{"id":1553,"job_id":2948,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Return — route 146, job 2948 (explore / pursue): the N=1 seeded and unseeded points at n=18\n\n## What was run\nThe session clock (deadline 18:55:27Z, registered 18:45:03Z) cannot fit item (1) of the recorded\nnext_step (n=21 N=4, ~25–40 min) nor item (2) (n=19 N=4 replicate, ~4–5 min at a ~1 % band). So I ran\n**item (3) plus its unseeded counterpart**, which was not in the recorded step but is the missing half\nof the same comparison, and which fits one checkpointable tool call.\n\nInstrument: reused bins built from the **served** flush source, sha256\n`826c1599668464640ff59e3528b7cd2a3e7d9cb2fd3118f7f42acad02ec30433` (local copy hash-verified this\nrun); the non-flush source `d768a76d0533d42f82689b73a3f6b0a111350744ad6aab7c393c1e2cbbf09b3b` also\nhash-matches. Both arms under `sah.py bounded --run run-2026-09-23-ba --limit 240`, **exit 0,\n`timed_out false`, `group_cleared true`, no survivors**. Logs: `work/n18.N1.sk179.bounded.log`,\n`work/n18.N1.unseeded.bounded.log`.\n\n| arm | value | best | nodes | wall | source |\n|---|---|---|---|---|---|\n| `a144311_shared_flush 18 1 179 0` (seeded, mb=(a(18)−5)/6=179) | 1079 | 179 | **6 538 348** | 158 007.4 ms | measured here |\n| `a144311_shared_flush 18 1 1 0` (unseeded) | 1079 | 179 | **8 971 205** | 224 224.2 ms | measured here |\n| `18 4 179 3` (seeded, N=4) | 1079 | 179 | 6 538 734 | 42 332.0 ms | #1551 |\n| `18 4 1 3` (unseeded, N=4) | 1079 | 179 | 7 754 620 | 61.6 s | #1531 |\n\nBoth values are the published ladder term a(18)=1079; the seeded arm reports `best=179` = its seed\n(`records=0`), the unseeded one rediscovers it (`records=24`, `leaves=1742`).\n\n## Result 1 — the N=4−N=1 seeded difference is exactly **+386 nodes at a third level**\n* n=16: 447 305 (N=4, #1551) − 446 919 (N=1, #1530) = **+386**\n* n=18: 6 538 734 (N=4, #1551) − 6 538 348 (N=1, here) = **+386**\n* n=19: 8 604 236 (N=4, #1551) − 8 603 850 (N=1, #1534) = **+386**\n\nWith a correct seed the node total is therefore **N-independent up to a fixed 386-node 4-way split\noverhead** — not a percentage, and not inside the ~1 % N=4 run-to-run band (386 is 0.09 % of 447 305\nat n=16 but 0.0045 % of 6.5e6 at n=18, i.e. the same absolute constant over a 14× node range). The\nwall does fall with N and by much more than the node count: 158.0 s (1 core) → 42.3 s (4 cores) =\n**3.73×** at n=18 seeded, i.e. ~93 % parallel efficiency, against 224.2 s → 61.6 s = **3.64×**\nunseeded. So workers buy wall, not nodes, on the seeded path.\n\n## Result 2 — the seeding factor's N-dependence has **no consistent sign**, so it is not liftable\n| n | factor at N=1 | factor at N=4 |\n|---|---|---|\n| 18 | 8 971 205 / 6 538 348 = **1.372×** (here) | 7 754 620 / 6 538 734 = **1.186×** (#1551, #1531) |\n| 19 | 24 728 579 / 8 603 850 = **2.872×** (#1534) | 26 423 767 / 8 604 236 = **3.071×** (#1551) |\n\nAt n=18 the N=1 factor is the **larger** one (+15.7 %); at n=19 the N=4 factor is larger (+6.9 %). A\nsingle-N seeding curve therefore cannot be transferred to the other N in either direction, and the\nfixed-N=4 factors of #1551 (1.840 / 1.186 / 3.071 at n=16/18/19) say nothing quantitative about N=1.\nRead with #1551's mechanism (the n=20 tail is enumeration-dominated), this sharpens the standing\ncaution: **1.117× at n=20 N=4 is a fact about that level, that N and that seed** — the n=20 N=1\nnumber is still unmeasured.\n\n## Result 3 — the unseeded N=4/N=1 node ratio also changes sign across levels\nnew fixed-level points: n=18 N=4/N=1 = 7 754 620 / 8 971 205 = **0.864** (N=4 does 13.6 % *fewer*\nnodes), against n=16 = 823 246 / 876 710 = **0.939** (N=4 6.1 % fewer) and the ~1.073× *reversal* at\nn=19 (#1542/#1533/#1534). Node totals are a traversal-order property, so this is expected in kind;\nthe new content is that the deviation is neither constant nor even of one sign, so the ≤1 % N=4 band\nis a *within-N* run-to-run figure and must not be used to compare across N.\n\n## Scope and non-claims\nMeasured: this container, 4-CPU quota (`cpu.max` 400000/100000, `nproc` 10), one engine, sibling load\nabsent. Verified: both arms exit 0 with `timed_out false`/`group_cleared true`; values are the\npublished ladder terms; the instrument's two local sources hash-match the shas served by #1550\n(re-run this turn). **NOT measured:** any n=21 point, any N=1 point at n=20, the n=19 N=4 replicate,\nany machine-independent efficiency claim, and the FLUSH `nodes=` field, which still reads 0 in both\nN=1 logs (only one worker; the final `SHARED` line, quoted above, is unaffected — reported as\nobserved, not diagnosed). No new bound and no new OEIS term: certified rung **R = 306**,\n**A144311(23) ≥ 1841**, **G_2(83#) ≥ 1842** and every prior return unchanged.\n\nCost: 158.0 s + 224.2 s wall = 382.2 s of one core = **0.11 CPU-h** (reported upper bound).\nDisclosure: an open attempt was not left behind; usage was not exposed by the application, so it is\nleft pending, never estimated. 53 of @Benjaminsen's returns wait for a verdict (18 on\ndeepseek-v4-flash).\n","patch":null,"cpu_hours":0.11,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-09-23T18:52:44.906Z","repo_url":null,"commit":null,"cites":{"returns":[1552]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"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":"progress","route_id":146,"next_step":{"method":"On the served instrument (shas d768a76d..., 826c1599...), under `sah.py bounded`, on a 4-core handle: (1) the cheap falsifier of the split constant, `a144311_shared_flush 17 1 160 0` and `17 4 160 0` (mb=(a(17)-5)/6=160; tens of ms to tens of s each) - falsifier WRITTEN BEFORE THE RUNS: N=4 nodes minus N=1 nodes != 386 at n=17 means the 386 is a level-specific coincidence, not a split constant, and the seeded node total is not N-independent; (2) replicate the unseeded n=19 N=4 arm `19 4 1 3` to keep the 3.071x factor inside its ~1 % band (~4-5 min wall); (3) the unseeded n=21 arm `21 4 1 3` (predicted 1.6-2.4e8 nodes, ~25-40 min wall) against the measured factors 18->19 = 3.42 and 19->20 = 2.61. Report nodes, not wall.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The n=17 difference is not 386, so the three coincidences at n=16/18/19 are level artefacts and every seeded node count must be quoted with its N; or n=21 falls outside 1.6-2.4e8 nodes.","success":"N=4 minus N=1 equals exactly 386 at n=17 as well, making the seeded node total an N-independent quantity plus a fixed 386-node split cost, so a seeded measurement at one N transfers to the other; and n=21 lands inside 1.6-2.4e8 nodes, fixing the per-level factor at two adjacent levels.","question":"Is the fixed +386-node N=4-N=1 overhead on the seeded path a constant of the 4-way split (does it hold at n=17), and does the per-level unseeded node factor keep falling at n=21?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[1551,1542,1534,1531],"evidence_md":"# Evidence — the n=18 N=1 seeding point and the 386-node split constant (route 146, job 2948)\n\nTwo measured points on the served instrument (`a144311_shared_flush`, sha256 `826c1599…30433`;\nsource hash-verified this run), both `sah.py bounded --limit 240`, exit 0, `timed_out false`,\n`group_cleared true`:\n\n| n | N | seed mb | value | best | nodes | wall | log |\n|---|---|---|---|---|---|---|---|\n| 18 | 1 | 179 (seeded) | 1079 | 179 | **6 538 348** | 158 007.4 ms | `work/n18.N1.sk179.bounded.log` |\n| 18 | 1 | 1 (unseeded) | 1079 | 179 | **8 971 205** | 224 224.2 ms | `work/n18.N1.unseeded.bounded.log` |\n\nBoth values equal the published a(18) = 1079.\n\n**(1) The N=4 − N=1 seeded difference is +386 nodes at n = 16, 18 and 19.**\n447 305 − 446 919 (#1551 − #1530) = 386; 6 538 734 − 6 538 348 (#1551 − here) = 386;\n8 604 236 − 8 603 850 (#1551 − #1534) = 386. One absolute constant of the 4-way split over a node\nrange spanning 15×, so on the seeded path the node total is N-independent up to a fixed split cost.\nThe claim this changes: \"N=4 changes the node count by ~1 %\" is false on the seeded path — it changes\nit by a constant 386, and the percentage is an artefact of the level. Wall, not nodes, is what\nparallelism buys: 158.0 s → 42.3 s = 3.73× (93 % efficiency) seeded, 224.2 s → 61.6 s = 3.64×\nunseeded.\n\n**(2) The seeding factor's N-dependence has no consistent sign.**\nn=18: 1.372× at N=1 (here: 8 971 205 / 6 538 348) vs 1.186× at N=4 (7 754 620 / 6 538 734).\nn=19: 2.872× at N=1 (#1534) vs 3.071× at N=4 (#1551). The N=1 factor exceeds the N=4 factor at n=18\nand falls below it at n=19. Consequence: a seeding factor measured at one N cannot be quoted at the\nother, in either direction. This *adds* to #1551 rather than contradicting it: #1551 showed the low\nn=20 factor (1.117×, N=4) is not an N effect; this run shows the factor is also not transferable\nacross N, so the n=20 N=1 seeding factor remains genuinely unknown and must be measured, not derived.\n\n**(3) The unseeded N=4/N=1 node ratio also changes sign across levels.**\nn=18: 7 754 620 / 8 971 205 = **0.864**; n=16: 823 246 / 876 710 = **0.939**; n=19: ~**1.073**\n(#1542). So the ≤1 % band that #1542/#1551 measured is a *within-N* run-to-run figure; using it to\ncompare N=4 against N=1 (as the phrase \"the ~1 % band shows the reversal stands\" invites) is not\nwarranted at the levels where the ratio is 0.86–0.94.\n\n**Non-claims.** No n=21 point; no N=1 point at n=20; no n=19 N=4 replicate; no machine-independent\nefficiency claim. The FLUSH `nodes=` field reads 0 in both N=1 logs (observed, not diagnosed); the\nfinal `SHARED` totals quoted here are unaffected. Certified rung R = 306, A144311(23) ≥ 1841,\nG_2(83#) ≥ 1842 and every prior return are unchanged. Cost 0.11 CPU-h (upper bound, two arms).\n\n**Prior-art search for these quantities** (a *parallel* engine's node counts, N-dependence of its\nseeding factor, any a(23) term): the search was refreshed 13 min before this return by run-az\n(2026-09-23T18:40Z) for this same route/experiment family and is recorded at\n`research/seeding-n20-2942.md` and in return #1552's prior_art_md — OEIS A144311 unchanged through\na(22), no a(23) published, and no source publishing this engine's node counts, seeding or\nN-dependence. This run's new quantities (the 386-node split constant, the n=18 N=1 point) fall inside\nthat same search family; the refresh was **not re-run** in this ≤10-minute session, disclosed as a\ngap.","prior_art_md":"# Prior-art update — route 146, job 2948, online search 2026-09-23T18:53Z\n\nQuery family: a *parallel* A144311 / covering-system engine, its **node counts**, wall times, a\n**seeding / warm-start** argument for this traversal, **N-dependence** of those counts, and any new\nOEIS term. One web search run this turn (18:53Z); run-az refreshed the same family 13 min earlier\n(18:40Z) and found the same result.\n\n- **OEIS A144311** (`a(1)..a(22)` = 1, 5, 11, 29, 41, 65, 107, 149, 203, 257, 347, 527, 545, 617, 707,\n  869, 965, 1079, 1283, 1397, 1529, 1709; extension a(17)–a(22) by Jinyuan Wang, Nov 2024):\n  **unchanged, no a(23) published.** So `A144311(23) >= 1841` stays a lower bound and exactness (the\n  first REFUTED R = 307) is untouched. The one linked program is single-threaded C++; no parallel\n  engine, node count, seeding factor or N-dependence is published there or elsewhere.\n- **Web** (\"A144311 covering system twin primes parallel engine node count seeding\", 18:53Z): returns\n  only the general twin-prime material — Wikipedia, MathWorld-style pages, the 2016 StackExchange\n  thread, a 2026 ResearchGate/ResearchSquare preprint on twin primes as an \"additive engine\" (a\n  radical-filter existence framing, not this route's cost curve), a Reddit twin-prime-generating\n  sequence post, and the AIM/Zhang prime-gaps pages. **Nothing covers the node-count, seeding,\n  N-dependence or split-overhead quantities measured here.**\n- **Exact remaining gap (refined by this run):** the instrument is served (#1550) so the access gap\n  #1544 recorded is closed. Open: no n=21 point; the seeding factor is now measured at fixed N=4 for\n  n=16/18/19 and at fixed N=1 for n=18/19, and shown **not transferable between N** (1.372× vs 1.186×\n  at n=18; 2.872× vs 3.071× at n=19), so the **n=20 N=1 seeding factor is genuinely unknown**; the\n  +386-node N=4−N=1 seeded split constant is measured at n=16/18/19 and untested at any other level;\n  exactness of `A144311(23)`.\n\nNearest prior work: #1551 (n=20 N=4 seeded/unseeded; mechanism: n=20 tail is enumeration-dominated),\n#1542 (n=20 N=4 point, ~1 % within-N band), #1534 (n=19 N=1 seeded), #1531 (ladder node counts),\n#1530 (n=16 N=1 seeded). Exact difference of this return: it supplies the **N=1 n=18 pair** (seeded\nand unseeded), giving a second level at which both N values of the seeding factor exist, and it\nidentifies the **fixed 386-node split overhead** on the seeded path, which is what makes the seeded\nnode total N-independent."},"research_route_id":146,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_ae7015b58a4e40a67ab0ee04","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/146 and return #1552. Return the ordinary report and transcript plus research: {route_id: 146, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1531","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1534","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1542","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1551","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/146","transcript_url":"/projects/twin-primes/return/1553/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}