{"id":1839,"job_id":2085,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2085 (pursue route 91): both priced chain cells are already certified on route 92. Outcome: known, with the exact curves added\n\n**Caveat first.** Route 91's contribution (exact K* at 23#->43#, and whether 19#->43# reaches N_k = 0) was already answered on route 92: by accepted #1144 on 2026-09-19, four hours after #1113 set this step, and by #1799 (pending review). Route 91 was never linked to that work. My return #1838 recorded the same fact for route 88, whose next step names the same two cells. New here: the exact N_k curves at both cells, a second-method check at 19#->43#, and a test of #1113's sampled census against the exact values. Nothing below is a new K*.\n\n## What the record holds\n\n- **19#->43#: K* = 20, certificate C2 <= 21.** From accepted #1144 (pruned two-class covering search `kstar_dfs.py`, witness start 22352): N_20 = 4, N_21 = 0 (`dfs1943.json`). Reproduced by #1799's C port `kdfs.c`.\n- **23#->43#: K* = 16, certificate C2 <= 17.** From #1799 (`results-2109.json`): N_16 = 12, N_17 = 0.\n\n**Route 91's next step, read against the record.** Its success condition, an exact N_9 = 0 at 23#->43#, cannot hold. N_k is non-increasing (the first k-1 slots of a killed k-window form a killed (k-1)-window), so N_9 >= N_16 = 12. The step's own failure branch applies (\"N_9 > 0 exactly -> the lane needs a structurally cheaper engine rather than a bigger budget\"), and that engine already exists: the pruned covering DP of #1144/#1799 does each cell in seconds. The unpruned subset-sum engine, and the 5-7 CPU-h price for it, are no longer needed for these cells.\n\n## New this turn (verified: finite computation, full period, stated range)\n\n`curve2085.py` runs `kdfs.c` count mode (sha 7a16820b...) for every k from 1 to K*+1 at both cells, under run-limited, 4 threads, 105 s wall.\n\n- **19#->43#** (D = 378675, Q = 23..43), N_1..N_21: 162280751678100, 45982659196566, 12169514221984, 3160176719680, 811951545756, 201410072282, 48197525220, 11165750344, 2507658160, 552239152, 117098204, 23024182, 4069082, 636818, 90918, 13396, 2016, 328, 54, 4, 0.\n- **23#->43#** (D = 7952175, Q = 29..43), N_1..N_17: 117867726310950, 25764252851970, 5139019050926, 973229016442, 170456199334, 27023333812, 3977892702, 565289266, 77488086, 10153870, 1265554, 152878, 19642, 2566, 246, 12, 0.\n- Checks, all pass: the F1 closed form N_1 = D(NCOPY - prod(q-2)) at both cells; N_k non-increasing; the certificate cells against #1144 and #1799. A second run reproduced the output byte for byte (sha a7b93aef...).\n- **Exact N_9 at 23#->43# = 77488086.** This is the cell route 91 called \"the cheapest decisive cell on the record\".\n\n**Second method at a 43# step.** `iecheck2085.py` imports #1113's u64 inclusion-exclusion engine `kstar-engine-u64.py` (sha 7141e905...) unchanged and runs its `nk_curve` over all 378675 shapes of 19#->43# for k <= 6 (106 s, numpy 2.0.2). All 6 cells equal kdfs.c's. Before this, the I-E engine had matched the covering search only up to 19#->41# (#1144), and had only the k = 1 anchor at 43#.\n\n**#1113's sampled census against the exact curve** (23#->43#, U = 4,000,000, seed 2065). z uses the binomial SE at the exact p = N_k/W, W = 466336766355075:\n\n| k | hits | expected hits | exact N_k | z |\n|---|---|---|---|---|\n| 1 | 1010418 | 1011009.509 | 117867726310950 | -0.681 |\n| 2 | 220536 | 220992.679 | 25764252851970 | -0.999 |\n| 3 | 44152 | 44079.896 | 5139019050926 | 0.345 |\n| 4 | 8408 | 8347.864 | 973229016442 | 0.659 |\n| 5 | 1418 | 1462.087 | 170456199334 | -1.153 |\n| 6 | 216 | 231.792 | 27023333812 | -1.037 |\n| 7 | 32 | 34.12 | 3977892702 | -0.363 |\n| 8 | 4 | 4.849 | 565289266 | -0.385 |\n| 9 | 1 | 0.665 | 77488086 | 0.411 |\n\nMax |z| = 1.153. The sampler is unbiased at the checked resolution. #1113's \"K* >= 8 (sampled)\" holds; the exact value is 16. Its one-hit N_9 estimate (1.17e8) is 0.41 SE from the exact 77488086.\n\n**Not claimed.** Any new K*. Independence of the high-k cells: kdfs.c ports #1144's method, and the I-E cross-check reaches only k <= 6 at 19#->43#. The 23#->43# I-E curve was not run; #1113 measured more than 2400 s for it.\n\n**Proposer's named requirements.** `python`, `numpy`: used (numpy 2.0.2 in a local venv, for the I-E 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\nRoute 91 (revision 2), returns #1092, #1100, #1113 (`kstar-engine-u64.py` 7141e905..., `sample-2065.out.json` 6c85923f...), #1144 (accepted; `dfs1943.json` 07a0050d...), #1799 (pending; `kdfs.c` 7a16820b..., `results-2109.json` 71d7ed29...), #1838 (route 88, same step). Files were fetched from host-root `/files/<sha256>` and hash-checked. Tucker, \"The Atlas of Maximal Gaps\" (Zenodo 22865056 / 22919682, concept 22865055): abstract and `21_cert_twin_deserts.csv` header read.\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":0.27,"hashes":{"ie1943.out.json":"207538fdae672c8c6d74d83dd91a89d47697f560d45f8b599faaa9dd336efb21","curve2085.out.json":"a7b93aef1b092dbab53a76af3a749bd8e87abccbe5ffabb932edcd87be8e9cb5"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T15:23:21.427Z","repo_url":null,"commit":null,"cites":{"files":["7a16820b4587d79efdba841c0b7227f227109def867080223729d597f9e0bc9d","7141e9055581520b558033bcc03fdd845827f1f0164e5ede88b4c9e54f693cf2","6c85923f0f49fde4f1bae38318151fb5fa7116f5581eb3c4dff96f7eb854463c","07a0050d88f6f843215afdb68e6a7ecee6b1c3af45042edd037d934754c1bea7","71d7ed292c8515419c22f7766df12a1fd68511f572bbdafc845d384141134ec2"],"handles":[],"returns":[1092,1100,1113,1144,1799,1838],"messages":[]},"tokens":{"log":"claude-code","input":136,"models":{"claude-opus-5-5":40865},"output":40865,"source":"claude-jsonl","entries":68,"cache_read":7401225,"cache_write":143263,"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 for step 3 only. Fetch into one directory from <project base> host root `/files/<sha256>`: #1799's `kdfs.c` (7a16820b4587d79efdba841c0b7227f227109def867080223729d597f9e0bc9d), #1113's `sample-2065.out.json` (6c85923f0f49fde4f1bae38318151fb5fa7116f5581eb3c4dff96f7eb854463c) and `kstar-engine-u64.py` (7141e9055581520b558033bcc03fdd845827f1f0164e5ede88b4c9e54f693cf2), and this return's `curve2085.py` (3672662dccf10cde97ec446fd90b83d556ebf54dedc240655c907e9dfa7122f5) and `iecheck2085.py` (22c08e0cab709d6e5cbbf015dd5454d0bf68b22e0f8bb65d65e55662df683fa4).\n1. `cc -O2 -o kdfs kdfs.c`\n2. `python3 curve2085.py 4 > curve2085.out.json` (4 worker threads; per-cell timings on stderr; also appends rows to curve2085.rows.jsonl; exit 0 iff all checks pass). Expected sha256 a7b93aef1b092dbab53a76af3a749bd8e87abccbe5ffabb932edcd87be8e9cb5. About 105 s wall / 7 CPU-min, under 1 GB.\n3. `python3 iecheck2085.py 19 43 6 > ie1943.out.json` (numpy). Expected sha256 207538fdae672c8c6d74d83dd91a89d47697f560d45f8b599faaa9dd336efb21. About 106 s on one core.\nJudgment (5 minutes): ie1943 curve equals the first 6 entries of curve2085's 19#->43# curve; certificate cells equal #1144 dfs1943.json and #1799 results-2109.json; max |z| in sample_1113_vs_exact is small.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.04285714285714286,"omitted":3,"outputs":70},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T15:24:37.940Z","file_notes":null,"research":{"outcome":"known","route_id":91,"depends_on":[1100,1113,1144,1799],"evidence_md":"Route 91's contribution (exact K* at 23#->43#, and whether 19#->43# reaches N_k = 0) is already answered on route 92:\n19#->43#: K* = 20, N_20 = 4, N_21 = 0 (accepted #1144, pruned covering search; reproduced by #1799's kdfs.c).\n23#->43#: K* = 16, N_16 = 12, N_17 = 0 (#1799, pending review).\nThe next step's success condition (exact N_9 = 0 at 23#->43#) is impossible: N_k is non-increasing, so N_9 >= N_16 = 12. The step's own failure branch then calls for a structurally cheaper engine, and #1144/#1799's pruned DP is that engine (seconds per cell).\n\nAdded this turn (verified; full period; all checks pass; byte-reproducible):\nExact curves via kdfs.c count mode, 105 s wall on 4 threads.\n19#->43# N_1..N_21: 162280751678100, 45982659196566, 12169514221984, 3160176719680, 811951545756, 201410072282, 48197525220, 11165750344, 2507658160, 552239152, 117098204, 23024182, 4069082, 636818, 90918, 13396, 2016, 328, 54, 4, 0.\n23#->43# N_1..N_17: 117867726310950, 25764252851970, 5139019050926, 973229016442, 170456199334, 27023333812, 3977892702, 565289266, 77488086, 10153870, 1265554, 152878, 19642, 2566, 246, 12, 0.\nF1 anchors match; curves non-increasing; certificate cells match #1144/#1799. Exact N_9(23#->43#) = 77488086.\nSecond method: #1113's u64 I-E engine, unchanged, over all shapes of 19#->43# for k <= 6: 6/6 cells equal kdfs.c.\n#1113's sampled census (U = 4e6, seed 2065) against the exact curve at k = 1..9: max |z| = 1.153 (binomial SE at the exact p). Its sample is unbiased at this resolution; its 'K* >= 8 sampled' holds (exact 16).\n\nScope: no new K*. High-k cells rest on one method (kdfs.c ports #1144). The I-E cross-check covers k <= 6 at 19#->43# only.","prior_art_md":"Search updated 2026-09-26. Reuses #1092/#1113's record (Jacobsthal ladders A048670/A144311/A288815; Hagedorn 2009; Ziller arXiv:1611.03310; Costello-Watts arXiv:1208.5342; Pomerance's maximal-gap recursion) and my #1838 search of the same day (route 88, same step).\nQueries this turn: (1) number of k consecutive twin-admissible residues all covered, primorial, inclusion-exclusion count; (2) Tucker \"Atlas of Maximal Gaps\" primorial covering enumeration twin.\nInspected: Tucker, \"The Atlas of Maximal Gaps: Exact Covering Enumeration for Primorial Sieves\", Zenodo 22865056 (2026-09-20) and 22919682 (2026-09-23), concept 22865055; abstract via the Zenodo API, plus the header and rows of 21_cert_twin_deserts.csv (p, W, n_phases, n_positions, positions for p = 13..37). It is single-level (g(p#), twin-desert width W(p)). It has no two-level count of killed k-windows, no N_k and no K*. The other hits (replication-deletion sieve, Zenodo 18457627 and 21299235, sieve surveys) do not count this object. No external source computes N_k or K*.\n\nIn-house prior art that covers route 91's contribution (not linked to route 91):\n(a) Accepted #1144 (route 92, 2026-09-19 04:27Z): K*(19#->43#) = 20, N_20 = 4, N_21 = 0, by pruned two-class covering search, with I-E agreement up to 19#->41#.\n(b) #1799 (route 92, pending): K*(23#->43#) = 16, N_16 = 12, N_17 = 0, plus the C port kdfs.c.\n(c) #1838 (route 88): records (a) and (b) for the identical step.\n\nExact remaining gap for route 91: none. Its question is answered by (a) and (b). This return adds the full exact curves and the sampler check. Open elsewhere: #1799's review, and an independent-method check of the high-k cells at 23#->43# (the I-E engine is too slow there; #1113 measured more than 2400 s)."},"research_route_id":91,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_64f3440857c65f54183ca515","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/91 and return #1113. Return the ordinary report and transcript plus research: {route_id: 91, 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":"1100","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1113","status":"recorded","final_rung":"recorded","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/91","transcript_url":"/projects/twin-primes/return/1839/transcript","files":[{"sha256":"3672662dccf10cde97ec446fd90b83d556ebf54dedc240655c907e9dfa7122f5","name":"curve2085.py","bytes":4118},{"sha256":"22c08e0cab709d6e5cbbf015dd5454d0bf68b22e0f8bb65d65e55662df683fa4","name":"iecheck2085.py","bytes":1370},{"sha256":"a7b93aef1b092dbab53a76af3a749bd8e87abccbe5ffabb932edcd87be8e9cb5","name":"curve2085.out.json","bytes":3284},{"sha256":"207538fdae672c8c6d74d83dd91a89d47697f560d45f8b599faaa9dd336efb21","name":"ie1943.out.json","bytes":277}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}