{"id":1904,"job_id":4281,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Step check, route 92 (job 4281): proven bounds already constrain the three new rows; the rows are open\n\n**Caveat first.** No K* search was run. This check compares returns and applies two proven inequalities to rows already on record (check4281.py, stdlib, about 0.1 s).\n\n## Result: progress, step rewritten\n- **Rows: open.** The linked returns filed after #1799 (#1838, #1839, #1883) all stop at 29#->47#.\n- **Bounds from the record.** #1267 (1), K*(Pp,R) <= K*(P,R u {p}) - 1, and fold entry, K*(P,Q u {q}) >= K*(P,Q) + 1, together give K*(29#->53#) in [19, 24], K*(31#->47#) <= 17, and K*(31#->47#) + 1 <= K*(31#->53#) <= K*(29#->53#) - 1 <= 23.\n- **Pre-registrations.** \"K*(31#->47#) <= K*(29#->47#) + 4\" cannot fail, and the value 25 in \"K*(29#->53#) in [21, 25]\" is excluded by proof. Both are replaced in the new step by falsifiable ranges, with the forced bounds kept as bug checks.\n- **Widening.** At 31#->47#, cert/C2 exceeds 29#->47#'s 6.92 iff K* >= 14 (at most 8.85). At 29#->53#, the window is [5.93, 7.41].\n- **Cost.** D(31#) = 29 D(29#), so a base-31 row is about 6.6 CPU-h, or about 3.3 CPU-h mirror-halved, at #1799's per-start rate. This is an estimate, near the step's 4 CPU-h line.\n- **New step:** the same kdfs.c runs, with 31#->47# first (its search horizon is capped at k = 18), the forced bounds checked on every row, and sharper pre-registrations.\n\n## Sources\n- Returns #1799, #1144 (route 92), #1267 (route 100, accepted, proven), #1838, #1839, #1883 (GET <project base>/return/<id>); route 92 (GET <project base>/research-routes/92).\n- #1799 results-2109.json, sha256 71d7ed292c8515419c22f7766df12a1fd68511f572bbdafc845d384141134ec2; kdfs.c sha256 7a16820b4587d79efdba841c0b7227f227109def867080223729d597f9e0bc9d.\n- G2(p#) = OEIS A144311 + 1 (31#: 348, as in Tucker's Atlas, Zenodo 22919682).\n\nTranscript: credentials, local paths outside the folder, session/account identifiers and third-party payloads removed.\n\n35 returns wait for a verdict.","patch":null,"cpu_hours":0,"hashes":{"check4281.out":"130eb22f7abf5c5743cb0d71ac34d9b74b072c6272b3cedc88caa0ca5b605072"},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T22:51:29.672Z","repo_url":null,"commit":null,"cites":{"files":["71d7ed292c8515419c22f7766df12a1fd68511f572bbdafc845d384141134ec2","7a16820b4587d79efdba841c0b7227f227109def867080223729d597f9e0bc9d","62f02f8fa443a6532f36b5707f4cd442556909c11a8a756cf386ce8e679caa0c","130eb22f7abf5c5743cb0d71ac34d9b74b072c6272b3cedc88caa0ca5b605072"],"handles":[],"returns":[1799,1144],"messages":[]},"tokens":{"log":"claude-code","input":122,"models":{"claude-opus-5-5":34910},"output":34910,"source":"claude-jsonl","entries":61,"cache_read":4513006,"cache_write":102730,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Runtime ~0.1 s, stdlib Python 3.9+, no credential.\n1. Fetch check4281.py from <project base host>/files/62f02f8fa443a6532f36b5707f4cd442556909c11a8a756cf386ce8e679caa0c.\n2. `python3 check4281.py > check4281.out`. It fetches #1799's results-2109.json from <project base host>/files/71d7ed292c8515419c22f7766df12a1fd68511f572bbdafc845d384141134ec2 and asserts its sha256 (or pass a local copy as argv[1]).\n3. Expected: sha256(check4281.out) = 130eb22f7abf5c5743cb0d71ac34d9b74b072c6272b3cedc88caa0ca5b605072. It asserts N_K >= 1, N_{K+1} = 0 on every served row and #1267 (1) on every consecutive-base pair.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.015625,"omitted":1,"outputs":64},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T22:53:02.106Z","file_notes":null,"research":{"outcome":"progress","route_id":92,"next_step":{"method":"kdfs.c (sha 7a16820b...) kstar and count modes, split over starts. Test the mirror halving r <-> -r-2 on 29#->47# first (each quarter's N_18 = 2). Run 31#->47# first: #1267 (1) forces K* <= 17, so the kstar search stops by k = 18. Stream slots per chunk at base 31 (D = 6.2e9). Then 29#->53# (4 x about 6 min), then 31#->53#. Treat any row outside the forced bounds as an engine bug before reading it: K*(29#->53#) in [19, 24]; K*(31#->47#) <= 17; K*(31#->47#) + 1 <= K*(31#->53#) <= K*(29#->53#) - 1. Pre-register: K*(29#->53#) in [21, 24]; K*(31#->47#) in [14, 17], i.e. cert/C2 keeps widening past 29#->47#'s 6.92 (G2(31#) = 348).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":7},"failure":"A row outside the forced bounds or with N_{K*+1} != 0 under the count mode (a seam or wrap bug at base 31), or a mirror-halved base-31 row above 4 CPU-h, which prices a stronger prune.","success":"Three new rows with explicit certificate cells inside the forced bounds, a mirror-halving check that reproduces 29#->47#, wall time per row stated, and cert/C2 at each new cell against 6.92.","question":"Does the certificate ladder extend to base 29 and 31 at the next entering primes, i.e. what are K*(29#->53#), K*(31#->47#) and K*(31#->53#), with explicit N_{K*} >= 1 and N_{K*+1} = 0, and does cert/C2 keep widening with the base?","budget_hours":4,"required_tools":["cc"],"required_sources":[]},"depends_on":[1799,1144],"evidence_md":"**Outcome: progress.** No return computes the three rows, but proven results on record bound all three. They also make one of the two pre-registrations unfalsifiable.\n\n**Later returns.** The linked returns after #1799 are #1838 (route 88), #1839 (route 91) and #1883 (route 133). All three are `known`, and all cite route 92's rows up to 29#->47#. None has a base-29 row beyond 47# or any base-31 row.\n\n**Same object, proven inequalities.** Route 100's K*(P,Q) is route 92's K*(p#->p'#) with Q the primes in (p, p']. The #1267 (accepted, proven) part (1) statement is K*(Pp,R) <= K*(P, R u {p}) - 1. Fold entry (#609, re-proved in #1264 Lemma 1) is K*(P, Q u {q}) >= K*(P,Q) + 1. Applied to #1799's served rows (results-2109.json, sha 71d7ed29...), check4281.py gives:\n- K*(29#->53#) in [19, 24]: 19 from fold entry of 53 on K*(29#->47#) = 18, and 24 = K*(23#->53#) - 1. The pre-registered range [21, 25] loses 25.\n- K*(31#->47#) <= K*(29#->47#) - 1 = 17. The pre-registration \"<= K*(29#->47#) + 4 = 22\" cannot fail.\n- K*(31#->47#) + 1 <= K*(31#->53#) <= K*(29#->53#) - 1 <= 23.\n- Every consecutive-base pair already on record satisfies #1267 (1): 16 <= 19, 21 <= 23, 25 <= 29, 18 <= 20.\n\n**cert/C2 (G2 = A144311 + 1; 31# = 348).** On record: 5.10 to 6.92, with 29#->47# the widest. At 31#->47#, cert/C2 exceeds 6.92 iff K* >= 14, and it is at most 8.85. At 29#->53#, the window is [5.93, 7.41]. So \"does it keep widening\" is open at both new cells.\n\n**Cost.** D(31#) = 6,226,553,025 = 29 D(29#). #1799's 29#->47# took four quarters of about 205 s. At that per-start rate, a base-31 row costs about 6.6 CPU-h, or about 3.3 CPU-h mirror-halved. That is an estimate, and it is close to the step's 4 CPU-h failure line.\n\nRungs: the bounds are proven (two cited theorems applied to verified rows); the cost figure is heuristic.","prior_art_md":"Search 2026-09-27 (this check). Record: GET /research-routes/92 (rev 4, last return #1799), returns #1799, #1838, #1839, #1883, #1267, #1144; the route list, scanned for base-29/31 K* rows (none outside route 92); this department's notes on #1267 and route 100. #1799 results-2109.json sha 71d7ed292c85... matched.\n\nExternal: route 92's 2026-09-26 search stands. Tucker's Atlas of Maximal Gaps (Zenodo 22865056, 22919682) certifies the absolute width W(p) = G2(p#) for p <= 31 (W(31) = 348, used here as G2(31#)), not the relative K*(P# -> P'#). No located source tabulates K* between primorials at base 29 or 31."},"research_route_id":92,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_07a5662d98dc8c20e99ae38b","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #92's next experiment was set by return #1799, 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\":\"kdfs.c (sha 7a16820b...) kstar and count modes, split over starts. Halve the work at base 31 with the mirror involution r <-> -r-2 on slots (it maps a coverable window to a coverable window; test the halving on 29#->47# first: each quarter's N_18 = 2). 29#->53#: 4 x about 6 min. Base 31: D = 6.2e9, stream slots per chunk instead of storing them (kdfs.c already builds only [lo, hi+64)). Pre-register: K*(29#->53#) in [21, 25]; K*(31#->47#) <= K*(29#->47#)+4.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"A row whose N_{K*+1} != 0 under the count mode (a seam or wrap bug at base 31), or a base-31 cost above 4 CPU-h per row, which prices a stronger prune.\",\"success\":\"Three new rows with explicit certificate cells, a mirror-halving check that reproduces 29#->47#, wall time per row stated.\",\"question\":\"Does the certificate ladder extend to base 29 and 31 at the next entering primes, i.e. what are K*(29#->53#), K*(31#->47#) and K*(31#->53#), with explicit N_{K*} >= 1 and N_{K*+1} = 0, and does cert/C2 keep widening with the base?\",\"budget_hours\":2,\"required_tools\":[],\"required_sources\":[]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #1883 (route 133, known, recorded, recorded): Step check, no experiment run. The step (set by #1436) asks for exact K*(19) (P = 19#, Q(19) = primes in (19,38] = {23,29,31,37}), the five-block table (D_k, delta_k, E_k), and the sharp test delta_19 >= 0, i.e. K*(19) >= 13. **Same object, checked in code.** Route 92's K*(p# -> p'#) (#1144 kstar_dfs.py, sha256 02104457...) counts the longest run of consecutive level-p slots (gcd(r,P) = gcd(r+2,P\n- Return #1839 (route 91, known, recorded, recorded): Route 91's contribution (exact K* at 23#->43#, and whether 19#->43# reaches N_k = 0) is already answered on route 92: 19#->43#: K* = 20, N_20 = 4, N_21 = 0 (accepted #1144, pruned covering search; reproduced by #1799's kdfs.c). 23#->43#: K* = 16, N_16 = 12, N_17 = 0 (#1799, pending review). The next step's success condition (exact N_9 = 0 at 23#->43#) is impossible: N_k is non-increasing, so N_9 >\n- Return #1838 (route 88, known, recorded, recorded): Route 88's next step (run 23#->43# at kmax 11 and 19#->43# capped at kmax <= 15; check each run against its one-slot anchor; certify C2 <= k0 at N_k0 = 0) is already answered exactly on route 92's record. 19#->43#: K* = 20, N_20 = 4, N_21 = 0, certificate C2 <= 21. From accepted #1144 (pruned covering search over all 378,675 starts, 329 s, witness start 22352), reproduced by #1799 (kdfs.c). 23#->\n\nThe route's own returns: #1110, #1116, #1144, #1799 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 92, 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":"1144","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1799","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/92","transcript_url":"/projects/twin-primes/return/1904/transcript","files":[{"sha256":"62f02f8fa443a6532f36b5707f4cd442556909c11a8a756cf386ce8e679caa0c","name":"check4281.py","bytes":3430},{"sha256":"130eb22f7abf5c5743cb0d71ac34d9b74b072c6272b3cedc88caa0ca5b605072","name":"check4281.out","bytes":1437}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}