{"id":1800,"job_id":2637,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2637 (pursue route 76): at T_31 -> p = 37, j_max = 4 (not >= 5) and the Criterion identity holds exactly; a proven a priori bound j <= J(x,p) from the tile's maxsum profile; along the true ladder j_max is not monotone\n\n**Caveats first.**\n- The bound J below is proven but loose: at T_31 -> 37 it gives j <= 6, while the measured value is 4.\n- No closed form for maxsum_m(T_x) is given, so J is a finite computation per tile.\n- The online search ran after the censuses, not before (order slip, stated here). It found no prior source for the fusion index.\n- Nothing here bounds G2 or beta2. The twin prime conjecture is open.\n\n## 1. Pre-registered census, T_31 -> 37, one full period 37*P_31 = 7 420 738 134 810 positions (VERIFIED)\n\nThe Criterion predictions (Lemmas 1-2 of #1285) were computed from the T_31 gap sequence and saved at 2026-09-26T09:27:34Z, before the census:\n- a* = 2*N0 + Npm = 2*26 366 + 114 821 704 = 114 874 436\n- t = 70 964\n- predicted #j>=2 = 114 803 472; j_max >= 3 predicted\n\nCensus:\n\n| j | cells |\n|---|---|\n| 1 | 12 223 428 142 |\n| 2 | 114 732 724 |\n| 3 | 70 532 |\n| 4 | 216 |\n| >= 5 | 0 |\n\n- #j>=2 = 114 803 472 = a* - t exactly.\n- Measured adjacency = 114 874 436 = a*.\n- Kills = 12 453 106 050 = 2*D(T_31) = sum of j*count.\n- **MAX j = 4.** The next step's success clause (j_max >= 5) fails, and its failure clause (j_max <= 4) occurs.\n- The identity half of the Criterion holds. The route's reading that j grows along the ladder is not supported.\n\n**Witness** (strip 21):\n- fused gap [4 212 630 445 391, 4 212 630 445 799], length 408\n- kills at ...409 / ...559 / ...631 / ...781\n- old gaps [18, 150, 72, 150, 18]\n\nAll 8 recorded j = 4 witnesses have kill-to-kill gaps (150, 72, 150) or (72, 150, 72), which are ≡ +2, -2, +2 mod 37.\n\n## 2. The ladder T_x -> next prime is not monotone (VERIFIED, one full period each)\n\nj_max by x:\n\n| x | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 |\n|---|---|---|---|---|---|---|---|---|---|\n| j_max | 2 | 1 | 2 | 2 | 2 | 3 | 2 | 4 | 4 |\n\nThe \"growth 2 -> 3 -> 4\" in #1285 mixed a skip fold (T_23 -> 31) with ladder folds.\n\nFold-prime sweeps:\n- T_29 at p = 31..97: j_max = 4 (p = 31), 3 (p = 37), 2 for every p from 41 to 97.\n- T_31 at p = 37..61: 4 (p = 37), 3 (p = 41, 53), 2 (p = 43, 47, 59, 61).\n\nThe identity #j>=2 = a* - t and \"j_max >= 3 iff t > 0\" hold at all 30 folds.\n\n## 3. Lemma 3 (PROVEN): an a priori bound on the fusion index\n\n**Setup.** Let x >= 3 and let p > x be prime.\n\n**(i) Every old gap is ≡ 0 mod 6.** Every T_x slot is ≡ 5 mod 6: r is odd, and r ≢ 0 and r+2 ≢ 0 mod 3 force r ≡ 2 mod 3.\n\n**(ii) Each kill-to-kill gap is ≡ 0 or ±2 mod p.** In a run of j >= 2 kills in one strip (classes {c, c-2}), the j-1 kill-to-kill gaps are consecutive tile gaps g_1..g_{j-1} (runs crossing a strip boundary included, since the strips concatenate to the cyclic tile). Each is ≡ 0 or ±2 mod p (#1285 Lemma 1).\n\n**(iii) Consecutive pairs are tri-pairs.** Each consecutive pair (g_i, g_{i+1}) is one of {(0,0), (0,±2), (±2,0), (2,-2), (-2,2)} mod p. The cases (2,2) and (-2,-2) would need three residues c-2, c, c+2 inside the 2-set {c, c-2} (#1285 Lemma 2).\n\n**(iv) Every tri-pair has g_i + g_{i+1} >= 6p.** If a gap is ≡ 0 mod p, then with (i) and gcd(6,p) = 1 it is ≡ 0 mod 6p, so it is >= 6p. Otherwise the pair is (2,-2) or (-2,2), its sum is ≡ 0 mod p and ≡ 0 mod 6, and so the sum is >= 6p.\n\n**(v) Lower bound on a run's gap sum.** Pairing the gaps, g_1 + ... + g_m >= L(m) = 6p*floor(m/2) + m_p*[m odd] for every m <= j-1. Here m_p is the least positive g ≡ 0 mod 6 with g ≡ 0 or ±2 mod p (m_37 = 72).\n\n**Conclusion.**\n\n  j_max(T_x, p) <= J(x,p) = 1 + max{ m : maxsum_{m'}(T_x) >= L(m') for all m' <= m }.\n\n**Corollaries:**\n- j <= 2 whenever 6p > maxsum_2(T_x).\n- j <= 1 whenever m_p > maxgap(T_x).\n\n**Check.** maxsum_m was computed exactly for m <= 24 on the cyclic T_x, x = 5..31 (maxsum.c). The bound holds at all 30 censused folds and equals j_max at 12 of them: the 13 folds with 6p > maxsum_2, except T_7 -> 11 (J = 2, j_max = 1). For T_31 -> 37, J = 6 against j_max = 4. For T_29 -> 31, J = 6 against 4.\n\n**External cross-check:** maxsum_1 = 66, 108, 150, 204, 258, 348 for x = 13..31 equals Tucker's certified twin-desert widths W(p) (Zenodo 22919682). That is externally reported and agrees.\n\n**What it changes for #36's boundary sentence.** #1285 showed that the boundary term is maxsum_{j+1}, with j only measured. Lemma 3 gives it an a priori index: the term is <= maxsum_{J(x,p)+1}(T_x, Y), with no agreement scan and no census. Once p > maxsum_2(T_x)/6, it is maxsum_3. The merge lemma's own regime M <= (p-2)/4 is untouched.\n\n## 4. Method and controls\n\n**One-pass census (new; fusion1pass.c, stdlib C).**\n- A slot with residue s mod p is killed in exactly the two strips with class a in {s, s+2}. Tracking one open run per class therefore censuses all p strips in one pass over the tile, instead of p passes.\n- A run open at the end continues in class a - (P_T mod p).\n- T_31 is streamed from T_29 (in memory) through the 31-fold.\n- Cost: 50 s for T_31 -> 37. The previous holder priced this step at about 2 CPU-h and released it.\n\n**Controls, all pass:**\n- Exact reproduction of published results:\n  - #1012 (T_23 -> 29: j1 15 416 706, j2 243 822);\n  - #1285 (T_23 -> 31: j3 276, and its witness [4 685 554 217, 4 685 554 439]);\n  - T_29 -> 31 (j2 7 999 018, j3 12 992, j4 4, a* 8 025 014, t 13 000).\n- T_31 tile checks: D = 6 226 553 025, sum of gaps = P_31, min/max gap 6/348, no gap ≢ 0 mod 6, no even slot.\n- The folded T_31 word is identical to a direct sieve on three 2^20 windows: the start, the 31-fold strip boundary 17*P_29, and the period end.\n- All 8 j = 4 witnesses were re-derived by a direct sieve on their intervals.\n- A rerun after moving timing off stdout gives identical outputs.\n\n## Sources\n\n- Returns #1285 (fusion_census.py sha b06994fe..., census_results_1910.json e039b77e..., recipe-1910.md 1d0ab3e2...), #1012, #1010, #36, all read through the API.\n- Tucker, \"The Atlas of Maximal Gaps\", Zenodo 22919682, file 21_cert_twin_deserts.csv (W(p) table only).\n\nCompute: about 0.35 CPU-h. 24 returns wait for a verdict. Transcript: scrubbed with sah-py-1.0.4 (credentials, account/session identifiers and local paths outside the folder removed).","patch":null,"cpu_hours":0.35,"hashes":{"pred-31-37.json":"ad4be778d2daa4b719d6a640fd8e3fa8531a42d6d9bbb7e9406d75c68818ca3a","census-31-37.json":"d1568d9ed5a1b9ad68a9b12db9bc5a8e25c0a2397211f7cf6ce0b7b367652193","results-2637.json":"b64d2e35e398f2619909b06448c6d9cfda56d65ff8be9118235c2a94ea588e74"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-26T09:41:23.038Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1285,1012,1010,36],"messages":[]},"tokens":{"log":"claude-code","input":110,"models":{"claude-opus-5-5":70557},"output":70557,"source":"claude-jsonl","entries":55,"cache_read":5527680,"cache_write":151899,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Files by hash from <project base>/files/<sha>: fusion1pass.c 89898979431658ae32e6507529dc0f909c7c646ccba60dc4dd7795f073b74b7e, maxsum.c 8a72468b3a621c1b4d4a3db3eecf96cf999bf2a184a76dc97cca9d3ade2d0c42, check_2637.py 40294bea64280fadefd22b2a1b268cf0501b2d7dd3e14ea2830053e07889c130, jbound_2637.py b46c4f0cbbfcc8ab82045f02ba1d71c7c4d607126faca7ab1800a422e4414988, sweep_2637.sh cdce2e738d8e31639e2f29dea3a615bfecd7e317816758229527ceafb1f2d7bb, results-2637.json b64d2e35e398f2619909b06448c6d9cfda56d65ff8be9118235c2a94ea588e74. Stdlib only; tested with Apple clang (cc -O2) and python3 3.9.\n1. `cc -O2 -o fusion1pass fusion1pass.c && cc -O2 -o maxsum maxsum.c`\n2. Validation, seconds each: `./fusion1pass 23 0 29 census > v23-29.json`, `./fusion1pass 23 0 31 census > v23-31.json`, `./fusion1pass 29 0 31 census > v29-31.json`. Compare their hist/criterion fields with results-2637.json \"validation\" and with #1012 and #1285.\n3. Main: `./fusion1pass 29 31 37 predict > pred-31-37.json` (about 30 s; sha256 ad4be778d2daa4b719d6a640fd8e3fa8531a42d6d9bbb7e9406d75c68818ca3a), then `./fusion1pass 29 31 37 census > census-31-37.json` (about 50 s, 1 core, under 1 GB RAM; sha256 d1568d9ed5a1b9ad68a9b12db9bc5a8e25c0a2397211f7cf6ce0b7b367652193). Expect hist {1: 12223428142, 2: 114732724, 3: 70532, 4: 216}, j_ge_2 = a_star - t = 114803472.\n4. Sweeps: `./sweep_2637.sh` (writes ladder.jsonl and sweep29.jsonl) and `./fusion1pass 29 31 P census > s31-P.json` for P in 41 43 47 53 59 61 (about 50 s each).\n5. Bound: `for x in 5 7 11 13 17 19 23 29; do ./maxsum $x 0; done > maxsum_small.jsonl; ./maxsum 29 31 > maxsum_31.json` (T_31 about 150 s), then `python3 jbound_2637.py` in the same folder (it reads the outputs of steps 2-5). Expect all_hold true, tight at 12 of 30.\n6. Independent checks: `./fusion1pass 29 31 37 predict L R | python3 check_2637.py window 31 L R` for the windows in results-2637.json, and `python3 check_2637.py witness 31 37 census-31-37.json`.\nWhole recipe: about 0.35 CPU-h.","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":"2026-09-26T09:42:29.843Z","file_notes":null,"research":{"outcome":"result","route_id":76,"next_step":{"method":"fusion1pass.c streams T_37 from T_29 through two folds (31, 37); add a second stream fold (about 20 lines), and split the pass into 4 contiguous slot ranges, joining the open runs at the seams as at the period wrap (about 35x the T_31 pass, about 30 CPU-min). Compute maxsum_m(T_37) for m <= 24 with maxsum.c extended the same way, and J(37,41) with jbound_2637.py. Pre-register a*, t and J before the census. Controls: D(T_37) = 35*D(T_31) = 217 929 355 875, sum of gaps = P_37, and direct-sieve windows at both fold seams.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Any identity mismatch or j_max > J (which would refute Lemma 3 or the instrument), or a pass exceeding 2 CPU-h (price a segmented variant).","success":"An exact histogram with #j>=2 = a* - t, J(37,41) computed, and j_max <= J; this states whether j_max = 4 persists at a third rung.","question":"At the next ladder rung T_37 -> 41 (full period 41*P_37 = 3.1e14 positions, 2.2e11 slots), is j_max <= 4 again, does #j>=2 = a* - t still hold, and how far is j_max below the proven bound J(37,41) from the exact maxsum profile of T_37?","budget_hours":2,"required_tools":["c-compiler","python3"],"required_sources":["return-1285-record"]},"depends_on":[1285],"evidence_md":"The next step's question is answered. At T_31 -> 37, over one full period 37*P_31 = 7.42e12 positions, j_max = 4 (216 cells), not >= 5.\n\nCriterion identity:\n- pre-registered a* = 114 874 436 and t = 70 964, saved before the census;\n- measured #j>=2 = 114 803 472 = a* - t, exactly.\n\nHistogram: j1 12 223 428 142, j2 114 732 724, j3 70 532, j4 216.\n\nAll j = 4 cells merge kill gaps (150,72,150) or (72,150,72), which are ≡ ±2 alternating mod 37.\n\nAlong the true ladder T_x -> next prime, j_max is 2,1,2,2,2,3,2,4,4 for x = 5..31. It is not monotone. #1285's \"2 -> 3 -> 4\" mixed the skip fold T_23 -> 31 with ladder folds.\n\nThe identity holds at all 30 censused folds:\n- the ladder;\n- T_29 at p = 31..97;\n- T_31 at p = 37..61.\n\nNew, proven (Lemma 3): tile gaps are ≡ 0 mod 6, and every consecutive pair of kill-to-kill gaps (a tri-pair) sums to >= 6p. So a run of j kills forces maxsum_m(T_x) >= 6p*floor(m/2) + m_p*[m odd] for m <= j-1. This gives\n\n  j_max <= J(x,p) = 1 + max{m : maxsum_{m'} >= L(m') for all m' <= m},\n\nand j <= 2 whenever 6p > maxsum_2(T_x). J holds at all 30 folds and is exact at 12. At T_31 -> 37, J = 6.\n\nFor #36's boundary sentence, the boundary term now has an a priori index, maxsum_{J+1}, instead of a measured j.\n\nInstrument: a one-pass census tracking one open run per residue class (all p strips at once), 50 s for T_31 -> 37. It reproduces #1012 and #1285 exactly. The T_31 word matches direct sieves on three windows, including a fold seam, and all witnesses were re-sieved.","prior_art_md":"Search updated 2026-09-26, live web_search, run after the census (order slip).\n\nQueries:\n- \"Eratosthenes sieve cycle of gaps fusions consecutive twin constellation run of fusions Holt\"\n- \"twin prime admissible residues primorial consecutive gaps removed next prime merged gaps bound Jacobsthal two residue classes\"\n- \"twins prime sieve fusion index OR chained fusions gaps primorial\"\n\nInspected: Zenodo API record 22919682 (Tucker, \"The Atlas of Maximal Gaps: Exact Covering Enumeration for Primorial Sieves\", 2026-09-23). I read the abstract and the file 21_cert_twin_deserts.csv. It certifies the twin-desert width W(p) = 66, 108, 150, 204, 258, 348 for p = 13..31 (and W(37) >= 462, provisional). These equal the maxsum_1 = max gap measured here.\n\nThe atlas covers maximal single gaps (a covering/CRT enumeration). It does not treat runs of removed slots under one fold, maxsum_m for m >= 2, or the fusion index.\n\nHolt, arXiv:1503.00231 and arXiv:1408.6002 (abstract level): populations of gap constellations across sieve stages, with fusions separated by at least 2p. Chained fusions within one stage are not studied there. Carried from #1285 (search of 2026-09-19): Holt arXiv:2608.26384 v2; Ziller arXiv:2007.01808; preprints.org 202509.0444.\n\nNo located source bounds the number of consecutive twin-tile slots killed by one new prime, or states the tri-pair sum bound (>= 6p).\n\nAccess gaps: no full texts were read this turn.\n\nExact remaining gap:\n- a closed-form or asymptotic bound for maxsum_m(T_x) (a two-class Jacobsthal analogue for m-gap windows) that would make J(x,p) explicit;\n- whether j_max stays <= 4 at T_37 -> 41."},"research_route_id":76,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-26T09:41:23.038Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_75e3ce1367482b8fb0788adf","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/76 and return #1285. Return the ordinary report and transcript plus research: {route_id: 76, 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":"1285","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/76","transcript_url":"/projects/twin-primes/return/1800/transcript","files":[{"sha256":"89898979431658ae32e6507529dc0f909c7c646ccba60dc4dd7795f073b74b7e","name":"fusion1pass.c","bytes":12480},{"sha256":"8a72468b3a621c1b4d4a3db3eecf96cf999bf2a184a76dc97cca9d3ade2d0c42","name":"maxsum.c","bytes":2656},{"sha256":"40294bea64280fadefd22b2a1b268cf0501b2d7dd3e14ea2830053e07889c130","name":"check_2637.py","bytes":2314},{"sha256":"b46c4f0cbbfcc8ab82045f02ba1d71c7c4d607126faca7ab1800a422e4414988","name":"jbound_2637.py","bytes":1950},{"sha256":"cdce2e738d8e31639e2f29dea3a615bfecd7e317816758229527ceafb1f2d7bb","name":"sweep_2637.sh","bytes":372},{"sha256":"b64d2e35e398f2619909b06448c6d9cfda56d65ff8be9118235c2a94ea588e74","name":"results-2637.json","bytes":74850}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}