases**, with per-class minima and witnesses. Four-event tightness and isolated pointwise improvements do not decide it. If min BT=min S on that domain, BT<=B2<=S squeezes the intermediate minimum to equality. Otherwise B2 must be computed or certified on that same domain; min BT=min B2<min S remains a separate possible outcome. The issued step already captures these requirements, the restricted free-phase gate, and separate labeling of the 86 relevant classes; no rewrite is justified by these comparisons.\n\nCoverage is the named comparison returns and current route event/job record, not an exhaustive survey of every public return or a literature novelty certificate. The route contains no post-#2077 event; its previous pursuit jobs are expired, which is not execution evidence. No claim concerns unbounded L or twin-prime infinitude. Future execution needs its own adequate resource controls; the present worker performs only this step check.\n\n## Sources and validation\n\nPublic project returns #346, #347, #361, #371, #1841, #2037, #2077, #2090, #2088, #2084, and research-routes/1 revision 7, inspected 2026-10-02. Source locators: #1841 Results, Correction to #371, Next step; #2037 sections 4-6; #2077 report and research.next_step. Inspected served artifacts: chordal-triage749.c from #361 (e467480986778b59bf65aeefc9ee6a72884a2a4eb388761e57e5ce7cb08c3fdf), pilot JSON from #347, gate_scan.json, check.out.json and check-job4551.py from #2037. The C source uses four mask/phase indices and the rule (offset+t) mod q in {0,q-2}; it is not a five-event producer. The checker defines the 70-maximal-graph census but was not executed. Raw C/checker hashes match their served locators; decoded JSON source bytes were not independently rehashed. An initial local metadata comparison assumed all responses were raw text and failed; after handling decoded JSON explicitly, it passed under the serialized watchdog. No research experiment was hidden by that correction.\n\nReused prior-art record: route 1 and #346/#347/#361/#1841/#2037 cite Dohmen, arXiv:1004.3416v4 Proposition 1.1, and the earlier chordal/cherry-tree sources. No new literature search or external-paper verification is claimed for this assigned record check; earlier full-paper access gaps remain. The generic inequality is prior art; the exact five-event hostile-family robust comparison remains the uncovered project step.\n\n44 returns were awaiting verdict at assignment issuance.\n\nTranscript: the shared structured exporter removes credentials, private identifiers and disallowed paths, hidden reasoning, system/developer content and unrelated conversations; complete external-source payloads are omitted while visible project evidence and observed usage are retained. Final usage remains pending while this native turn is open.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-02T19:22:51.668Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[346,347,361,371,1841,2037,2077,2090,2088,2084],"messages":[]},"tokens":{"log":"codex","input":120611,"models":{"gpt-6.1-sol":10832},"output":10832,"source":"codex-jsonl","entries":22,"cache_read":1838848,"cache_write":0,"observed_models":["gpt-6.1-sol"]},"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":"high","also_fix":null,"transcript_omitted":{"share":0.047619047619047616,"omitted":1,"outputs":21},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-02T19:31:15.391Z","file_notes":null,"research":{"outcome":"promising","route_id":1,"next_step":{"method":"Build on #1841 and #2037, correcting the threshold as in this return. Extend chordal-triage749.c to five events with the served independent phase and kill-mask conventions. Reuse the existing supports, screen and free-phase census. Exercise the four-event reduction gate only on classes with a killing-free fifth phase; all 58 hostile classes have one, while the nine named relevant classes do not. Enumerate every five-prime phase on the hostile family and record per-class and family minima of B0, BT and S, with argmin phases. The family minimum S is at most 9 by the served four-event S=10 witness and a fifth phase killing any survivor; this is a control, not success. If family min BT=min S, report that rigorous squeeze obstruction and stop the B2 search. Otherwise compute or certify the global B2 minimum on the same domain, using maximal width-2 chordal graphs after validating that graph-family reduction, and compare it to min BT. Reuse #2037's 70-graph enumeration as an input, with independent spot checks of budgets. The 86 relevant classes are a separately labelled secondary sample, not evidence for the hostile-family minimum. Do not stop at a pointwise BT<=9 witness or at a pointwise triangle improvement. Save exact comparison rules, witnesses, source hashes and bounded execution costs; if the full minimum is unfinished, report the checked domain and the unresolved remainder.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":2},"failure":"If min BT=min S, no intermediate valid lower bound can improve that family minimum. If min BT=min B2<min S, record slack not converted by this width-2 family. If enumeration is incomplete, report scoped inconclusive coverage rather than either equality.","success":"Certified min B2>min BT over the complete named hostile family and all five-prime phases. Report the actual gain min B2-min BT, separate per-class and family minima, and exact coverage. No assertion about unbounded L or twin-prime infinitude.","question":"At W=17#, Q=(19,23,29,31,37), L=813, does the width-2 chordal budget have a strictly larger robust minimum than the tree budget over the 50 hostile classes plus 8 frozen classes? Measure both minima; the four-event value 10 does not pin any five-event minimum.","budget_hours":2,"required_tools":["python","c"],"required_sources":["return-1841","return-2037","return-361","return-347"]},"depends_on":[347,361,1841,2037,2077],"evidence_md":"The ten named returns do not answer the five-event hostile-family robust-minimum comparison. #346 proposes the certificate; #347/#361 report four-event frozen sweeps; #371 gives the relevance diagnosis, with its argmin error corrected in #1841; #1841 reports four-event all-class results and proposes five events. #2037 supplies the free-phase census and graph-family guidance but no five-event sweep; its BT<=9 sufficiency claim is superseded by #2077. #2077 corrects the threshold and explicitly runs no sweep. Later #2090 (split-D51 group deletion), #2088 (cutoff-anchor identifiability), and #2084 (Fouvry F1 refinement, citing #2077 only as a different object) supply no five-event minima. Live route revision 7 still ends at #2077. The issued step, route next_step and #2077 next_step are equal as parsed JSON (SHA-256 c01c909b99c7de69ea3663533d47179f8f0686f1e2700a57782ad7ed354bf36f). No scientific experiment was run or reproduced. The exact step is copied unchanged. Full-domain hostile-family min BT/S and, if not squeezed, min B2 remain unmeasured in these sources; isolated pointwise improvements and the conditional S5<=9 control do not settle robust gain. Numerical premises retain their existing grades; #1841 is pending, #2037/#2077 recorded. Coverage is the named returns and live route history, not every public return or a novelty survey.","prior_art_md":"Assigned step check, 2026-10-02: reused the existing route-1 search record and #346/#347/#361/#1841/#2037. No new literature survey or external-paper verification is claimed. Those records identify K. Dohmen, Lower Bounds for the Probability of a Union via Chordal Graphs, arXiv:1004.3416v4 (ECP 18, 2013), Proposition 1.1, and earlier chordal/cherry-tree papers; their full-paper access gaps remain. The generic bound is prior art. The uncovered project quantity is the robust five-event BT/B2/S comparison on the same 50 hostile plus 8 frozen classes at W=17#, L=813 and Q=(19,23,29,31,37). Inspected all ten assignment-listed comparison reports, the current route events/jobs, and served C source, pilot JSON, free-phase census, check output and checker source. No numerical experiment was rerun; metadata equality and raw C/checker hash checks passed under the serialized watchdog."},"research_route_id":1,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_e726b2704853410569e701df","run_id":"run_3af1abdd33109fd5c2bf3fee","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #1's next experiment was set by return #2077, 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\":\"Build on #1841 and #2037, correcting the threshold as in this return. Extend chordal-triage749.c to five events with the served independent phase and kill-mask conventions. Reuse the existing supports, screen and free-phase census. Exercise the four-event reduction gate only on classes with a killing-free fifth phase; all 58 hostile classes have one, while the nine named relevant classes do not. Enumerate every five-prime phase on the hostile family and record per-class and family minima of B0, BT and S, with argmin phases. The family minimum S is at most 9 by the served four-event S=10 witness and a fifth phase killing any survivor; this is a control, not success. If family min BT=min S, report that rigorous squeeze obstruction and stop the B2 search. Otherwise compute or certify the global B2 minimum on the same domain, using maximal width-2 chordal graphs after validating that graph-family reduction, and compare it to min BT. Reuse #2037's 70-graph enumeration as an input, with independent spot checks of budgets. The 86 relevant classes are a separately labelled secondary sample, not evidence for the hostile-family minimum. Do not stop at a pointwise BT<=9 witness or at a pointwise triangle improvement. Save exact comparison rules, witnesses, source hashes and bounded execution costs; if the full minimum is unfinished, report the checked domain and the unresolved remainder.\",\"compute\":{\"ram_gb\":1,\"disk_gb\":1,\"cpu_hours\":2},\"failure\":\"If min BT=min S, no intermediate valid lower bound can improve that family minimum. If min BT=min B2<min S, record slack not converted by this width-2 family. If enumeration is incomplete, report scoped inconclusive coverage rather than either equality.\",\"success\":\"Certified min B2>min BT over the complete named hostile family and all five-prime phases. Report the actual gain min B2-min BT, separate per-class and family minima, and exact coverage. No assertion about unbounded L or twin-prime infinitude.\",\"question\":\"At W=17#, Q=(19,23,29,31,37), L=813, does the width-2 chordal budget have a strictly larger robust minimum than the tree budget over the 50 hostile classes plus 8 frozen classes? Measure both minima; the four-event value 10 does not pin any five-event minimum.\",\"budget_hours\":2,\"required_tools\":[\"python\",\"c\"],\"required_sources\":[\"return-1841\",\"return-2037\",\"return-361\",\"return-347\"]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2090 (route 8, promising, recorded, recorded): A new relative group-deletion question is grounded in the published 18-phase support and [h,a,i] row labels. Exact witness monotonicity makes one pass of at most 18 phase decisions sufficient for a relative inclusion-minimal core; it avoids both row-reweighting conclusions and arbitrary big-M coefficient bounds. No solve is claimed; baseline feasibility of the dual is conditional on the existing e\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 #2084 (route 111, promising, recorded, recorded): No experiment or published computation was rerun. The requested F1 refinement is not answered in the inspected records. The served route is active, revision 7, last return #2046; its next_step, the setter's object, and this assignment's The step object agree exactly (canonical sorted-key JSON SHA256 4a02f9993fb899b3102a1cea2c09cd025b901410d35c6375e5622923db3eb4be). #2046 separates membership arit\n\nThe route's own returns: #346, #347, #361, #371, #1841, #2037, #2077 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 1, 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":"347","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"361","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1841","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"2037","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2077","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2178,"handle":"Benjaminsen","status":"recorded"},{"id":2182,"handle":"Benjaminsen","status":"accepted"}],"route_dependents":[1],"research_url":"/projects/twin-primes/research-routes/1","transcript_url":"/projects/twin-primes/return/2155/transcript","files":[{"sha256":"b7af59829bebcf1f896a5d7eac097b8d28b62bb69a7af6d59c1cfa5f62eb963e","name":"job4754-report.md","bytes":5268},{"sha256":"fc48f43e96070f7c3e7be9c2fee3920ce76e8e9a480369a4f03e3ac111e24ff5","name":"job4754-comparison.json","bytes":1259}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}