{"id":1422,"job_id":2806,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Triage of route 132: a plain CDCL encoding of A144311 reproduces the ladder to n = 12 but its UNSAT cost grows ~4-6x per prime; blocked as specified\n\n**Outcome: blocked (scoped obstruction).** The encoding is correct. Its cost is the problem: the measured growth rules out the n = 13..17 gate within the route's 1.5 h budget, and n = 25 by many orders of magnitude.\n\n## What was run (measured)\n- Encoding (file f6786c7b…, `gate.py`): one-hot residue r_p per prime p <= p_n (sequential-counter exactly-one). For each position i in [0,R), one clause OR_p [r_p = 1-i or r_p = -1-i mod p]. **The route's \"free start s vs fixed window\" uncertainty does not arise.** By CRT the start is absorbed into the free residues, so no start variable is needed. Solver: PySAT 1.9.dev15, CaDiCaL 1.9.5 (`cd19`), 1 core, under a process-group time/CPU/memory limit.\n- Gate: OEIS A144311 has offset 1: a(1..22) = 1,5,11,…,1709. The route's list has an extra leading 1, but its n = 13..17 values (545..965) are correct. For **n = 1..12**, every R = a(n) was SAT with a lifted and independently checked witness, and every R = a(n)+1 was UNSAT (file 2f6ae019…).\n- UNSAT(a(n)+1) wall-clock seconds / conflicts: n=9 0.35 s/39k; n=10 1.93 s/195k; n=11 11.2 s/912k; n=12 48.8 s/3.08M. Ratio per added prime: 5.6, 5.8, 4.3.\n- An exact mod-6 reduction (file 5074eda5…, `reduced.py`). Primes 2 and 3 leave exactly one class mod 6 uncovered, and translating that class is absorbed by the p >= 5 residues. So a(n) = 6M(n)+5, where M(n) is the longest run k = 0..M-1 covered by the classes s_p ± 6^{-1} (mod p), p = 5..p_n. This also explains OEIS's comment that a(n) = 5 mod 6. It cuts the positions 6x and verified SAT/UNSAT for n = 9..11 (SAT at n = 12), but **did not reduce UNSAT cost**: n=11 took 24.7 s/1.97M with cd19, and Glucose4 was slower (101.9 s). n = 12 UNSAT was cut off by the time limit.\n\n## Extrapolation (heuristic, not measured)\nAt ×4-6 per prime from 48.8 s at n = 12: n = 17 ≈ 1-2 days of core time; n = 23..25 ≈ 10^8-10^11 s. For scale, Wang's 2024 exhaustive program already reached a(22) = 1709. So plain CDCL is behind custom search even on the gate, and cannot certify a(25) (R = 2454/2455) under any sane budget.\n\n## Scope\nMeasured only for this encoding family (direct one-hot, with or without the mod-6 reduction), CaDiCaL 1.9.5 and Glucose 4, single core, no symmetry breaking beyond the mod-6 reduction, no cube-and-conquer. The negative is about this instrument, not about SAT in general.\n\n44 returns wait for a verdict.","patch":null,"cpu_hours":0.13,"hashes":{"gate.py":"f6786c7b9ee03830183ac80a520158d71fca193af62ee3896e663877a3294a2e","reduced.py":"5074eda560a66351348539cac51b7a0e5f07b15da0c9a1694241cd4b7c3b0b72"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-22T21:54:04.109Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1415,1411,1405],"messages":[]},"tokens":{"log":"claude-code","input":74,"models":{"claude-opus-5-5":24898},"output":24898,"source":"claude-jsonl","entries":37,"cache_read":1894729,"cache_write":65869,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe (job 2806)\nPython 3.13 with `pip install --target ./pylib python-sat` (1.9.dev15 used). Put gate.py and reduced.py (files above) next to ./pylib.\n1. `python3 gate.py 1 12`: one JSON line per (n,R). Expect sat=true with witness_ok=true at R = a(n), and sat=false at a(n)+1. n = 12 UNSAT takes about 50 s on one core.\n2. `python3 reduced.py 9 11 cd19,g4`: same verdicts on the mod-6 reduced instance (m = floor(R/6)).\nTimes are machine-dependent; verdicts are deterministic. Reference output: runs.jsonl (2f6ae019…).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":39},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"runs.jsonl (2f6ae019…): UNSAT a(n)+1 = 0.35, 1.93, 11.2, 48.8 s for n = 9..12; reduced encoding n = 11 took 24.7 s.","statement":"A direct one-hot CNF encoding of the A144311 covering CSP (with or without the exact mod-6 reduction), solved by CaDiCaL 1.9.5 or Glucose 4, has UNSAT cost growing about 4-6x per added prime (n = 9..12). The n = 13..17 gate exceeds the 1.5 h budget, and n = 25 is out of reach by many orders of magnitude.","assumptions":"Single core; no symmetry breaking beyond mod-6 and no reflection breaking; no cube-and-conquer, XOR/pseudo-Boolean or BDD reasoning; extrapolation from 4 data points.","revisit_when":"An encoding or solver setup (symmetry breaking, cube-and-conquer, or a proof-logging exhaustive search such as Wang's program with certificates) shows UNSAT growth of at most about 2x per prime on n = 10..14, or a certificate format for existing exhaustive search is proposed."},"route_id":132,"depends_on":[],"evidence_md":"Encoding validated (measured): SAT at a(n) with checked witnesses and UNSAT at a(n)+1 for n = 1..12 (OEIS offset 1). The free-start concern is void because CRT absorbs the start into the residues. UNSAT cost with CaDiCaL 1.9.5: 0.35 s, 1.93 s, 11.2 s, 48.8 s for n = 9..12 (ratio 4.3-5.8 per prime; conflicts 39k -> 3.08M). An exact mod-6 reduction (a(n) = 6M(n)+5; cover k = 0..M-1 by s_p ± 6^{-1} mod p, p >= 5) cuts positions 6x but gave no speedup (n = 11 UNSAT 24.7 s); Glucose4 is slower. Extrapolated (heuristic): n = 17 gate about 1-2 core-days, n = 25 about 1e8-1e11 s. The route's weakest assumption (tractability at n = 25) is refuted for this encoding class, and the pre-registered n = 13..17 gate does not fit its 1.5 h budget. Files: gate.py f6786c7b…, reduced.py 5074eda5…, runs.jsonl 2f6ae019….","prior_art_md":"Search 2026-09-22: \"SAT solver Jacobsthal function computation covering consecutive integers primes\"; OEIS A144311 page. No SAT/CDCL treatment of Jacobsthal-type functions was found; the existing work uses custom exhaustive or greedy search. Inspected: OEIS A144311 (offset 1; a(8)-a(16) Alekseyev 2009, a(17)-a(22) J. Wang 2024 C++ program; comment a(n) = 5 mod 6). Hagedorn, Math. Comp. 2009 (h(n) for n < 50; h(49) took about 3 months on 30 computers: the scale that exact search needs). Ziller & Morack arXiv:1611.03310 (algorithms, primes <= 251). Ziller & Morack arXiv:1706.03668 (paired Jacobsthal function for primorials, primes up to 73; closest prior art for the twin-prime ± setting; its mapping to A144311 was not checked beyond the abstract). Costello & Watts arXiv:1208.5342 (computational upper bounds on h). Remaining gap: no certified UNSAT for any a(n)+1 beyond published search, and none at n >= 23. The plain encoding here is not the instrument for it."},"research_route_id":132,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_da55f23c995cabb5136f4e91","run_id":"run_9b61e87450f9e980d3ee2df4","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/132 and return #1415. Return the ordinary report and transcript plus research: {route_id: 132, 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/132","transcript_url":"/projects/twin-primes/return/1422/transcript","files":[{"sha256":"f6786c7b9ee03830183ac80a520158d71fca193af62ee3896e663877a3294a2e","name":"job2806_gate.py","bytes":2206},{"sha256":"5074eda560a66351348539cac51b7a0e5f07b15da0c9a1694241cd4b7c3b0b72","name":"job2806_reduced.py","bytes":2195},{"sha256":"2f6ae019563a62074a2195a9a4b1197c73740d39f06e51cb22cbcd16ae755422","name":"job2806_runs.jsonl","bytes":4068}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}