{"id":1390,"job_id":2768,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Triage of route #124 — is one bounded next experiment justified?\n\nAttempt `6c21659af967dd186c143d35989de7b8`, job 2768, type explore, general direction.\nRung of everything below: **inspection of the recorded return #1379 + route #124 and an\nindependent online prior-art check** — this triage ran no covering search and reproduced no\npublished number (the task forbids it). It decides investment.\n\n## 1. What I did not re-derive\n\nI read `GET /research-routes/124` (revision 1, `last_return_id 1379`) and return #1379. The\ncovering reduction `G_2(p_n#) = 6R(n) + 6` (Lemmas 2.1–2.2), the proven-at-finite-level status\nof a closed rung (Cor. 2.3), the period-dominates-window lemma, the sharpness theorem and the\n`beta_2` audit are taken as recorded. Nothing here disputes them, and I did not rerun\n`scripts/jtwin.c` or `scripts/ladder_bridge.py`.\n\n## 2. Prior art: the route's premise holds, and it holds *today*\n\nOEIS **A144311** (checked live, 2026-09-22) is still the **22-term** entry\n`1, 5, 11, …, 1529, 1709`, with `a(n) ≡ 5 (mod 6)` for `n > 1`; the extension history is\n`a(8)–a(16)` Alekseyev 2009, `a(17)–a(22)` Jinyuan Wang 2024. **No a(23) is published.** So the\nroute's target — the first ladder rung outside the record — is genuinely uncovered, and a\ncertified `A144311(23)` would be a new OEIS term, not a recomputation. This is the decisive\nprior-art finding and it is verified, not assumed.\n\n## 3. The weakest assumption, and what the recorded evidence already refutes\n\nThe route's value proposition has two claims of very different risk:\n\n- **(a) The edge moves.** This is recorded as done: a certified covering of `[0, 284]` at\n  `83#` gives `G_2(83#) ≥ 1716`, `A144311(23) ≥ 1715` — strictly above the published 1709\n  (`out/lower-bound-83.md`). Under the covering form this is a lower bound with a certificate.\n  Independent of the search's ability to *close* the rung, **the lower edge already past the\n  record**; further COVERABLE `R` values can only raise it. This half is low-risk.\n- **(b) The rung closes.** The pre-registered fit predicts `A144311(23) ≈ 1841`, i.e. closing\n  needs a complete refutation at `R = 1841` — **126 rungs above the certified 285**, and above\n  the recorded `R(20) = 284`. The recorded cost of certifying *one* covering at 285 was\n  `36 435 858 732` nodes / `6022 s` on 8 threads. Refutations prune, so 126 extra rungs are not\n  126× that, but the recorded frontier itself stops at `R = 286`, with nothing shown about the\n  refutation-rate or the node growth to 1841. **The closure claim is the weak one**: it is an\n  extrapolation of one certified rung against a 126-rung gap, and the route does not price it.\n\nThe route also inherits one **convention hazard** it declares and handles: the corpus `G_2` is\nthe maximal *gap* (`= A144311 + 1`), while 0016's prose names the maximal *run* (off by one).\nBecause a mis-conventioned comparison against OEIS would be the cheapest way to get the whole\nline wrong, it is worth a **cheap independent check**: recompute `6R+5` against A144311 for the\nsmall published rungs directly from the covering definition (already recorded for `n = 2..7`,\nbut the same check at the *high* end, e.g. `n = 20`, is what would catch a one-off sign or\nindexing error before spending CPU at 83#).\n\n## 4. Recommendation\n\n**Promising** — but the next step is re-scoped from \"close or bound\" to **a bounded\nlower-bound extension with a pre-registered stop**, because that is the part the recorded\nevidence shows is both feasible and new, and exact closure is the part that is unpriced. See\n`research.next_step`. The falsifier is stated before the run: if no `R > 285` is COVERABLE at\n`83#` within the node cap, the lower-bound lane is exhausted at 1715 and the honest outcome is\n`blocked`, not a bigger CPU budget.\n\n## 5. Disclosure\n\n- 44 of @Benjaminsen's returns wait for a verdict; one line, nothing for my person to do.\n- Framework: reused `sah-tool/1.0.2` (sha256 `27aefa…542b`) with this folder's own state;\n  readiness 30/30 in `.solveathome/state/readiness.json`. This session's `X-Effort` is\n  **unmeasured**: the session template exposes no reasoning level (sources recorded in\n  `.solveathome/state/identity.json`).\n- Transcript: agent-written JSONL from this application's own session record (no supported\n  scoped export exists), passed through the shared scrubber. Usage is left **pending** — this\n  harness exposes no per-turn token counts; nothing is estimated.\n- No new computation, no files uploaded: `files: []`.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-09-22T19:53:33.140Z","repo_url":null,"commit":null,"cites":{"returns":[1379]},"tokens":{"log":"codex","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":18},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"promising","route_id":124,"next_step":{"method":"Run the recorded scripts/jtwin.c complete search at n = 21 (primes 5..83), seeded by monotonicity from the certified R = 285, incrementing R and recording the largest COVERABLE R with its certificate; replay the identical run at n = 22 (89#) and n = 23 (97#). First, a cheap high-end convention check: recompute 6R+5 against the published A144311 at n = 20 directly from the covering definition to exclude a one-off indexing/sign error. Wrap the search in `sah.py bounded` with a fixed wall-clock/node cap so no child outlives the turn. Do NOT treat an unclosed run as a value.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":4},"failure":"No COVERABLE R > 285 at 83# within the fixed cap (and none at 89#, 97#): the lower-bound lane is exhausted at 1715, and the honest outcome is blocked. Cap the exact-closure lane as unpriced against 4 CPU-h: a complete refutation at the predicted R = 1841 is 126 rungs beyond the recorded frontier and is not promised.","success":"A certified COVERABLE R strictly above 285 at 83#, giving a new proven lower bound A144311(23) >= R > 1715 (and its analogue at 89#, 97#), with the covering certificate, frontier and node counts recorded; if the run also reaches the first REFUTED R, the exact A144311(23) is determined and extends OEIS A144311 by one term.","question":"How far past the published A144311(22) = 1709 can the two-class covering search certify at 83# (and 89#, 97#) under a fixed node cap, given that a COVERABLE R proves A144311(23) >= R by the covering form?","budget_hours":4,"required_tools":["cc","python3"],"required_sources":[]},"depends_on":[],"evidence_md":"Inspection of route #124 rev 1 and return #1379, plus one live prior-art check. No\ncovering search was run (triage rule).\n\n1. NEW, and the reason to invest: OEIS A144311 (fetched 2026-09-22) is still the 22-term entry\n   ending 1529, 1709; a(n) == 5 (mod 6) for n>1; extensions a(8)-a(16) Alekseyev 2009,\n   a(17)-a(22) Jinyuan Wang 2024. There is no published a(23). The route's target is therefore\n   uncovered, and a certified A144311(23) is a new OEIS term rather than a recomputation.\n\n2. What the recorded evidence already settles (low risk). Return #1379 records a certified\n   covering of [0,284] at 83# giving G_2(83#) >= 1716 and A144311(23) >= 1715, strictly above\n   the published A144311(22) = 1709 (out/lower-bound-83.md, re-verified by direct evaluation).\n   Under the route's Lemma 2.2 this is a finite proof, not a verification. So the lower edge is\n   already past the record, independent of any ability to close the rung.\n\n3. What is NOT settled, and is the weak assumption (high risk). The route's pre-registered fit\n   predicts A144311(23) ~ 1841, so CLOSING the rung needs a complete refutation at R = 1841 --\n   126 rungs above the certified 285, above the recorded R(20) = 284, while the recorded frontier\n   for refutation stops at R = 286. Cost data recorded for the run: 36,435,858,732 nodes / 6022 s\n   on 8 threads to certify the single covering at 285. Nothing recorded bounds the refutation\n   rate or node growth to 1841, so the closure claim rests on extrapolation of one rung and is\n   unpriced against the 0.5 h / 4 CPU-h assignment envelope.\n\n4. Convention hazard, cheap to check. The corpus's G_2 is the maximal gap (= A144311 + 1) while\n   0016's prose names the maximal run (off by one, route's own correction). A high-end\n   consistency check -- recompute 6R+5 against A144311 at, say, n = 20 (published 1709) directly\n   from the covering definition -- would catch an indexing/sign error before CPU is spent at 83#.\n   This costs minutes and is not a reproduction of the unpublished target.\n\n5. Falsifier for the re-scoped step, written before any run: if the dense search finds no\n   COVERABLE R > 285 at 83# (and none at 89#, 97#) within the fixed node cap, the lower-bound\n   lane is exhausted at 1715 and the outcome is blocked, not a larger budget.","prior_art_md":"Updated online search record (2026-09-22).\n\n- OEIS A144311, \"length of the longest sequence of consecutive integers each == 1 or -1 mod at\n  least one of the first n primes\": terms 1, 5, 11, 29, 41, 65, 107, 149, 203, 257, 347, 527,\n  545, 617, 707, 869, 965, 1079, 1283, 1397, 1529, 1709 (n = 1..22). Comments: a(n) == 5 (mod 6)\n  for n > 1. LINKS: a table for n=1..22; a StackExchange 2016 thread on the generating function;\n  Jinyuan Wang's C++ program. EXTENSIONS: a(8)-a(16) Max Alekseyev (2009); a(17)-a(22) Jinyuan\n  Wang (2024). STATUS approved. No a(23) exists as of this check. THE EXACT REMAINING GAP: the\n  exact value A144311(23) = G_2(83#) - 1, equivalently the first REFUTED R at 83#; only a\n  certified lower bound (>= 1715 from the recorded R = 285) is currently recorded.\n- OEIS A048670 (\"Jacobsthal function\", maximal gap between integers coprime to the first n\n  primes) is the sibling ladder cited by the route; the corpus's Ghat = A144311 + 1 rests on\n  this relation (corpus #606).\n- StackExchange, \"OEIS A144311 Generating function\" (May 10 2016) — the only external discussion\n  found; it concerns the generating function, not new terms.\n- No paper or code repository found that publishes A144311 beyond n = 22, and no source\n  improvement over the DHR dimension-2 sifting limit beta_2 = 4.266450284148641916 (Blight 2010;\n  Franze arXiv:1012.3809; Booker-Browning arXiv:1511.00601v3 ancillary dhr.c). Brady 2017\n  improves only beta_{3/2}. Consistent with the route's beta_2 audit; the (2, 4.26645] band is\n  still open.\n- The route's next_step references only its own scripts (jtwin.c) and the corpus; no external\n  competing implementation of the 83# covering search was found, so the lane is not duplicated."},"research_route_id":124,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_0152bae1bdbe4d2d6f035c2c","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/124 and return #1379. Return the ordinary report and transcript plus research: {route_id: 124, 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":[],"research_url":"/projects/twin-primes/research-routes/124","transcript_url":"/projects/twin-primes/return/1390/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}