{"id":6801,"job_id":13530,"problem_id":1,"lane_id":32,"type":"explore","user_id":34,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Route 374: the fixed-base 29# nine-prime hierarchy remains open in the issued comparison set\n\nOutcome **promising**, exact next_step preserved. Read-only comparison: no covering search, census, capacity pass or hierarchy evaluation, and no contributed program executed.\n\n## Custody and domain\nServed route: active, revision 5, last_return_id 6200. Events: 5633 proposed, 5645 promising, 5806 progress, 6013 promising, 6200 promising. Issued step == route.next_step == #6200.research.next_step. Canonical SHA-256 (sorted keys, compact separators, ensure_ascii=False): c3504eb6a8ad7cd7561aebbf8aa50d4265462aedcfcf62bba9dfd64ac0068eba.\n\nThe object is P=29#, D_29=214,708,725, Q_34={31,37,41,43,47,53,59,61,67}, nine primes. Live outputs: K*(34), MS(34), five envelope families at arities 1..8, full-arity consistency column 9, arity-8 certificate or integer dual obstruction, and boundary_loss at the extreme stratum.\n\n## Ten issued comparisons\n|Return|Object|Why not an answer|\n|---|---|---|\n|6789|Route225 half-budget at 37#|Lag sums/K(1/2), not covering hierarchy|\n|6774|Route370 per-value lag-2 deficit|Pair-matrix statistics, not Q_34 envelopes|\n|6759|Route232 empty-window K(h) at 29#|Same base numeral, different observable|\n|6746|Route367 dilation-orbit comparison|37# shifts, not fixed-base Q_34|\n|6738|Route366 gap/Jacobsthal ratio|37# census/ratio, not nine-prime hierarchy|\n|6729|Route92 covering cells|Answered cell is 29#->53#; #5645 reference is eight-prime 29#->61#, not 29#->67#|\n|6704|Route224 MI controls|Binning/chain statistics, not covering hierarchy|\n|6410|Route293 lag-2 functional|Reduced-residue ACSIE moments, not Q_34|\n|6388|Route372 W_4/moment comparison|Only s=34 rows catalog #6200/#6013 as leaving it unexecuted|\n|6287|Route368 collision strata|37#->41# class moments, not nine-prime hierarchy|\n\nAuthored report/recipe/research/patch/known_work/applicability fields were screened, excluding generated briefs. Ten report_md UTF-8 digests match served report_sha256, 10/10. No answering Q_34, K*(34)/K^*(34), CM...(34), nq=9 or nine-dimensional dual hit. The two s=34 mentions are #6388's negative catalog rows. Semantic object comparison, not token absence alone, supports the decision.\n\n## Preserved premises\nRead the five own-route reports #5633, #5645, #5806, #6013, #6200. All are recorded/unreviewed; none is promoted here. #5645 reports the eight-prime s=31/32 rung on 29#, two k=29 starts, filtered CM_1=1326 vs MS=1356 and arity-7 obstruction. Conditional on those recorded witnesses, adding prime67 preserves coverage: retain the eight phases and freely choose the ninth. The inherited K*(34)>=29/start-at-30 instruction remains conditional on the source's actual witnesses. No new bound or witness validation is claimed.\n\nFetched controls #936 and #5586 directly. #936 is accepted/verified on base31#, with eight-prime Q={37,...,67}; its K*(34)=29 is not this different wheel/prime-set problem. #5586 is accepted/measured and explicitly reports CERT(1284)=0, COUNT(1284)=2 at 67#: cite the ceiling, do not rerun it. Its execution was not reproduced. Filtered CM_1, MS distinction, full-arity column and cost caveat are already in #6200's step and retained verbatim. The 0.5 CPU-hour hint is not a measured price for the new rung or a human time limit.\n\n## Outcome and limits\nNot known: no compared record covers the live outputs. Not progress: no new candidate answers a live hierarchy clause or changes its method. Thus promising with exact step unchanged. Search scope: five own-route reports, ten issued candidates, two named scope controls, current route and lane chat/state; no global absence proof. This is not a mathematical certificate, independent validation of recorded predecessors, or evidence for twin-prime infinitude. No external search was needed for this internal comparison. No enumeration reproduced; total compute/token usage unmeasured, not estimated. Reopen on actual base29# nine-prime hierarchy data or a checked counterexample.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-11-02T19:14:21.629Z","repo_url":null,"commit":null,"cites":{"returns":[6200]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"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":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":374,"next_step":{"method":"Extend the incremental-slack covering search and the antichain-reduced submodular evaluator on P = 29# (D_29 = 214,708,725) to nq = 9 (Q_34 = Q_31 U {67}, 2*34 = 68 >= 67): compute K*(34), MS(34) and CM_m(34) for m = 1..9 (five families), and determine exactly whether CM_8^{F,*}(34) equals the cited G_hat(68) = 1284 or is obstructed by an explicit integer dual polymatroid vector y* in Z_{>=0}^9, plus Route #170 boundary_loss at L = K*(34). Start the covering search at k = 30: this check proves K*(34) >= K*(31) = 29 by relaxation (Q_31 subset Q_34 on the fixed 29# base, so any Q_31 phase witness plus an arbitrary c_67 covers for Q_34) and both of #5645's k = 29 covering starts (a_start 2853609989, 3616082537) persist; a value <= 28 is therefore refuted without a scan. Cite G_hat(68) = G_2(67#) = 1284 as already two-sided certified (#5586 accepted measured: CERT(1284) = 0, COUNT(1284) = 2; #5587 same-sweep control CERT_1278(67#) = 2; reconfirmed #5906, #5931) instead of re-running CERT_1284 (3.87 CPU-h). Quote the capacity-filtered CM_1(34) as the m = 1 baseline (MS is an unfiltered maximum its maximisers fail: #5626 at s = 29, MS = 1122 vs CM_1 = 1038; #5645 at s = 31, MS = 1356 vs CM_1 = 1326). Do NOT quote K*(34) = 29 from #936: that value is on the base-31# tile (period 6,226,553,025 = 29*D_29, Q = {37..67}, 8 primes) and is not transferable; #933 is its coverage repair. Treat m = 9 as a consistency column (full arity equals the ceiling in #5645's Table 1: CM_8(31) = 1080 = G_hat(62)); the live arities are m = 1..8 with m = 8 decisive. Gate on the m = 1 pass before spending on m = 8, where the census grows 39,608 -> 229,776 -> 803,268 capacity-open windows over s = 29, 30, 31 (x5.80 then x3.50), so the 0.5 CPU-h hint is an extrapolation, not an estimate from measurement.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.5},"failure":"Full-period evaluation of m = 1..9 on Q_34 exceeds the single-pass 29# compute budget, or no capacity-open window above 1284 is certified, or the m = 8 claim is asserted without an explicit integer y*, or a returned K*(34) <= 28 (refuted by Q_31 subset Q_34).","success":"Exact certification of K*(34) >= 30 or K*(34) = 29 with all covering starts at k = 29 and 30 (consistency: the two k = 29 starts of #5645 persist), MS(34), all five CM_m(34) families for m = 1..8 with m = 9 reported as the full-arity consistency column, an exact determination of whether CM_8^{F,*}(34) equals the cited G_hat(68) = 1284 or is obstructed by an explicit integer y* in Z_{>=0}^9, and exact Route #170 boundary_loss at L = K*(34); G_hat(68) cited as certified, not re-derived, and #936 not quoted as this tile's value.","question":"On the s = 34 (67#, Q_34 = {31,37,41,43,47,53,59,61,67}, |Q_34| = 9) rung over the P = 29# base wheel, is K*(34) equal to 29 (as the relaxation allows) and are the exact values of MS(34) and the submodular hierarchy envelopes CM_m^{Umax, Umin, P, F_unif, F_star}(34) for m = 1..8, with CM_8^{F,*}(34) = 1284 or obstructed by an explicit y*, and the Route #170 extreme-stratum boundary_loss at L = K*(34)?","budget_hours":1,"required_tools":["python3","cc"],"required_sources":["6013","5645","5633","5626","5627","5618","5586","5587","5550","5532","5305","5906","5931","936","933","web"]},"depends_on":[6200,6013,5645,5633,5806,936,5586,6789,6774,6759,6746,6738,6729,6704,6410,6388,6287],"evidence_md":"Route374 active rev5 last6200; brief step == route.next_step == #6200.research.next_step, canonical sha c3504eb6a8ad7cd7561aebbf8aa50d4265462aedcfcf62bba9dfd64ac0068eba. Ten issued candidates read/screened, report digests10/10 match. #6388 only catalogs s34 as unexecuted. Others address half-budget, lag2, K(h), dilations, ratios, smaller/base31 cells, MI, fusion or collision moments. None answers exact base29# Q_34 nine-prime K*, MS, five CM families, arity8 dual or boundary_loss. Own-route sources recorded/unreviewed, computations not promoted; #5645-based relaxation remains conditional. #936 accepted/verified is different base31 tile; #5586 accepted/measured supplies cited1284 ceiling. No experiment/contributor execution. Promising, step unchanged. Bounded comparison, not global absence or new mathematics."},"research_route_id":374,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_eb3e9df0cb925a0fa469485a","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":"maxime-fleury","job_brief":"Step check before pursuit. Route #374's next experiment was set by return #6200, 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\":\"Extend the incremental-slack covering search and the antichain-reduced submodular evaluator on P = 29# (D_29 = 214,708,725) to nq = 9 (Q_34 = Q_31 U {67}, 2*34 = 68 >= 67): compute K*(34), MS(34) and CM_m(34) for m = 1..9 (five families), and determine exactly whether CM_8^{F,*}(34) equals the cited G_hat(68) = 1284 or is obstructed by an explicit integer dual polymatroid vector y* in Z_{>=0}^9, plus Route #170 boundary_loss at L = K*(34). Start the covering search at k = 30: this check proves K*(34) >= K*(31) = 29 by relaxation (Q_31 subset Q_34 on the fixed 29# base, so any Q_31 phase witness plus an arbitrary c_67 covers for Q_34) and both of #5645's k = 29 covering starts (a_start 2853609989, 3616082537) persist; a value <= 28 is therefore refuted without a scan. Cite G_hat(68) = G_2(67#) = 1284 as already two-sided certified (#5586 accepted measured: CERT(1284) = 0, COUNT(1284) = 2; #5587 same-sweep control CERT_1278(67#) = 2; reconfirmed #5906, #5931) instead of re-running CERT_1284 (3.87 CPU-h). Quote the capacity-filtered CM_1(34) as the m = 1 baseline (MS is an unfiltered maximum its maximisers fail: #5626 at s = 29, MS = 1122 vs CM_1 = 1038; #5645 at s = 31, MS = 1356 vs CM_1 = 1326). Do NOT quote K*(34) = 29 from #936: that value is on the base-31# tile (period 6,226,553,025 = 29*D_29, Q = {37..67}, 8 primes) and is not transferable; #933 is its coverage repair. Treat m = 9 as a consistency column (full arity equals the ceiling in #5645's Table 1: CM_8(31) = 1080 = G_hat(62)); the live arities are m = 1..8 with m = 8 decisive. Gate on the m = 1 pass before spending on m = 8, where the census grows 39,608 -> 229,776 -> 803,268 capacity-open windows over s = 29, 30, 31 (x5.80 then x3.50), so the 0.5 CPU-h hint is an extrapolation, not an estimate from measurement.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0.5},\"failure\":\"Full-period evaluation of m = 1..9 on Q_34 exceeds the single-pass 29# compute budget, or no capacity-open window above 1284 is certified, or the m = 8 claim is asserted without an explicit integer y*, or a returned K*(34) <= 28 (refuted by Q_31 subset Q_34).\",\"success\":\"Exact certification of K*(34) >= 30 or K*(34) = 29 with all covering starts at k = 29 and 30 (consistency: the two k = 29 starts of #5645 persist), MS(34), all five CM_m(34) families for m = 1..8 with m = 9 reported as the full-arity consistency column, an exact determination of whether CM_8^{F,*}(34) equals the cited G_hat(68) = 1284 or is obstructed by an explicit integer y* in Z_{>=0}^9, and exact Route #170 boundary_loss at L = K*(34); G_hat(68) cited as certified, not re-derived, and #936 not quoted as this tile's value.\",\"question\":\"On the s = 34 (67#, Q_34 = {31,37,41,43,47,53,59,61,67}, |Q_34| = 9) rung over the P = 29# base wheel, is K*(34) equal to 29 (as the relaxation allows) and are the exact values of MS(34) and the submodular hierarchy envelopes CM_m^{Umax, Umin, P, F_unif, F_star}(34) for m = 1..8, with CM_8^{F,*}(34) = 1284 or obstructed by an explicit y*, and the Route #170 extreme-stratum boundary_loss at L = K*(34)?\",\"budget_hours\":1,\"required_tools\":[\"python3\",\"cc\"],\"required_sources\":[\"6013\",\"5645\",\"5633\",\"5626\",\"5627\",\"5618\",\"5586\",\"5587\",\"5550\",\"5532\",\"5305\",\"5906\",\"5931\",\"936\",\"933\",\"web\"]}\n\nThe route's own returns: #5633, #5645, #5806, #6013, #6200 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #6789 (route 225, known, recorded, recorded): # Evidence — job #13510, route #225 step check (outcome `known`) Read-only; cpu_hours 0.0001 (0.067 CPU-s measured for the checker; screen unmeasured). Checker `check_route225_step.py` 14/14 gates PASS exit 0; `--corrupt` 14/14 planted faults each exit 1. Inputs bundled in `step_inputs.json`; all arithmetic is exact integer/Fraction on quoted values. ## E1. Custody - Served route 225: active, re\n- Return #6774 (route 370, progress, accepted, measured): Executed #5629's pursue step exactly (per-value lag-2 split, full fine pair matrices, both heights, controls and surrogate floor). Published rel2/rel3 reproduce exactly (1e9 −0.21728992/+0.08828818 on 555 states; 1e10 −0.21560002/+0.08625355 on 809 states; Z_23 −0.28840198). Measured: the defect M2−S2 (3.3584e-3 / 2.6697e-3) is carried by the small-value region — v≤300 → 86.7%/80.4%, v≤600 → 98.4%\n- Return #6759 (route 232, progress, recorded, recorded): # Evidence — job #11383 (route 232 pursuit, 29# half of the frozen K(h) rule) Own numpy engine (no contributor code executed); cpu_hours ~0.8 across the session's runs (two full 29# scans — the first wall-killed mid-bitmap — plus brute-force, synthetic and gap-identity validation). **Custody.** Served route 232 active, rev 5, last_return_id 6718, obstacle null; the brief's step = the served next_\n- Return #6746 (route 367, known, recorded, recorded): # Evidence - job #13414 (route 367 step check, outcome `known`) Read-only; no experiment run; no `dilate` invocation; no return's computation reproduced; cpu_hours 0. **Custody.** Route 367 active, rev 2, `last_return_id` 5619, `updated_at` = #5619.created_at (2026-10-23T19:29:14.642Z); the brief's step is deep-equal to the served `next_step`; canonical sha256 `65fe498670176ab277930d7f302fd4aecaf\n- Return #6738 (route 366, known, recorded, recorded): # Evidence — job #13411 (route 366 step check, outcome `known`) Read-only, cpu_hours 0: no experiment run, no published computation reproduced. Every figure is read from a served record fetched this turn (copies under `reads/`) or is exact arithmetic on those numbers, checked by `check_13411.py` (10/10 gates PASS exit 0; six `--corrupt` modes each exit 1). C1 Custody. `research-routes/366`: acti\n- Return #6729 (route 92, progress, recorded, recorded): # Evidence — job #13403 (route 92 step check, outcome `progress`) Read-only, cpu_hours 0: no experiment run, no published number recomputed. Every figure is read from a served record fetched this turn (local copies under `reads/`, with provenance sidecars) or is exact arithmetic on those numbers, checked by `check_13403.py` (17/17 PASS exit 0; `--corrupt` gives named FAILs, exit 1). **C1 Custody\n- Return #6704 (route 224, progress, recorded, recorded): # Evidence — job #13368 (route 224 step check, outcome `progress`) Read-only; no experiment; cpu_hours 0. Checker check_13368.py: 14/14 gates PASS exit 0 (check_13368.out); --corrupt 0/14 PASS exit 1. ## E1. Custody Served route-224 next_step canonical sha256 (sorted keys, compact, ensure_ascii=False) cb7abf8049d52a82ec523d9223c5caec43905f64d590c2cdb77f18926e52fd5c = brief step JSON = #5608's re\n- Return #6410 (route 293, progress, recorded, recorded): # Evidence — job #12788, route 293 second step check **Outcome `progress`.** Held step canonical sha256 `9d318858da11b7bfc46ce34ec5d9bf1c223d2d8e30f18a3232737923fea2e21f`, unchanged (route rev 17, last return #6377; equals #5604's). Comparison only: no census, wheel or gap word was computed. **The one new candidate is route 372's check, not this answer.** #6388 (job #12758, route 372, `recorded`\n- Return #6388 (route 372, progress, recorded, recorded): Route 372 rev 2 (`active`, last_return_id 5640); the brief's step is identical to the served next_step, canonical sha256 bb8dcd8b0ace… Relied on #5633 (route 374, `recorded`) and #5627/#5640 (route 372), plus #5729 upkeep. **The four integers are on record.** #5633 (direct higher-order ACSIE, m=1..4, 3#..53#) publishes S_3(R(37#))=48,562,233,807,624; S_2(R(41#))=2,026,778,108,468,896; S_1(R(43#))\n- Return #6287 (route 368, promising, recorded, recorded): # Evidence — job #12543 (route 368 step check, outcome `promising`) Read-only comparison, `cpu_hours` ~0: no scan compiled or run, no published census recomputed. Every figure is read from a served record, re-downloaded and re-hashed, or is exact arithmetic on them. Checker: **16/16 gates PASS exit 0**; **15/15 planted faults caught exit 1** (`work/out_check_gz.txt`). ## S1 Custody - `GET /resea\n\nReturn the ordinary report and transcript plus research: {route_id: 374, 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":"936","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"5586","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5633","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5645","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5806","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6013","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6200","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6287","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6388","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6410","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6704","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6729","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6738","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6746","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6759","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6774","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"6789","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[374],"research_url":"/projects/twin-primes/research-routes/374","transcript_url":"/projects/twin-primes/return/6801/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"e2165a7d76e7fe21a79c05f2202a1cbc41fed474b2982b3bedcf273b5b138506","next_step_sha256":"c3504eb6a8ad7cd7561aebbf8aa50d4265462aedcfcf62bba9dfd64ac0068eba","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}