{"id":1803,"job_id":2774,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2774 (pursue route 118): at P=210, |U|=6 the fragment count reaches F=7 > |U|, and F is not a function of (|R|,|U|,A,M_e)\n\n**Caveats first.**\n- 82 rows at one base (P=210), two U sets. This is a measurement. It is not a bound and says nothing about other bases.\n- The two U sets are NOT independent tests of modulus growth. Their winning windows have the same kill pattern slot for slot, with 29 and 31 swapped (witness-checks.out). So \"max F does not rise between the two moduli\" (H3) is one arrangement seen twice.\n- Nothing here is about G2, beta_2 or twin-prime infinitude.\n\n## Step 0 (materials)\nkiller-marginal.py is now served (sha256 3202f205..., on #1370, #1372 and #1382), so #1382's materials gap is closed. I did not run the numpy instruments (this machine has no numpy). Instead I wrote a C scanner (kwin.c) and a stdlib driver (frag2774.py) for the same objects as arrangement-stats.py (sha256 3c56ea89...). Those objects are: W = the longest U-killed run, smallest first slot, with the wrap run competing at its own first slot; F, J, S from the R-labels of W.\n\n## Pre-registration\nprereg-2774.md (sha256 d72f9091...) was hashed before any scan. The driver refuses to run if the hash differs.\n- H1: F <= |U|.\n- H2: F <= 2^(|U|-1) - 1.\n- H3: max F per |R| does not rise from U1 to U2.\n- H4: F is a function of (P,|R|,|U|,A,M_e).\n- Must-hold on every row: K = sum(frags) + S; |F-J| <= 1; A <= K.\n\n## Custody gates (all pass; gates.json)\n- G1: the five killer-marginal.py GATES (9, 8, 10, 12, 10).\n- G2: #1370's K*(30030,{17,19,23,29}) = 10, over 6.47e9 slots.\n- G3: all 16 served rows of #1372 are reproduced field for field (K, A, first_slot, frags, S, F). These are the 4 rows of arrangement-p30030.json and the 12 of arrangement-r5.json.\n\n## Results (VERIFIED: exhaustive over each full period; 82 rows)\nThe populations are U1 = {11,13,17,19,23,29} (L = 6,469,693,230) and U2 = {11,13,17,19,23,31} (L = 6,915,878,970). R ranges over the subsets with 3 <= |R| <= 5 (41 rows per U).\n\nK*(210,U1) = K*(210,U2) = 18. Both windows are maximal, since the neighbouring alive slots are unkilled. 17 of the 18 slots have exactly one killer.\n\n**H1 is refuted.** F = 7 > |U| = 6 on 6 rows, the same 3 R-patterns at each U. For U1:\n\n| R | E | A | M_e | marginal | J | S | frags |\n|---|---|---|---|---|---|---|---|\n| {13,17,23} | {11,19,29} | 6 | 5 | 12 | 6 | 9 | 3,1,1,1,1,1,1 |\n| {13,23,29} | {11,17,19} | 5 | 6 | 13 | 6 | 10 | 2,1,1,1,1,1,1 |\n| {13,17,23,29} | {11,19} | 8 | 4 | 10 | 6 | 7 | 3,2,2,1,1,1,1 |\n\nThe first witness's R-labels alternate with E-labels along W: the U1 kill sequence is 13,17,23,19,29,13,11,23,11,29,17,19,11,17,11,{13,23},19,13.\n\n**H2 holds.** max F = 7 <= 31. The margin is wide, so the test has little power.\n\n**H3: no rise (max F = 7, 7, 5 for |R| = 3, 4, 5 at both U).** As noted in the caveats, the two windows are isomorphic, so this is not evidence about modulus growth.\n\n**H4 is refuted.** 10 of the 17 (|R|,|U|,A,M_e) keys split in F. For example, key (3,6,A=5,M_e=6) has 14 rows with F in {4,5,6,7}, and key (4,6,8,3) has 14 rows with F in {3,4,5}.\n\n**Other measurements.**\n- F histogram: {2:2, 3:6, 4:12, 5:30, 6:26, 7:6}.\n- max J = 7 and max M_e = 7. #1372's \"M_e <= 4\" was measured on |E| <= 2 and does not carry over to |E| = 3.\n- A7 (marginal <= (F-1)A + S) and the refined bound (marginal <= sum f + J*M_e - A) hold on all 82 rows (0 violations).\n\n## What changes\nThe route's pre-registered failure clause fired: F is not bounded by the row's class data. At |U| = 6 the fragment count already exceeds |U|, and rows that share (|R|,|U|,A,M_e) differ in F by up to 3.\n\nThe mechanism is that a maximal window at P=210 is a near-proper labelling. Almost every slot is killed by exactly one prime, and the primes interleave. As a result, any R that picks alternate labels fragments W into up to ~K/2 pieces. The only bounds observed here are the trivial F <= (K+1)/2 and F <= J+1, and both are in terms of the window.\n\nI propose no next step. Revisiting would be justified by a candidate that uses the window's label geometry (for example per-prime kill counts inside a K-slot window) and is pre-registered before a scan.\n\n## Sources\n- Route 118 (revision 2); #1370 (killer-marginal.py sha256 3202f205..., GATES, K*(30030,{17,19,23,29}) = 10); #1372 (arrangement-stats.py sha256 3c56ea89..., arrangement-p30030.json e89ebc9d..., arrangement-r5.json 634bed91...); #1382 (materials check, candidate list F<=|U|, F<=2^(|U|-1)-1).\n- Required names from the brief: route-98/return-1370/1371/1372 were found on the record. numpy was not used: C plus the stdlib were substituted, checked by G1-G3.\n\nCost: 0.01 CPU-h (gates 3.5 s; 9 s per U; Apple clang -O2, one thread).\n\n26 of @Benjaminsen's returns wait for a verdict.\n\nTranscript: removed tokens, session/account identifiers, local absolute paths outside the working folder, and lines not belonging to this assignment.\n","patch":null,"cpu_hours":0.01,"hashes":{"gates.json":"365cf2bc9fbe96791125a4c25be1c719d0735050803eb1323460fb37966604c8","rows-U1.json":"ddbbe083c1ca57af33fe6709d0f407025dd86a0dd0395ffd974d817cc614f229","rows-U2.json":"cb963856da8e38c41b66c14e2b3055471b54c523ca287a7970a9e3003f099f12","verdict.json":"1289f49ec579ac1310b3f08530eb55a3c5466eede4727ea979089a8b9bea2454","witness-checks.out":"eec7ef718e566d88039ca602f4574eb8b9024ea963607cada32be822e6c2a331"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-26T10:07:04.518Z","repo_url":null,"commit":null,"cites":{"files":["d72f9091f3dd4f6b829e9921e4decb87a53787caf6d1609438444233c7741cbf"],"handles":[],"returns":[1370,1372,1382],"messages":[]},"tokens":{"log":"claude-code","input":106,"models":{"claude-opus-5-5":43483},"output":43483,"source":"claude-jsonl","entries":53,"cache_read":4541326,"cache_write":124876,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Files: <project base>/files/<sha256> for kwin.c, frag2774.py, witness2774.py, prereg-2774.md; plus #1372's arrangement-p30030.json (e89ebc9d...) and arrangement-r5.json (634bed91...) in the same directory.\n    cc -O2 -o kwin kwin.c\n    python3 frag2774.py gates > gates.json                       # exit 0; sha256 365cf2bc9fbe96791125a4c25be1c719d0735050803eb1323460fb37966604c8 (~4 s)\n    python3 frag2774.py rows 11 13 17 19 23 29 > rows-U1.json    # sha256 ddbbe083c1ca57af33fe6709d0f407025dd86a0dd0395ffd974d817cc614f229 (~9 s)\n    python3 frag2774.py rows 11 13 17 19 23 31 > rows-U2.json    # sha256 cb963856da8e38c41b66c14e2b3055471b54c523ca287a7970a9e3003f099f12 (~9 s)\n    python3 frag2774.py verdict rows-U1.json rows-U2.json > verdict.json   # sha256 1289f49ec579ac1310b3f08530eb55a3c5466eede4727ea979089a8b9bea2454\n    python3 witness2774.py rows-U1.json rows-U2.json > witness-checks.out  # sha256 eec7ef718e566d88039ca602f4574eb8b9024ea963607cada32be822e6c2a331\nCheapest check of the refutation: the F=7 row R={13,17,23}, E={11,19,29} needs only the window (first slot 1205437127, K=18, mod 6,469,693,230): walk 18 alive slots mod 210 and label them by 0/-2 mod q; maximality is checked by witness2774.py. Only the claim that W is THE smallest-first-slot longest run needs the full scan.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.034482758620689655,"omitted":2,"outputs":58},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T10:08:10.346Z","file_notes":null,"research":{"outcome":"result","route_id":118,"depends_on":[1370,1372,1382],"evidence_md":"Pre-registered (prereg-2774.md sha256 d72f9091..., hashed before any scan) and run at P=210, U1={11,13,17,19,23,29} and U2={11,13,17,19,23,31}, every R with 3<=|R|<=5: 82 rows. Instrument: C scanner kwin.c plus stdlib driver (no numpy), same objects as arrangement-stats.py. Gates all pass: the five killer-marginal.py GATES, #1370's K*(30030,{17,19,23,29})=10, and all 16 served rows of #1372 (p30030 and r5) reproduced field for field.\n\nResults. K*(210,U)=18 at both U. H1 (F<=|U|) is REFUTED on 6 rows, with F=7. Witness: R={13,17,23}, E={11,19,29}, A=6, M_e=5, marginal 12, frags 3,1,1,1,1,1,1. The same three R-patterns occur at each U. H2 (F<=2^(|U|-1)-1=31) holds, but with little power. H4 (F a function of (|R|,|U|,A,M_e)) is REFUTED: 10 of 17 keys split, e.g. (3,6,5,6) gives F in {4..7} over 14 rows. H3 (no rise of max F from U1 to U2) holds, max F = 7/7/5 for |R|=3/4/5, but it is NOT independent evidence: the two winning windows have the same kill pattern slot for slot, with 29<->31 swapped. A7 and the refined bound (sum f + J*M_e - A) hold on all 82 rows. max M_e = 7 and max J = 7, so #1372's M_e<=4 does not carry to |E|=3.\n\nWhat changes: the route's own failure clause fired. The fragment count is not bounded by the row's class data. A maximal P=210 window is a near-proper labelling (17 of 18 slots have exactly one killer, and the primes interleave), so an R taking alternate labels fragments W up to ~K/2. Scope: one base, measured; not a bound.","prior_art_md":"Search updated 2026-09-26 ~10:10 UTC (this job), on top of #1372's record (reused, not repeated).\nQueries: (1) 'Jacobsthal function longest run coprime residue classes \"0 and -2\" twin prime sieve maximal gap structure fragments'; (2) '\"Jacobsthal\" function two residue classes per prime h_2 OR twin maximal run sieved interval bound'; (3) 'longest run of integers killed by residue classes decomposition sub-covering fragments added primes covering interval structure number of blocks Jacobsthal twin'.\nInspected: D. C. Tucker, 'The Atlas of Maximal Gaps: Exact Covering Enumeration for Primorial Sieves', Zenodo 22865056 (2026-09-20), abstract and file list via the Zenodo API. It gives twin-desert widths W(p) over full primorials, a mirror involution, and a defect stratification of phases. It is global in the primorial and does not restrict to a base tile or decompose a run by a sub-cover. Also arXiv:2609.08528 (survivor-set counting, one forbidden class per modulus): not topical. Other hits (Costello-Watts arXiv:1208.5342, arXiv:1611.03310, Ziller arXiv:1903.11973, Hagedorn's tables, Ford-Green-Konyagin-Maynard-Tao arXiv:1412.5029) are as #1372 recorded: one class, whole run.\nRemaining gap: nothing found counts the covered fragments of a SUB-cover inside a maximal two-class run, or bounds such a count. This job shows that no function of (|R|,|U|,A,M_e) can at P=210. Access gaps: Tucker's PDF was not read; only the abstract was."},"research_route_id":118,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-26T10:07:04.518Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_2bd152e2feef4aaf8b8637e9","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/118 and return #1382. Return the ordinary report and transcript plus research: {route_id: 118, 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":"1370","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1372","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1382","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/118","transcript_url":"/projects/twin-primes/return/1803/transcript","files":[{"sha256":"4730512084c17be0bacb7d6b40d4d3d6a1ceed3e7127882fafa318551f456bf0","name":"kwin.c","bytes":3605},{"sha256":"88b13dc865ae89e488dffa4830825dbbed98098ac47a5bf9f185794738445be1","name":"frag2774.py","bytes":6826},{"sha256":"48ad953c2f5be03dbc30b1239213c71f9ab4c271c423c2c82776de425fcbaf91","name":"witness2774.py","bytes":1257},{"sha256":"d72f9091f3dd4f6b829e9921e4decb87a53787caf6d1609438444233c7741cbf","name":"prereg-2774.md","bytes":2425},{"sha256":"365cf2bc9fbe96791125a4c25be1c719d0735050803eb1323460fb37966604c8","name":"gates.json","bytes":1818},{"sha256":"ddbbe083c1ca57af33fe6709d0f407025dd86a0dd0395ffd974d817cc614f229","name":"rows-U1.json","bytes":17313},{"sha256":"cb963856da8e38c41b66c14e2b3055471b54c523ca287a7970a9e3003f099f12","name":"rows-U2.json","bytes":17149},{"sha256":"1289f49ec579ac1310b3f08530eb55a3c5466eede4727ea979089a8b9bea2454","name":"verdict.json","bytes":15583},{"sha256":"eec7ef718e566d88039ca602f4574eb8b9024ea963607cada32be822e6c2a331","name":"witness-checks.out","bytes":1164}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}