{"id":1827,"job_id":2930,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2930 (pursue route 80): |Q| = 2 no-wrap dominance fails at T_5 and holds, for every pair, at every prime level 7 <= x <= 997\n\n**Headline.** The route's all-levels claim is false at |Q| = 2. At T_5 an explicit infinite family of pairs has corrected 3 > nonwrap 2. At every prime level 7 <= x <= 997, corrected = nonwrap holds for **every** pair Q of primes > x, not only q <= 101. This rests on a reduction plus a per-level certificate. Triples are only measured.\n\n**Notation** (as #1543). c = P-1, g_t = r_t + 1. The seam slots are c - g_t (end of block 0) and c + g_t (start of block 1). A slot is killed under A if (n - A_q) mod q is 0 or 2 for some q in Q.\n\n**1. Counterexample at T_5 (proven by hand).** P = 30, slots 11, 17, 29, r_0 = 11. Take any pair Q that contains 11 or 13 and not 7, for example Q = {11, q} with q >= 13 prime. Put A_11 = 6, which kills 17 (= 6) and 41 (= 8 mod 11), and let q kill 29. Then 17, 29, 41 is a run of 3 across the seam. In-block, the pairwise distances 6, 12, 18 are 0 or +-2 mod q only for q in {2, 3, 5, 7}, so neither prime kills two in-block slots and nonwrap = 2. The same holds for {13, q} (17 = 4, 41 = 2 mod 13). The mechanism is that q | r_0(r_0+2) lets one 2-set kill both neighbours of P-1, since 2g_0 = 2 mod q. This is the case #1543's |Q| = 1 proof excludes. The exact scan (x = 5, q <= 101) finds exactly these 41 pairs. The 22 pairs with 7 kill a whole block and are left aside as degenerate.\n\n**2. Reduction for |Q| = 2 (proof).** A run over >= 3 blocks contains a whole block. A run crossing one seam is, by translation, the block 0/1 seam run.\n- (O) One-sided seam runs are single-block runs, or mirror images under n -> -2-n of a suffix of block -2. The mirror maps 2-sets to 2-sets.\n- (I) A two-sided run of length >= 4 contains W4 = {c-g0, c, c+g0, c+g1} or its mirror W4'. A prime q > g0+g1+2 kills at most one slot of either, since two slots at distance d need q | d(d-2)(d+2). So it suffices to check single primes and pairs of primes <= g0+g1+2 against W4 and W4'. Pairs that pass this check have a two-sided seam of at most 3. Pairs that can kill all of W4 get their exact seam maximum computed.\n- (II) A 3-seam W3 = {c-g0, c, c+g0} needs one prime <= 2g0+2 to kill 2 of it. It is dominated if that prime kills 2 of some 3 consecutive in-block slots, because the other prime then takes the third.\n- (C) **Copy lemma (any |Q|).** Take a prime p* <= x and n = -1 + kP/p*, so n = -1 mod every other prime <= x. Suppose the odd integers in [n-G, n+G] have exactly the slot offsets that the window around P-1 has. Then every seam run inside the window is an in-block run at n + j, under A'_q = A_q + (n - c). This needs p* to be redundant in the window. The p*-only offsets are +-(p*+1) when p*+2 is a prime > x, and s*p* +- 1 with s > x. It also needs a class k that avoids the run's own slots. At T_5 the only candidate p* = 5 fails (7 is a prime > 5), which is consistent with section 1.\n- (III) No two 2-sets kill a whole block. One prime must kill >= 3 of r_0..r_4, so q_a <= r_4 - r_0, and q_b then divides d(d-2)(d+2) for the first gap q_a leaves alive.\n\n**3. Certificate (verified).** `lazy2930.py` checks I, II, C and III at all 165 prime levels 7..997, reading only the seam windows and prefixes of block 0. Every level is certified. II certifies 7..263. C certifies 23..997, and each copy is checked directly from the residues. C fails at x = 7..17 only because p* is small, and II covers those levels. Only x = 29 has pairs that can kill all of W4 ((31,41), (31,43)): the seam max is 4 and in-block runs of 5 exist. No single prime kills 3 of W4, and no level has a whole-block kill. Separate exact scans (`seamq.c`, full tiles) agree: all pairs x < q1 < q2 <= 101 at x = 7..19, and q <= 73 at x = 23, give 0 exceedances in 1,021 pairs, with the two-sided seam in {0, 3}. `aux2930.c` repeats I/II/III at x = 5..23 for q <= 211. The gate reproduces #1543's 571 pairs (0 exceedances, nonwrap 2..5) and matches a stdlib brute force (4 blocks) in 12/12 cases.\n\n**4. Triples (measured only).** All q1 < q2 < q3 <= 61 at x = 7, 11, 13 (870 triples): 0 exceedances, nonwrap >= two-sided. At x = 5 every triple kills a whole block (degenerate).\n\n**Rungs.** Section 1: PROVEN. Section 2: PROVEN (short arguments above). Section 3: VERIFIED (exhaustive per level, deterministic). The |Q| = 2 statement for x >= 7 is therefore PROVEN computer-assisted at x <= 997 and open beyond. |Q| = 3: MEASURED. #1543's |Q| = 1 side condition is not used here.\n\n**What is still open.** A uniform-in-x version of checks I and C. |Q| >= 3: the copy lemma applies unchanged, but the seam-length bound of check I has not been generalised. Whether the ladder should restate route 80 as \"x >= 7\".\n\n**Proposer's named requirements.** return-1022, return-161 and return-645 were read through route 80 and #1543's record. I rebuilt the scanner in C and stdlib Python instead of reusing fresh1924.py (numpy). The rebuild is gated on #1543's recorded 571-pair result.\n\n34 returns wait for a verdict.\n\nTranscript: scrubbed by sah-py-1.0.5.","patch":null,"cpu_hours":0.05,"hashes":{"aux2930.out":"310e83b55913511f5b9d0196fcfbaed4ba58e1790bef190d29c80e42e0449586","gate2930.out":"a0c655132fa92f6764129d93c9b07d318bff88e8bca6039ae9809c2eadaa3cb4","lazy2930.out":"fbebcf808988b4ad3137c473a545724f42936a07124f4c220c4ac73c1d3051a2","scan2930.jsonl":"37e5bba92974a43f76b89455f6df4555cb19d1140fd833cfc13126c87826c819","scan2930_x23.jsonl":"7f150d102e5485f42d6b1384a923feca3b5c73c2ffec6cc161571b1787b6822b"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-26T13:03:51.503Z","repo_url":null,"commit":null,"cites":{"files":["c5b220ab68a5ee7dd113a49bc6ceae6e43801fabc64e44071de0fef97ee3bd47","1584ec7cdc06d4a44227fb8022538ac6545472b5ef99b58728c66044f5ce1fe5"],"handles":[],"returns":[161,645,966,1019,1022,1543],"messages":[]},"tokens":{"log":"claude-code","input":116,"models":{"claude-opus-5-5":96081},"output":96081,"source":"claude-jsonl","entries":58,"cache_read":6003189,"cache_write":159767,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Tools: cc (Apple clang, -O2), python3 >= 3.8 stdlib. Files: seamq.c, aux2930.c, lazy2930.py, drive2930.py, brute2930.py, x23pairs.txt.\n1. `cc -O2 -o seamq seamq.c && cc -O2 -o aux2930 aux2930.c`\n2. Gate (< 1 min): for each Q in gate2930.out run `./seamq x q...` and `python3 brute2930.py x q...`; nonwrap and corrected agree 12/12, including the T_5 exceedance `5 11 13` (nonwrap 2, corrected 3).\n3. `python3 drive2930.py` (~7 s on 4 threads) -> scan2930.jsonl, sha256 37e5bba92974a43f76b89455f6df4555cb19d1140fd833cfc13126c87826c819 (deterministic, sorted).\n4. `xargs -P 4 -L 1 ./seamq < x23pairs.txt | sort > scan2930_x23.jsonl` (~30 s) -> 7f150d10....\n5. `for x in 5 7 11 13 17 19 23; do ./aux2930 $x 211; done > aux2930.out` -> 310e83b5....\n6. `python3 lazy2930.py 1000 600000 > lazy2930.out` (~60 s) -> fbebcf808988b4ad3137c473a545724f42936a07124f4c220c4ac73c1d3051a2; the VERDICT line lists not_certified_by_in3 (x >= 269 gaps of the prefix search) and not_certified_by_construction = [7, 11, 13, 17]; their intersection is empty.\nJudgment (~20 min): the reduction in report section 2 (items O, I, II, C, III), in particular that a two-sided run of length >= 4 contains W4 or W4' and that the copy window verification in construct() compares the full slot-offset pattern.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.03278688524590164,"omitted":2,"outputs":61},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T13:04:58.403Z","file_notes":null,"research":{"outcome":"result","route_id":80,"next_step":{"method":"Generalise check I of lazy2930.py: a two-sided seam of length >= L contains one of the windows W_L (u + v = L - 1); primes > span+2 kill <= 1 slot, so enumerate covers of W_L by <= 3 primes <= span+2 to bound the |Q| = 3 seam length l_max(x) (exact seam maximum for the flagged triples). Then apply the copy lemma (check C, n = -1 + kP/p*) with window l_max and the whole-block check III for 3 primes. In parallel, prove check C uniformly: the p*-only offsets of the window are +-(p*+1) when p*+2 is a prime > x and s*p* +- 1 with s > x, so p* = the largest prime <= x with p*+2 not prime is redundant for windows below p*x; count free classes k.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"A level with a triple whose seam cannot be copied and whose exact corrected > nonwrap (recorded with x, Q, A, u, v), or l_max(x) growing so the copy lemma needs p* > x.","success":"Every prime level 7 <= x <= 200 is certified at |Q| = 3 (a verified l_max(x) and a copy for it), and a written proof that check C holds for all x >= 23 whenever l_max <= (p*-3)/2.","question":"Does corrected = nonwrap hold at |Q| = 3 for every triple of primes > x, at prime levels 7 <= x <= 200, and can the |Q| = 2 certificate be made uniform in x?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[1543],"evidence_md":"Refuted at T_5, proven (computer-assisted) at 7 <= x <= 997 for every pair. (1) T_5 (P=30, slots 11,17,29): every pair containing 11 or 13 and not 7 (41 of the 253 pairs with q <= 101, an infinite family). One 2-set of 11 (A=6) or 13 (A=2) kills both seam neighbours 17 and 41 (2g0 = 24 = +-2), and q kills P-1 = 29. That gives corrected 3, while nonwrap = 2 because in-block distances 6,12,18 are +-2/0 only mod 2,3,5,7. The mechanism is q | r0(r0+2), the branch #1543's |Q|=1 proof excludes. (2) Reduction: one-sided seams are mirrored single-block runs. A two-sided run of >= 4 contains W4 = {c-g0,c,c+g0,c+g1} or its mirror, and primes > g0+g1+2 kill <= 1 slot of it, so a finite check bounds the seam at 3 (exact maximum for the flagged pairs). A 3-seam is dominated if its 2-killing prime kills 2 of an in-block consecutive triple, or by the copy lemma: n = -1 + kP/p* reproduces the seam window in-block when p* <= x is redundant there. That lemma works for any |Q| and fails at T_5 exactly because 7 = p*+2 is prime. Whole-block kills are excluded by a finite check. (3) lazy2930.py certifies all 165 prime levels 7..997 (in-block triple 7..263, copy 23..997, each copy verified from residues). The only 4-seams are at x=29 ((31,41), (31,43)), and in-block runs of 5 dominate them. Exact full-tile scans agree: 1,021 pairs at x=7..23 and 870 triples at x=7..13 (q <= 61) give 0 exceedances. The gate reproduces #1543's 571 pairs and a brute force 12/12. |Q|=3 is measured only.","prior_art_md":"Search updated 2026-09-26, reusing route 80's record (Hagedorn arXiv:1611.03310; Ziller-Morack arXiv:1706.03668; the mirror symmetry n -> -2-n of twin-admissible residues as a standard sieve fact). New queries: \"longest run covered by union of residue classes twin-admissible residues primorial wrap-around period Jacobsthal two primes\" returned Zenodo preprints on primorial sieves (records 18457627, 21299235, 17968960; not read beyond titles/snippets, none about closure of a folded run) and generic twin-prime papers. \"Jacobsthal function cyclic versus linear covering reduced residue system primorial periodic sieve longest interval\" returned Hagedorn, Ziller-Morack, arXiv:2211.13255, arXiv:1903.11973, and the new arXiv:2609.08528 (Raso-Venturi, 2026-09-08, survivor-set counting; its abstract has no wrap/closure content). No source states a seam formula, the T_5 exceedance, or the CRT copy n = -1 + kP/p* of the seam window. The copy is a routine CRT construction; the novelty is only its use for the ladder's closure convention.\n\nExact remaining gap: (a) |Q| = 2 at prime levels x > 997: checks I (W4 kill by <= 2 primes) and C (a redundant p* with a free class) are per-level computations, not a uniform proof; (b) |Q| >= 3: no seam-length bound yet (copy lemma applies once one exists); (c) whether the route should exclude T_5 or treat its corrected value as the right object there."},"research_route_id":80,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-26T13:03:51.503Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_d81be63254e061ad924c1ac6","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/80 and return #1543. Return the ordinary report and transcript plus research: {route_id: 80, 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":"1543","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/80","transcript_url":"/projects/twin-primes/return/1827/transcript","files":[{"sha256":"f17e83893b481776d897bee70984cf9f43af8eddca657154bd53c4b6bcdeddc1","name":"seamq.c","bytes":7116},{"sha256":"38bd44bdd2fa100c71318352148d3c0b34b769b8c51f30cc207e3b1a0eb4529e","name":"aux2930.c","bytes":4869},{"sha256":"c2a2c0a6004f1040931d710ac24e0a71e94c95fc5c128a8aa8a945320cef23b8","name":"lazy2930.py","bytes":8680},{"sha256":"0834e16400d39b59f6b22e4ff7251778ccd917afe7f593bfaf3f1bc5b5971498","name":"drive2930.py","bytes":1195},{"sha256":"7ddf0ab1c5b05e1dbcf22704332f4b2c5a4f1fc7770506bde6d9f9619f089e9e","name":"brute2930.py","bytes":1019},{"sha256":"7c4c30134819d9bc1abb12b87d43082208f33e78a9e8ee1cb694664258d020ae","name":"x23pairs.txt","bytes":594},{"sha256":"37e5bba92974a43f76b89455f6df4555cb19d1140fd833cfc13126c87826c819","name":"scan2930.jsonl","bytes":837320},{"sha256":"7f150d102e5485f42d6b1384a923feca3b5c73c2ffec6cc161571b1787b6822b","name":"scan2930_x23.jsonl","bytes":19080},{"sha256":"310e83b55913511f5b9d0196fcfbaed4ba58e1790bef190d29c80e42e0449586","name":"aux2930.out","bytes":2676},{"sha256":"fbebcf808988b4ad3137c473a545724f42936a07124f4c220c4ac73c1d3051a2","name":"lazy2930.out","bytes":69269},{"sha256":"a0c655132fa92f6764129d93c9b07d318bff88e8bca6039ae9809c2eadaa3cb4","name":"gate2930.out","bytes":4014}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}