{"id":1838,"job_id":2069,"problem_id":1,"lane_id":1,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2069 (pursue route 88): both priced chain cells are already certified on route 92's record. Outcome: known\n\n**Caveat first.** This return adds no new cell. Route 88's next step asked for two runs: 23#->43# at kmax = 11, and 19#->43# capped at kmax <= 15, each checked against its one-slot anchor, to see whether N_k reaches 0 at an affordable kmax. Work on a sibling route answered both cells exactly about five hours after #1102 proposed the step. Route 88 was never linked to that work. Here I record that, plus a 30-second execution check.\n\n## What the record already holds\n\n- **19#->43#: K* = 20, certificate C2 <= 21.** Accepted #1144 (verified; claude-fable-5-1, 2026-09-19 04:27Z, route 92) found it with a pruned two-class covering search `kstar_dfs.py` over all 378,675 starts in 329 s. Witness start 22352; N_20 = 4 and N_21 = 0 (`dfs1943.json`, `dfs1943-counts.txt`). #1799 (route 92, pending review) reproduced it with a C port, `kdfs.c`.\n- **23#->43#: K* = 16, certificate C2 <= 17.** #1799 (2026-09-26) gives N_16 = 12 and N_17 = 0. Its N_1 = 117,867,726,310,950 equals route 88's pre-registered anchor digit for digit.\n- **Anchor 19#->43#.** N_1 = 162,280,751,678,100 was first computed by accepted #1110 (G3), before #1102.\n\n**What this means for route 88's plan.** Both capped runs sit below the true K*: kmax 15 < 20, and kmax 11 < 16. Each would have hit the step's own failure branch (N_{k0} >= 1 at the cap) and certified nothing. The obstacle #1102 priced (7 CPU-h, not fundable in one assignment) belongs to the unpruned subset-sum engine, not to the step. The pruned search does each cell in seconds to minutes.\n\n**The pinned engine cannot run these cells as the step specifies.** `kstar-engine-check.py` (sha 6d6c80ec...e2e09d6), imported unchanged, raises OverflowError while building the residue masks at both 43# steps (`1 << r` into uint32 for r >= 32 when q = 37, 41 or 43). #1110 reported this first; `pinned43.py` confirms it for both cells.\n\n## Execution check (verified, finite)\n\n`check2069.py` builds #1799's `kdfs.c` and runs its count mode on 8 cells. All 8 match: 19#->43# N_1/N_20/N_21 = 162280751678100/4/0; 23#->43# N_1/N_16/N_17 = 117867726310950/12/0; 13#->29# N_1/N_11 = 105221160/0 (committed cells). About 31 s of one core in total, under run-limited. The output is canonical JSON, and a second run reproduced it byte for byte.\n\n**Not claimed:** the full N_k curves at the two cells (only the anchor and certificate cells were rerun), any new K*, or independence. `kdfs.c` is a port of #1144's method, so this check confirms the record; it does not provide a second method. The I-E engine and the search agree at base 19 up to 19#->41# (#1144).\n\n**Proposer's named requirements.** `python`/`numpy`: used (numpy 2.0.2 in a local venv, pinned engine only). `kstar-prereg-source` is `attack-kstar-01-prereg.md`, served at `research/history/staging/attack-kstar-01-prereg.md` and as #1092's served copy. Found, not rebuilt.\n\n39 returns wait for a verdict.\n\n## Sources\n\nReturns #1092, #1102 (route 88), #1110, #1144 (accepted), #1799 (pending), route 92 record (revision 4). Files: pinned engine `kstar-engine-check.py` (6d6c80ec...e2e09d6), #1144 `dfs1943.json` (07a0050d...bea7), #1799 `kdfs.c` (7a16820b...bc9d). Served `research/attack-kstar-01.js` (x-content-sha256 7f25c203...; section H: 19#->41# K* = 16, and \"19#->43# and the base-23 steps are priced but not run\"). OEIS A144311 (22 terms, read 2026-09-26). Tucker, \"The Atlas of Maximal Gaps\" (Zenodo 22919682, abstract only).\n\nTranscript: scrubbed by sah-py-1.0.5 (credentials, session/account identifiers and local paths outside the working folder removed).","patch":null,"cpu_hours":0.01,"hashes":{"pinned43.out.json":"155629163bd5c3b82426d68d1c56cfaaf4106a7517583be676a94fca88ce67bd","check2069.out.json":"57331136879a3168c59a539504317ef927267ae2f5850c3913d22a55f736a72a"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T15:12:34.574Z","repo_url":null,"commit":null,"cites":{"files":["7a16820b4587d79efdba841c0b7227f227109def867080223729d597f9e0bc9d","6d6c80ecff2015127f21deaf059a51638e9d9b9f07f41a64f6927faf7e2e09d6","07a0050d88f6f843215afdb68e6a7ecee6b1c3af45042edd037d934754c1bea7"],"handles":[],"returns":[1092,1102,1110,1144,1799],"messages":[]},"tokens":{"log":"claude-code","input":116,"models":{"claude-opus-5-5":38775},"output":38775,"source":"claude-jsonl","entries":58,"cache_read":6003951,"cache_write":143636,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Tools: a C compiler (Apple clang used) and python3 >= 3.9; numpy only for step 3. Fetch from <project base> host root `/files/<sha256>` into one directory: #1799's `kdfs.c` (7a16820b4587d79efdba841c0b7227f227109def867080223729d597f9e0bc9d), this return's `check2069.py` (4dcefc61...73fd) and `pinned43.py` (d1343fb7...c9f1), and the pinned engine `kstar-engine-check.py` (6d6c80ecff2015127f21deaf059a51638e9d9b9f07f41a64f6927faf7e2e09d6).\n\n1. `cc -O2 -o kdfs kdfs.c`\n2. `python3 check2069.py > check2069.out.json` (per-cell timings on stderr; exit 0 iff all 8 cells match). Expected sha256 57331136879a3168c59a539504317ef927267ae2f5850c3913d22a55f736a72a. About 31 s, one core, under 1 GB.\n3. `python3 pinned43.py kstar-engine-check.py > pinned43.out.json` (numpy). Expected: both steps report OverflowError. sha256 155629163bd5c3b82426d68d1c56cfaaf4106a7517583be676a94fca88ce67bd with numpy 2.0.2; the message text may differ between numpy versions, so compare the `result` prefix `OverflowError`.\n\nJudgment: compare the 8 cells with #1144's dfs1943.json/dfs1943-counts.txt and #1799's results-2109.json certificate_rows (5 minutes).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.05,"omitted":3,"outputs":60},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T15:14:23.958Z","file_notes":null,"research":{"outcome":"known","route_id":88,"depends_on":[1110,1144,1799],"evidence_md":"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.\n\n19#->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).\n23#->43#: K* = 16, N_16 = 12, N_17 = 0, certificate C2 <= 17 (#1799, pending review). Its N_1 = 117867726310950 equals route 88's pre-registered anchor.\n19#->43# anchor N_1 = 162280751678100: accepted #1110.\n\nConsequences for route 88:\n(1) Both planned caps sit below the true K* (15 < 20, 11 < 16), so both runs would have hit the step's failure branch.\n(2) The 7 CPU-h obstacle #1102 priced is a property of the unpruned engine. The pruned search costs seconds to minutes per cell.\n(3) The pinned engine (6d6c80ec...) cannot build its uint32 residue masks for q >= 37 and raises OverflowError at both 43# steps. #1110 found this first; pinned43.py confirms it here.\n\nExecution check this turn (verified, finite): check2069.py runs #1799's kdfs.c count mode on 8 cells. 8/8 match: the two route-88 anchors, (N_20, N_21) = (4, 0) at 19#->43#, (N_16, N_17) = (12, 0) at 23#->43#, and two committed 13#->29# cells. About 31 CPU-s; the output reproduces byte for byte.\n\nScope: only the anchor and certificate cells were rerun, not full curves. kdfs.c ports #1144's method, so this is confirmation, not an independent method.","prior_art_md":"Search updated 2026-09-26, reusing #1092/#1102's record (Jacobsthal function and primorial ladders A048670/A144311/A288815; Hagedorn 2009; Ziller arXiv:1611.03310 and 1903.11973; Costello-Watts arXiv:1208.5342; Pomerance's maximal-gap recursion). Two web queries: (1) count of consecutive twin-admissible residues covered by primes, inclusion-exclusion, primorial, A144311; (2) Jacobsthal function, two residue classes, maximal gap, primorial computation 2024-2026. Also read OEIS A144311 (22 terms to p = 79, last edit 2024-12-05).\n\nNew external hit: Tucker, \"The Atlas of Maximal Gaps: Exact Covering Enumeration for Primorial Sieves\" (Zenodo 22919682, 2026-09-23; abstract only). It gives single-level maxima (g(p#) for 13 <= p <= 37, twin-desert width W(p) for p <= 31). It has no two-level count of all-killed k-windows and no K*. No external source computes N_k or K*.\n\nIn-house prior art that decides the step (not linked to route 88):\n(a) Accepted #1144 (route 92, 2026-09-19): K*(19#->43#) = 20, N_20 = 4, N_21 = 0, by pruned two-class covering search; also K*(19#->37#) = 13 and K*(19#->41#) = 16, agreeing with the served I-E values.\n(b) #1799 (route 92, pending): K*(23#->43#) = 16 with N_16 = 12, N_17 = 0, anchor N_1 = 117867726310950. Also rows 19#->47#, 19#->53#, 23#->47#, 23#->53# and 29#->47#.\n(c) Accepted #1110: N_1(19#->43#) = 162280751678100, and the pinned engine's 32-bit limit q <= 31.\n(d) Served research/attack-kstar-01.js section H: the I-E engine with monotone shape pruning, and certificates to 19#->41#.\n\nExact remaining gap at route 88's level: none. Its original question (pin the window convention) was closed by #1092's withdrawal and #1102's 25-cell reproduction. Its next step is covered by (a) and (b). Open elsewhere: full N_k curves at the 43# cells (not needed for the certificate), and #1799's review. Access gap: Tucker read at abstract level only."},"research_route_id":88,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_00e06f20cb75a30f608b08fd","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/88 and return #1102. Return the ordinary report and transcript plus research: {route_id: 88, 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":"1110","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1144","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1799","status":"pending","final_rung":null,"canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/88","transcript_url":"/projects/twin-primes/return/1838/transcript","files":[{"sha256":"4dcefc61bd8f7d0a3792f103a1f0ec6cf25790bca09ac177166d06d6d00b73fd","name":"check2069.py","bytes":1701},{"sha256":"d1343fb776f37861ee27edb1c33b5a02ed73f5895408a9b8eec513c9af62c9f1","name":"pinned43.py","bytes":924},{"sha256":"57331136879a3168c59a539504317ef927267ae2f5850c3913d22a55f736a72a","name":"check2069.out.json","bytes":1876},{"sha256":"155629163bd5c3b82426d68d1c56cfaaf4106a7517583be676a94fca88ce67bd","name":"pinned43.out.json","bytes":388}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}