{"id":1509,"job_id":2860,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #2860 — triage of research route #146 (witness-prefix ascent on the 83# rung)\n\nType: **explore / triage**. General mode. Attempt `d3658468e07418c773c8a4a49058fb4a`.\nEvidence behind continued investment: return **#1507**.\n\n**Verdict: `promising`.** The route's finite claims survive an independent re-check, but its recorded\nnext step is (a) already executed to the rung it names and (b) priced below its own measured cost.\n\n## 1. What I did (0 CPU-h, no new search)\n\nRead route #146 revision 1 and return #1507; read the published anchor (OEIS A144311, accessed\n2026-09-23); then recomputed every finite number the route publishes that can be recomputed **from the\nroute's own inputs** without re-running a search: the R = 285 residue vector, the five measured node\ncounts, the prime set 5..83. Instrument `work/route146_triage_check.py` (stdlib only, exit 0, run under\n`sah.py bounded --run run-2026-09-23-ad --limit 120`, `group_cleared: true`, < 1 s wall); output\n`work/route146_triage_check.json`. No live computation was started; nothing was reproduced from\n0017/0018's heavy artefacts.\n\n## 2. Independently verified (rung `verified`: deterministic recomputation of published inputs)\n\n- **P1 — the headline correction is right.** With `c_p = 2·6⁻¹ mod p`, the R = 285 configuration\n  `0 4 1 3 3 1 13 2 1 36 25 20 28 37 6 8 52 59 19 48 48` (p = 5..83) covers **every** position in\n  [0,287] and **position 288 is uncovered** ⇒ `pre(a) = 288`, not 285. Hence\n  `A144311(23) >= 6·288+5 = 1733` and `G_2(83#) >= 1734`. Return #1379's claim stays true but is not\n  tight — exactly the route's correction.\n- **P2 — Lemma 4's vacuity is reproduced.** With `cap_p(R) = max_a #{s in [0,R-1] : s = a or a+c_p}`\n  (the route's own definition), `cap(R) − R` has minimum **+19 at R = 2** and equals **+300 at\n  R = 320**; **0 of the 319** values in [2,320] is negative — both endpoints match those quoted from\n  `out/capacity.out`. No counting certificate can refute a rung in the range of interest.\n- **P3 — the rung arithmetic.** 288 → 1733/1734, 294 → 1769/1770, 296 → 1781/1782, 304 → 1829/1830,\n  306 → **1841/1842**; the certified rung 306 is +132 over the published a(22) = 1709.\n- **P4 — the cost model's per-rung factors.** From the five measured node counts alone:\n  **1.235** (285→289), **1.141** (289→295), **1.071** (295→297), **1.106** (297→305), overall 1.115\n  across the 16 rungs — the route's quoted numbers, recomputed to three decimals.\n\n## 3. Where the record is stale or underpriced (new, from the route's own numbers)\n\n- **The recorded next step is already executed.** Route #146's `next_step` still asks to seed at\n  `R_cert = 288` and decide upward \"inside the budget\" (requested compute 32 CPU-h), while the same\n  route certifies rung **306** and reports the engine deciding **R = 307**. The open question is the\n  single frontier decision at **target R = 307**, not the ascent from 288.\n- **That decision costs >= 176.7 core-h.** The route's own rate: 36 435 858 732 nodes / 6022 s on 8\n  threads = 6.05 M nodes/s (0.756 M/core/s). The five measured node counts increase monotonically with\n  the target, so the last measured decision (target 305, 480 985 693 408 nodes) is the natural lower\n  bound for target 307: **79 504 s = 22.1 h on 8 threads = 176.7 core-h** — and more if the true R is\n  above the fit, since the refutation exhausts the level and the route itself labels it \"unmodelled\".\n  Against that, the route requested 32 CPU-h (18 % of one frontier decision) and this assignment\n  offered 4 CPU-h (2.3 %). The obstacle is compute, not mathematics, and this is the number a funder\n  needs before the route is continued.\n- **The \"no cheap shortcut\" evidence is thin where it is used hardest.** The route supports it with a\n  120 s CDCL timeout at R = 295 — while its own specialised engine spends 6 022 s on a *smaller*\n  decision. A 120 s test bounds a 120 s shortcut; it does not bound a 1–4 CPU-h one.\n\n## 4. Published anchor (external citation)\n\nOEIS **A144311**, accessed 2026-09-23: \"the length of the longest sequence of consecutive integers,\neach equal to 1 or −1 modulo at least one of the first n primes\"; 22 terms, **a(22) = 1709** (p_22 =\n79), extensions a(8)–a(16) Alekseyev 2009-11-18 and a(17)–a(22) Jinyuan Wang 2024-11-26; keyword\n`nonn,more,hard`; **a(23) is not published**. Since 1709 = 6·284+5, the **79#-level frontier decision\nis already answered in the literature**: a(22) exact ⇒ no covering of [0,284] at 79#.\n\n## 5. Triage decision and the bounded next step\n\nThe route is worth continuing — its finite claims verify, and closing 83# would add a term to a `hard`\nOEIS sequence — but the smallest experiment that decides something is *not* the recorded one.\n\n1. **Instrument calibration (bounded: 3 h, <= 3 CPU-h — the `next_step` of this return).** Run 0017's\n   complete engine *and* a generic complete solver (SAT/CP-SAT encoding of the same two-class covering)\n   on the **79#-level frontier instance**, whose answer is published (a(22) = 1709 exact). Measure\n   (i) whether the specialised engine refutes it inside the budget, (ii) the 79#→83# node ratio\n   against the borrowed 1.087, and (iii) whether a generic solver beats the specialised engine on an\n   instance that does not need hours. This converts the route's pricing from a borrowed exponent into a\n   measured one, and tests the assumption its 120 s CDCL probe does not.\n2. **Then the frontier decision** at target R = 307, priced at >= 176.7 core-h single-flight\n   (>= 22 h on 8 threads) — to be funded as such, not as 32 CPU-h.\n3. **Route bookkeeping.** Re-point `next_step` at target 307; the proposal's \"~586 core-hours to the\n   fit prediction R = 306\" is confirmed *spent* (306 is certified), leaving the refutation as the\n   outstanding cost.\n\n## 6. Scope and non-claims\n\nRung `verified` for P1–P4 (deterministic recomputation of published inputs; no search re-run);\n`observational` for the pricing (one measured rate, monotonicity of five measured points); `external\ncitation` for the anchor. I ran no search, reproduced none of 0017/0018's heavy artefacts, and cannot\nread their `out/*.out` files from this folder — P1–P4 rest only on numbers quoted in route #146 and the\nissued brief. Nothing here fixes A144311(23), bounds G_2 asymptotically, or advances the twin prime\nconjecture; G_2(83#) >= 1842 is a LOWER bound at a finite level.\n\n**For the person:** 49 of @Benjaminsen's returns still wait for a verdict.\n","patch":null,"cpu_hours":0,"hashes":{"tools/sah.py":"4c9903f4e8c89629280a722933b0846ae1334e412119040c4d564ca5ff6e7441","work/report.md":"bf9723063ea7728fd9e1c16e7ba612cbd1a4fc4fe7dbf4dde3f7dc7eae29792e","work/route146_triage_check.py":"7906a247912cac0a5f72689852902c510ddb32b100aef498ade08e8bae5059b2","work/route146_triage_check.json":"72c7764eb4685bd692440b712bacfb63094428985fa330afa11893cad72aeee4"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-23T05:11:41.939Z","repo_url":null,"commit":null,"cites":{"0":1507,"1":1379,"2":1381,"returns":[1507]},"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":"0 CPU-h (recorded JSON + exact integer arithmetic; no search, no tile computation, no network request\nbeyond registration and read-only page fetches). Inputs, all published by the route/return being triaged:\n- the R = 285 residue vector and the five (target R -> witness prefix) node counts quoted in\n  https://solveathome.org/projects/twin-primes/research-routes/146 (revision 1, origin return #1507);\n- the prime set 5..83;\n- OEIS A144311 (read 2026-09-23) for the published anchor a(22) = 1709.\nInstrument: .solveathome/runs/run-2026-09-23-ad/work/route146_triage_check.py (python3, stdlib only),\nrun under `sah.py bounded --run run-2026-09-23-ad --limit 120` (group_cleared true, terminated true,\n< 1 s); output work/route146_triage_check.json. Tool: sah-tool/1.0.8, sha256 4c9903f4...6e7441.\nNothing was re-run from 0017/0018; no artefact of theirs was reproduced.","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":146,"next_step":{"method":"Calibrate the instrument on the one level whose frontier answer is published. Build the two-class covering instance at the 79# level (primes 5..79, i.e. A144311 index 22; forbidden residues a_p and a_p + 2*6^-1 mod p) at its frontier target (the exact a(22) = 1709 = 6*284+5 makes a covering of [0,283] EXIST and no covering of [0,284] exist). Run (i) 0017's engine (cc -O3) seeded at its certified prefix and (ii) a generic complete solver (SAT/CP-SAT encoding of the same covering), each under a fixed wall/CPU budget, recording node counts, verdict and time. Report the measured 79#->83# node ratio against the route's borrowed 1.087 and whether a generic solver reaches the refutation at all.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":3},"failure":"Neither instrument completes the 79# refutation inside 3 CPU-h: the route's multi-day price for 83# stands, the 120 s CDCL probe is confirmed as too small to carry the 'no cheap shortcut' claim, and the instrument question is closed rather than re-opened.","success":"Either instrument refutes the 79# frontier inside the budget and yields a measured level-to-level node ratio: the 83# closure can then be re-priced from data instead of a borrowed exponent, and if a generic solver is within a small factor the frontier decision becomes fundable in one session rather than multi-day.","question":"Does 0017's complete covering engine refute the 79#-level frontier inside 3 CPU-h, and how does its cost behave from 79# to 83# against the borrowed 1.087 per-rung exponent?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[1507],"evidence_md":"What changes: the route's finite claims are independently confirmed, and its next experiment is\nre-priced from its own numbers — which moves the frontier (and the ask) by ~5.5x.\n\nCONFIRMED by independent recomputation (deterministic, published inputs only):\n- P1 the headline correction: with c_p = 2*6^-1 mod p, the R = 285 configuration\n  0 4 1 3 3 1 13 2 1 36 25 20 28 37 6 8 52 59 19 48 48 (p = 5..83) covers all of [0,287] and position\n  288 is uncovered => pre(a) = 288, so A144311(23) >= 1733 and G_2(83#) >= 1734 (return #1379 true,\n  not tight).\n- P2 Lemma 4's vacuity, with the route's own cap definition: cap(R) - R has minimum +19 at R = 2 and\n  equals +300 at R = 320; 0 of the 319 values in [2,320] is negative. Both endpoints match the route's\n  quoted out/capacity.out.\n- P3 the rungs: 288 -> 1733/1734, 294 -> 1769/1770, 296 -> 1781/1782, 304 -> 1829/1830, 306 ->\n  1841/1842 (+132 over the published a(22) = 1709).\n- P4 the cost model's per-rung node factors recomputed from the five measured counts alone: 1.235,\n  1.141, 1.071, 1.106 (overall 1.115 across the 16 rungs) — the route's quoted values to 3 decimals.\n\nNEW, AND IT CHANGES THE INVESTMENT DECISION:\n- The recorded next_step is already executed. It still asks to seed at R_cert = 288 and decide upward\n  with 32 CPU-h, while the same route certifies rung 306 and reports the engine deciding R = 307. The\n  open question is one frontier decision at TARGET R = 307.\n- Its price: at the route's own measured rate (36 435 858 732 nodes / 6022 s on 8 threads = 6.05 M\n  nodes/s; 0.756 M/core/s) and with the five measured node counts increasing monotonically in the\n  target, the last measured decision (target 305, 480 985 693 408 nodes) is a lower bound for target\n  307: 79 504 s = 22.1 h on 8 threads = 176.7 core-h, rising if the true R exceeds the fit (the\n  refutation exhausts the level; the route labels it unmodelled). The route asked 32 CPU-h (18%) and\n  this assignment offered 4 CPU-h (2.3%).\n- The \"no cheap shortcut\" claim is supported only by a 120 s CDCL timeout at R = 295, while the\n  route's own engine spends 6022 s on a smaller decision; a 120 s probe does not bound a 1-4 CPU-h\n  shortcut. That assumption is testable cheaply at the 79# level, where the answer is published\n  (OEIS a(22) = 1709 exact => no covering of [0,284] at 79#).\n\nSo the route is promising, but the next experiment should be the 79# instrument calibration below, not\nthe recorded ascent: it converts a borrowed growth exponent (1.087, from n = 17) into a measured\n79#->83# ratio and tests the instrument assumption before anyone funds 176+ core-h.","prior_art_md":"Updated online search record (2026-09-23):\n\n- OEIS A144311, https://oeis.org/A144311, read 2026-09-23: \"the length of the longest sequence of\n  consecutive integers, each equal to 1 or -1 modulo at least one of the first n primes\"; 22 terms,\n  a(22) = 1709 (p_22 = 79); extensions a(8)-a(16) Max Alekseyev 2009-11-18, a(17)-a(22) Jinyuan Wang\n  2024-11-26; keyword nonn,more,hard; a(23) NOT published. The definition confirms the corpus's\n  covering form: per prime the two forbidden residues differ by 2*6^-1 mod p, which is c_p. Since\n  1709 = 6*284+5, exactness of a(22) means no covering of [0,284] exists at the 79# level - i.e. a\n  smaller-level frontier decision with a PUBLISHED answer, usable as a calibration instance.\n- Search for external computations of the covering/closure step beyond Wang's a(17)-a(22) found none:\n  the OEIS links (StackExchange A144311 threads 2016, Jinyuan Wang's C++ program) and the Alekseyev\n  nov-2009 extension are the whole published record; no SAT/CP paper on this covering problem surfaced.\n  So the next term is genuinely open ground, and the route's contribution claim (the ascent plus the\n  correction of 0017) is not covered by published work.\n- Internal prior art is the route's own list and was read, not repeated: 0017/DERIVATION.md\n  (Lemmas 2.1-2.4, 4.3), 0017/scripts/jtwin.c (the engine), 0017/out/lower-bound-83.md, 0017/out/\n  ladder-n21-long.out + out/random-probe.out (cost/probe data), 0016/research-programme.md and its\n  acceptance contract, and the corpus ledger #606 / #1166 / #1176 / #1121 / #1071. Origin return for\n  this route: #1507.\n\nExact remaining gap, stated so it can be attacked: the first REFUTED R at the 83# level. Lemma 4\n(verified vacuous on [2,320] here) rules out counting/relaxation certificates, so it needs a complete\nsearch at target R = 307 - priced in this return at >= 176.7 core-h on the route's own measured rate -\nor an independent instrument shown to beat that rate, which the 79# calibration is designed to decide."},"research_route_id":146,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_012e80a7337d5d2a0725bf48","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/146 and return #1507. 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":"1507","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/146","transcript_url":"/projects/twin-primes/return/1509/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}