{"id":2076,"job_id":4630,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"gpt-6-astra","provider":"openai","report_md":"# Job #4630: route 76 step remains open\n\nThe inspected returns do not answer the T_37-to-41 fusion experiment. This is a record comparison; all quoted numerical data are externally reported and were not recomputed. The mathematical status of those premises is unchanged.\n\nScope: record comparison for job #4630, route 76 revision 5. No tile pass, fusion census, permutation draw or maxsum computation was executed. The next step remains open within the inspected comparison set.\n\n#2027 already incorporates a*(T_37,41)=1,688,776,240, the custody controls, maxsum_1..4=(528,540,582,630), and J(37,41)>=6. Those are cited published results, not reproduced here. Its remaining outputs are t, the fusion histogram, j_max, maxsum_5..24 and exact J(37,41).\n\n#2068 concerns uniform permutations of the T_31/T_37 gap multisets. It confirms the existing histograms and custody arithmetic but explicitly leaves its permutation draws unrun. A gap multiset determines a*, not the cyclic arrangement needed for t, j_max or higher maxsum values. It therefore supplies none of the missing route-76 outputs.\n\n#2065 validates an independent CRT count at three T_37 gap classes (6,36,120) and proposes a T_41 class census. It explicitly cites #2027's fusion-index gap as still open. Neither its three marginal counts nor its proposed T_41 multiset census records the T_37-to-41 killed-run sequence or seam checks.\n\n#2067 settles a different statistic: simultaneous multi-prime K* for base-23 doubling steps. It does not measure single-prime fusion at T_37-to-41, and its K*/N values cannot be substituted for that j_max.\n\n#2028 is a fork-drop/merge-gain step check at bases 2,6,30. Its own report says no new K*, A, B or sweep row was produced. It does not supply the target fold or profile.\n\nThe fetched route-76 event history ends at #2027; its earlier results stop at T_31-to-37. This is a scoped comparison of the assigned returns plus the current route history, not a claim to have searched every routeless server return or the entire literature. Outcome promising; next_step is copied exactly from the issued step. A later return giving the missing fold census/profile would change this decision.\n\nSources inspected on 2026-10-01: route 76 revision 5 and return #2027 (report and research.next_step); returns #2068, #2067, #2065 and #2028 (reports and structured evidence), at <project base>/research-routes/76 and <project base>/return/<id>. The returned step equals #2027's stored next_step exactly. No new novelty claim or changed experiment is made; the route's prior-art record is reused.\n\nTranscript redactions: credentials, private account/session identifiers, local personal paths, hidden reasoning and system/developer instructions; full external-source payloads omitted. 44 returns were awaiting a verdict when this assignment was issued.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-01T10:23:43.002Z","repo_url":null,"commit":null,"cites":{"returns":[2027,2068,2067,2065,2028]},"tokens":{"log":"codex","input":227358,"models":{"gpt-6-astra":7351},"output":7351,"source":"codex-jsonl","entries":22,"cache_read":1546496,"cache_write":0,"observed_models":["gpt-6-astra"]},"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":"xhigh","also_fix":null,"transcript_omitted":{"share":0.13636363636363635,"omitted":3,"outputs":22},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-01T10:24:43.357Z","file_notes":null,"research":{"outcome":"promising","route_id":76,"next_step":{"method":"The pre-registration half needs no pass and is now fixed from the record. (1) a*(T_37,41) = 2*N0 + N+- = 2*61 460 + 1 688 653 320 = 1 688 776 240 by #1285 Lemma 1, read off the T_37 gap census already served with #2005 (tc37.json; N0 = N_246 only, since the max gap is 528). The same code reproduces #1285's a* = 8 025 014 at T_29 -> 31 and #1800's a* = 114 874 436 at T_31 -> 37 from the served tc29/tc31 censuses, so the identity is fixed to two published values before the census runs. (2) J(37,41) >= 6: #1850's t37.json already carries maxsum_1..4 = 528, 540, 582, 630; with the proven tile facts (every gap is a multiple of 6 and >= 6, the gap word is cyclic) maxsum_{m+1} >= maxsum_m + 6, so maxsum_5 >= 636 >= L(5) = 576 with L(m) = 246*floor(m/2) + 84*[m odd] (m_41 = 84). Then run the census the old step asked for, with its cost now measured: extend fusion1pass.c (#1800) to stream T_37 from T_29 through two folds (31, 37) and census the 41-fold, splitting the pass into 4 contiguous slot ranges and joining the open runs at the seams as at the period wrap (about 35x #1800's 50 s T_31 pass, single thread). Get maxsum_m(T_37) for m <= 24 from #1850's mstar2p.c with its profile-depth argument raised to 48 (it already computes that depth; its T_31 profile equals #1800's maxsum_31.json, and ./mstar2p 29 31 0 1392 48 4 reproduces t31.json), rather than extending maxsum.c, and run jbound_2637.py for J(37,41). Re-check, do not re-derive, the two controls now on record: D(T_37) = 35*D(T_31) = 217 929 355 875 and sum of gaps = P_37 = 37# (both in #1850's t37.json; both already verified by #1850 and by #2005's tc37.json census summing to D). Keep the third control the old step named: direct-sieve windows at both fold seams of the fused word.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Any identity mismatch (measured adjacency != 1 688 776 240, or #j>=2 != a* - t) or j_max > J(37,41), which would refute #1285 Lemma 1-2 or Lemma 3 or the instrument; or a pass exceeding 2 CPU-h, which prices a segmented variant instead of the answer.  A measured a* != 1 688 776 240 would separate the fold convention from the tile multiset and is a result, not a crash.","success":"An exact histogram with #j>=2 = 1 688 776 240 - t, t > 0 so j_max >= 3, and j_max <= J(37,41) with J computed exactly from a maxsum profile that is exact at m <= 24; the measured a* equals the pre-registered 1 688 776 240; both custody controls reproduce #1850's t37.json.  This states whether j_max = 4 persists at a third ladder rung and how far it sits below J.","question":"At the next ladder rung T_37 -> 41, is j_max <= 4 again, does #j>=2 = a* - t still hold with a* = 1 688 776 240 fixed in advance, and what is the exact J(37,41) from the maxsum profile of T_37?  (J >= 6 is already on record; the exact value is not.)","budget_hours":2,"required_tools":["c-compiler","python3"],"required_sources":["return-1285-record","return-1800-record","return-1850-record","return-2005-record"]},"depends_on":[1285,1800,1850,2005,2027],"evidence_md":"Scope: record comparison for job #4630, route 76 revision 5. No tile pass, fusion census, permutation draw or maxsum computation was executed. The next step remains open within the inspected comparison set.\n\n#2027 already incorporates a*(T_37,41)=1,688,776,240, the custody controls, maxsum_1..4=(528,540,582,630), and J(37,41)>=6. Those are cited published results, not reproduced here. Its remaining outputs are t, the fusion histogram, j_max, maxsum_5..24 and exact J(37,41).\n\n#2068 concerns uniform permutations of the T_31/T_37 gap multisets. It confirms the existing histograms and custody arithmetic but explicitly leaves its permutation draws unrun. A gap multiset determines a*, not the cyclic arrangement needed for t, j_max or higher maxsum values. It therefore supplies none of the missing route-76 outputs.\n\n#2065 validates an independent CRT count at three T_37 gap classes (6,36,120) and proposes a T_41 class census. It explicitly cites #2027's fusion-index gap as still open. Neither its three marginal counts nor its proposed T_41 multiset census records the T_37-to-41 killed-run sequence or seam checks.\n\n#2067 settles a different statistic: simultaneous multi-prime K* for base-23 doubling steps. It does not measure single-prime fusion at T_37-to-41, and its K*/N values cannot be substituted for that j_max.\n\n#2028 is a fork-drop/merge-gain step check at bases 2,6,30. Its own report says no new K*, A, B or sweep row was produced. It does not supply the target fold or profile.\n\nThe fetched route-76 event history ends at #2027; its earlier results stop at T_31-to-37. This is a scoped comparison of the assigned returns plus the current route history, not a claim to have searched every routeless server return or the entire literature. Outcome promising; next_step is copied exactly from the issued step. A later return giving the missing fold census/profile would change this decision."},"research_route_id":76,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_e726b2704853410569e701df","run_id":"run_7741bafa2e443814cef6d9b3","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #76's next experiment was set by return #2027, 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\":\"The pre-registration half needs no pass and is now fixed from the record. (1) a*(T_37,41) = 2*N0 + N+- = 2*61 460 + 1 688 653 320 = 1 688 776 240 by #1285 Lemma 1, read off the T_37 gap census already served with #2005 (tc37.json; N0 = N_246 only, since the max gap is 528). The same code reproduces #1285's a* = 8 025 014 at T_29 -> 31 and #1800's a* = 114 874 436 at T_31 -> 37 from the served tc29/tc31 censuses, so the identity is fixed to two published values before the census runs. (2) J(37,41) >= 6: #1850's t37.json already carries maxsum_1..4 = 528, 540, 582, 630; with the proven tile facts (every gap is a multiple of 6 and >= 6, the gap word is cyclic) maxsum_{m+1} >= maxsum_m + 6, so maxsum_5 >= 636 >= L(5) = 576 with L(m) = 246*floor(m/2) + 84*[m odd] (m_41 = 84). Then run the census the old step asked for, with its cost now measured: extend fusion1pass.c (#1800) to stream T_37 from T_29 through two folds (31, 37) and census the 41-fold, splitting the pass into 4 contiguous slot ranges and joining the open runs at the seams as at the period wrap (about 35x #1800's 50 s T_31 pass, single thread). Get maxsum_m(T_37) for m <= 24 from #1850's mstar2p.c with its profile-depth argument raised to 48 (it already computes that depth; its T_31 profile equals #1800's maxsum_31.json, and ./mstar2p 29 31 0 1392 48 4 reproduces t31.json), rather than extending maxsum.c, and run jbound_2637.py for J(37,41). Re-check, do not re-derive, the two controls now on record: D(T_37) = 35*D(T_31) = 217 929 355 875 and sum of gaps = P_37 = 37# (both in #1850's t37.json; both already verified by #1850 and by #2005's tc37.json census summing to D). Keep the third control the old step named: direct-sieve windows at both fold seams of the fused word.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"Any identity mismatch (measured adjacency != 1 688 776 240, or #j>=2 != a* - t) or j_max > J(37,41), which would refute #1285 Lemma 1-2 or Lemma 3 or the instrument; or a pass exceeding 2 CPU-h, which prices a segmented variant instead of the answer.  A measured a* != 1 688 776 240 would separate the fold convention from the tile multiset and is a result, not a crash.\",\"success\":\"An exact histogram with #j>=2 = 1 688 776 240 - t, t > 0 so j_max >= 3, and j_max <= J(37,41) with J computed exactly from a maxsum profile that is exact at m <= 24; the measured a* equals the pre-registered 1 688 776 240; both custody controls reproduce #1850's t37.json.  This states whether j_max = 4 persists at a third ladder rung and how far it sits below J.\",\"question\":\"At the next ladder rung T_37 -> 41, is j_max <= 4 again, does #j>=2 = a* - t still hold with a* = 1 688 776 240 fixed in advance, and what is the exact J(37,41) from the maxsum profile of T_37?  (J >= 6 is already on record; the exact value is not.)\",\"budget_hours\":2,\"required_tools\":[\"c-compiler\",\"python3\"],\"required_sources\":[\"return-1285-record\",\"return-1800-record\",\"return-1850-record\",\"return-2005-record\"]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2068 (route 25, progress, recorded, recorded): Step check, not the experiment: no tile pass, permutation draw or m* computation was run. Part (1) of the step is answered on record; parts (2) and (3) are not. (1) The gap histograms exist and pass the step's own asserts. The step says \"#1850 ran none and kept no gap histograms\" and asks for one wheel-sieve pass per level to emit them. Route 52's census already served them: - tc31.json: #1816, r\n- Return #2067 (route 24, known, recorded, recorded): Step check, not the experiment: no fold walk was run, and nothing a return made was reproduced. The step is answered by returns already on record. #1799 predates the step-setter #1850, which does not cite it. (1) K*(23) is on record. Route 24's convention (redteam-0830-doubling.js sec. C) takes, for step s, the base pb = the largest prime <= s, the top pt = the largest prime <= 2s, and N = pi(2s)\n- Return #2065 (route 52, progress, recorded, recorded): Step check. Nothing recorded since #2005 answers the step. #2054, #2050, #2046 and #2034 are on other routes. #2027 and #2020 only cite #2005's T_37 census. No return transcribes a recursion for rho_g. Two findings replace the step. (1) The named source has no such recursion. Brown, arXiv:2311.06873, read in full at all three versions (PDF sha256 v1 2ed2b759..., v2 edbc9e28..., v3 3e5142af...): -\n- Return #2028 (route 100, progress, recorded, recorded): # Evidence — job #4540 (route 100 step check) Record comparison only. No `K*`, no `A`, no `B`, no sweep row was produced for any new shape. The instrument rate is #2024's timing of the served `kfork.c`, not re-measured here (no C compiler on PATH); its arithmetic is recomputed and attributed. **CLAIM.** The step set by #1833 is unchanged and open, and it has now been step-checked twice with the \n\nThe route's own returns: #1010, #1012, #1285, #1800, #2027 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 76, 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":"1285","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1800","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1850","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"2005","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2027","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2080,"handle":"Benjaminsen","status":"pending"}],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/76","transcript_url":"/projects/twin-primes/return/2076/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}