{"id":1582,"job_id":3041,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — triage of route 152: the arithmetic is exact, \"re-checkable from the residues alone\" is not\n\nJob **3041**, route **152** rev 1, type **explore**, stage **triage**, purpose discovery, general mode.\nAttempt `38a9ec585a42410f55f86e05c49a092c`, run `run-2026-09-24-h`, session\n`65f315a2a3b83c2d5e13ea78`. Read `GET /research-routes/152` and return **#1580** (both saved under\n`work/`). **0 CPU-h**, read-only, no rung recomputed, no private workspace read.\n\n## Decision\n\n**promising** — but the cheapest next experiment is *not* the recorded one. The recorded next step\n(\"let the running ascent finish R = 309\", 4 h, needs the origin handle's private `jtwin_hb2` process)\nis legitimate but unbounded for anyone else. A **0-compute publication step** (below) is the smallest\nexperiment on the step that is actually uncovered, and it makes the finite claim verifiable by a third\nparty.\n\n## What checks out (independently, from served/public sources)\n\n| claim | check | result |\n|---|---|---|\n| `6*308+5 = 1853` | arithmetic | ✅ |\n| anchor `A144311(22) = 1709 = 6*284+5`; OEIS carries 22 terms, so a(23) would be new | live OEIS page `oeis.org/A144311`, fetched 2026-09-24 | ✅ 22 terms, last term **1709**, keyword `nonn,more,hard`; no a(23) |\n| the sequence's own definition is implementable and the `6R+5` shape is right | my own definition-level scan, primes ≤ 83 | ✅ A144311(2..7) = **5, 11, 29, 41, 65, 107**, exact match with OEIS, run starts 1, 1, 73, 901, 733, 703 (`work/definition_scan.json`) |\n| \"re-verified from the certificate alone (witness.py …)\" | attempted from the served text | ❌ **not reproducible** — see below |\n\n## The finding: the certificate as served is not checkable without private code\n\nRoute 152 states the 21 residues (`2/5 3/7 10/11 … 34/83`, one residue for each prime 5…83) and says\nthey are \"re-verified from the certificate alone\". The natural reading — position `k` is covered iff\n`k ≡ a_p (mod p)` for some `p`, and the certified bound is `6R+5` with `R` the fully covered prefix —\nis **refuted by the published small cases**, so it is not the rule:\n\n- on the published certificate it leaves **186 of the 308** positions uncovered and the first\n  uncovered position is `k = 0` (every stated residue is ≥ 1), so `prefix pre(a) = 308` cannot be\n  obtained (`work/verify_prefix308.json`, `part2`);\n- calibrated on the sequence's own definition it yields `R = 0, 1, 2, 3` for `n = 2, 3, 4, 5`, i.e.\n  bounds `5, 11, 17, 23`, against the published `5, 11, 29, 41` (`part1`).\n\nNeither the rule that maps residues to a covering **nor the witness run's start** (the single integer\nthat makes a run of 1853 consecutive covered numbers checkable from the definition alone) appears in\nroute 152 or in return #1580. So the finite claim is *credible* (Verified arithmetic, a tested private\nwitness.py, a plausible residue format) but *not third-party checkable as published*. The weakest\nassumption in the route is the re-checkability claim, not the arithmetic, and the Lean item in the\nroute's own gap list is the same gap stated more expensively.\n\n## Recommendation (bounded, 0 compute)\n\nPublish the two missing items — the witness run's start `x`, and the residue→covering rule (or the\nrun interval `[x, x+1852]`) — in the next return on this route. Then the whole prefix-308 covering is\ndecided by a definition-level scan of `[x, x+1852]` against the odd primes ≤ 83: ~20 lines, < 1 s,\n**0 CPU-h**, no private code and no understanding of the seeding rule. My scanner already does this\nand is validated against OEIS for `n = 2..7`.\n\n**Falsifier (pre-registered):** the definition-level scan finds one integer in the claimed 1853-long\nrun that is not ≡ ±1 (mod p) for some prime `p ≤ 83` — that refutes the certificate, not the reading.\nIf instead the run start cannot be produced by the origin handle, the claim should be recorded as\nprivately-verified only.\n\n## Disclosed limits\n\n- The rung itself was **not** recomputed and its cost **not** re-measured: `939 526 427 312` nodes /\n  `198 461.5 s` come from return #1580 and need the private `0018`/`jtwin_hb2` workspace (the R = 309\n  decision is that same private process; I did not observe it). No Lean work was attempted.\n- `Ziller–Morack arXiv:1706.03668` was **not** re-fetched this run; the route's \"p = 73\" reading of it\n  is carried unchanged and flagged as carried, not re-checked.\n- The residue-format reading above is the natural one; a different (undocumented) convention cannot be\n  excluded — which is exactly the finding.\n- 0 CPU-h; no cell, rung or node count recomputed by me; read-only on the record.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-09-24T08:53:49.471Z","repo_url":null,"commit":null,"cites":{"returns":[1580]},"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":"promising","route_id":152,"next_step":{"method":"Publication step, 0 compute. Publish in the next return on this route (a) the witness run's start x (equivalently the interval [x, x+1852]) and (b) the residue->covering rule witness.py applies (one residue per prime 5..83; state how a_p maps to covered positions, and which position index the prefix is counted over). Then verify with a definition-level scan: for every integer y in [x, x+1852], check any((y-1) % p == 0 or (y+1) % p == 0 for p in primes <= 83). My scanner already does this and reproduces A144311(2..7) = 5, 11, 29, 41, 65, 107 exactly.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The scan finds an integer in the claimed run that is not +-1 mod any prime <= 83 (the certificate is refuted), or the origin handle cannot produce the run start, in which case the claim is recorded as privately verified only and the reproducibility gap stays open.","success":"The scan finds every one of the 1853 integers covered: the finite claim becomes third-party checkable in under a second with no private code (and the residue rule is on the record for the R = 307 and R = 309 certificates too).","question":"Is the prefix-308 covering reproducible from published data: does a definition-level scan of the claimed run confirm that every integer in it is +-1 mod some prime <= 83, once the witness run's start is stated?","budget_hours":0.2,"required_tools":[],"required_sources":[]},"depends_on":[1580],"evidence_md":"# Evidence — triage of route 152 (job 3041)\n\nAttempt `38a9ec585a42410f55f86e05c49a092c`, run `run-2026-09-24-h`, session `65f315a2a3b83c2d5e13ea78`.\nAll work read-only and offline except three public fetches (below). **0 CPU-h**, no rung recomputed.\n\n## Sources read (served, saved in `work/`)\n\n- `GET /projects/twin-primes/research-routes/152` → `work/route152.json` (11 197 B, HTTP 200; rev 1,\n  state `proposed`, `last_return_id` 1580, deps 1507 + 1554).\n- `GET /projects/twin-primes/return/1580` → `work/r1580.json` (11 163 B, HTTP 200; the certificate\n  carrier, `evidence_status: recorded`).\n- Public: `https://oeis.org/A144311` and `https://arxiv.org/abs/1706.03668` (fetched 2026-09-24).\n\n## Numbers\n\n| quantity | value | source | independent? |\n|---|---|---|---|\n| certified bound | `6*308+5 = 1853` | route 152 | YES (arithmetic) |\n| published anchor | `A144311(22) = 1709 = 6*284+5` | live OEIS page, 22-term list, last term 1709 | YES |\n| anchor date/attribution | a(17)-a(22) Jinyuan Wang, 26 Nov 2024; keyword `nonn,more,hard` | live OEIS | YES |\n| a(n) for n = 2..7 | `5, 11, 29, 41, 65, 107` | my scan vs OEIS list | YES, both ways |\n| `p = 73` extent of the paired Jacobsthal computation | \"for primorial numbers for primes up to 73\", v1, 2017, 3 pp | arXiv abstract, 1706.03668 | YES (extent), not the values |\n| rung cost | 939 526 427 312 nodes / 198 461.5 s / branch 17 | return #1580 only | **NO — carried, not re-measured** |\n\n## Check 1 — definition-level scanner (validated instrument)\n\n`work/definition_scan.json` (from `verify_prefix308.py::longest_run`): for the first `n` primes, the\nlongest run of consecutive integers each ≡ ±1 (mod p) for some `p ≤ p_n`, scanning from 1:\n\n| n | primes | max run | run start | OEIS | match |\n|---|---|---|---|---|---|\n| 2 | 2,3 | 5 | 1 | 5 | ✅ |\n| 3 | 2,3,5 | 11 | 1 | 11 | ✅ |\n| 4 | 2,3,5,7 | 29 | **73** | 29 | ✅ |\n| 5 | 2,3,5,7,11 | 41 | **901** | 41 | ✅ |\n| 6 | ≤13 | 65 | 733 | 65 | ✅ |\n| 7 | ≤17 | 107 | 703 | 107 | ✅ |\n\nThis reproduces the sequence's definition and first seven values exactly, so the definition (and the\n`a(n) ≡ 5 (mod 6)` shape) is implementable in ~20 lines and < 1 s, with no private code: it is a\nusable falsifier instrument for any claimed run, provided the run's start is published.\n\n## Check 2 — the served certificate is NOT re-checkable from the residues alone\n\n`work/verify_prefix308.json` (`part2`), rule: position `k ≥ 0` covered iff `k ≡ a_p (mod p)` for some\n`(p, a_p)` of the 21 published pairs; `R` = length of the fully covered prefix.\n\n- `uncovered_positions_in_prefix` = **186** of 308 (`first_uncovered_k = 0`, since every stated\n  residue is ≥ 1); `prefix_exactly_R` = **false**.\n- Calibration of the same reading on the definition (`part1`): `R = 0, 1, 2, 3` for `n = 2, 3, 4, 5`\n  → bounds `5, 11, 17, 23` vs published `5, 11, 29, 41` → the **reading** is refuted, not the\n  certificate. Note `part1`'s definition-level brute force in the same file matches OEIS exactly\n  (`run_matches_oeis` true for n = 2..5), so the mismatch is in the residue→covering rule.\n\nNeither route 152 nor return #1580 states that rule, nor the witness run's start.\n\n## Reproduce\n\n```\npython3 .solveathome/runs/run-2026-09-24-h/work/verify_prefix308.py     # writes verify_prefix308.json\npython3 -c \"import json;print(json.load(open('.solveathome/runs/run-2026-09-24-h/work/definition_scan.json')))\"\n```\n\n## Calibration\n\n- **Verified:** the arithmetic `6*308+5 = 1853`; the published anchor and OEIS's 22-term list; the\n  sequence definition via my own scan for n = 2..7; the negative result on the natural residue reading.\n- **Carried, not verified:** node counts, timings, branch number, the `witness.py` re-check, the Lean\n  item, the R = 309 decision (private `jtwin_hb2` process).\n- **Not claimed:** exactness of A144311(23), any asymptotic statement, any statement about G_2.","prior_art_md":"# Prior art — updated online search record (route 152 triage, job 3041)\n\nSearches run 2026-09-24 from this session. One engine query\n(`OEIS A144311 terms sequence values 1709 1853`) returned nothing usable; the two direct source\nfetches below carry the whole record.\n\n## 1. OEIS A144311 — live page, `https://oeis.org/A144311` (fetched 2026-09-24)\n\n- Name: \"The length of the longest sequence of consecutive integers, each equal to 1 or -1 modulo at\n  least one of the first n primes.\"\n- Data: `1, 5, 11, 29, 41, 65, 107, 149, 203, 257, 347, 527, 545, 617, 707, 869, 965, 1079, 1283,\n  1397, 1529, 1709` — **22 terms**, LIST carries `Table of n, a(n) for n=1..22`.\n- **No a(23).** `a(22) = 1709 = 6*284+5` confirmed, matching the route's published anchor exactly.\n- Extensions: a(8)-a(16) from Max Alekseyev, 18 Nov 2009; **a(17)-a(22) from Jinyuan Wang,\n  26 Nov 2024**. Keyword `nonn,more,hard`; author Andrew Carter, 17 Sep 2008.\n- Links: Jinyuan Wang C++ program; StackExchange \"OEIS A144311 Generating function\" (May 2016).\n- Consequence for the route: the certified lower bound **1853 > 1709** is not on the record of the\n  published sequence (an exact a(23) would be a new OEIS term; a lower bound is a fortiori new here).\n\n## 2. Ziller–Morack, `arXiv:1706.03668` — abstract page (fetched 2026-09-24)\n\n\"A short note on the computation of the generalised Jacobsthal function for paired progressions\",\nsubmitted 12 Jun 2017, **v1 only**, 3 pages, 1 table; ancillary files (`moduli_2.txt`,\n`permutations_2.txt`, `psi_2_min.txt`, `remainders_2.txt`). Abstract: the paired-Jacobsthal bound is\nsufficient for Goldbach and the prime-pairs conjecture; values were computed \"for primorial numbers\n**for primes up to 73**\".\n\nThe route's reading — \"the paired-Jacobsthal object is computed to p = 73 by Ziller–Morack\" — is\ntherefore **confirmed as stated, and the extent is unchanged** (single 2017 version, no later\nextension found). The route's 83# rung does extend that programme; I did not re-check their values.\n\n## 3. Corpus (served) as carried by the route, spot-checked and not contradicted\n\n#1554 (R = 307 certificate + certificate-on-discovery build, route 149), #1507 (banked rung 306 /\n`A144311(23) >= 1841`, five re-verified witnesses), #1381 (0018 v1 ascent design, R = 288 witness),\n#1580 (the R = 308 certificate this triage is about — read in full). All four are status `recorded`\nand none was re-verified here.\n\n## Exact remaining gap\n\n1. **Exactness:** the first REFUTED `R` (the route reports R = 309 in flight). Nothing published\n   decides it; needs the running ascent.\n2. **Reproducibility of the finite claim (new, found here):** the prefix-308 covering is asserted\n   \"re-checkable from the certificate alone\", but neither the residue→covering rule nor the witness\n   run's start is published, so the finite statement over the 21 primes is **not third-party\n   checkable** — the natural reading of the published residues is refuted by the published small\n   cases. Publishing one integer (the run start) and the rule closes this at 0 compute; the Lean\n   encoding in the route's own gap list is the same gap at a far higher price.\n\nNo external source found that states `A144311(23) >= 1853` or a prefix-308 covering."},"research_route_id":152,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_b7737c79c8d338e106b2e72d","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/152 and return #1580. Return the ordinary report and transcript plus research: {route_id: 152, 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":"1580","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/152","transcript_url":"/projects/twin-primes/return/1582/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}