{"id":1802,"job_id":2709,"problem_id":1,"lane_id":1,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2709 (pursue route 67): at T31, R_loose(q) <= 3 for every prime 31 <= q <= 349, so the refined correction is at most a four-term sum and LMAX = 8 truncates nothing at T31\n\n**Caveats first.**\n- This bounds the NUMBER of correction terms in #159's inequality at one tile, T31. It says nothing about their values, the exponent, the margin or G2, and it states no asymptotic.\n- The rung above T31 (T37, D = 2.18e11 slots) is not measured here except through #159's two quoted fields.\n- The seven T31 spectrum controls come from my own department's earlier return (#1800, same model, different algorithm). The external controls are the T29 values from #1244/#1347, #159's fields as read by #971, and Tucker's certified W(31).\n\n## 1. Census on T31 (VERIFIED, exhaustive, one full period)\n\nPre-registered in prereg.md (sha256 007a6b66...) before any run.\n\n**Object.**\n- T31 = the twin-slot tile of level 31: P = 31# = 200 560 490 130, and D = prod_{3<=q<=31}(q-2) = 6 226 553 025 slots.\n- It is streamed as the fold of an in-memory T29 gap word by 31, with the cyclic (true periodic) gap word.\n- Every gap is coded per prime q: g mod q = 0, +2, -2, or other.\n\n**Statistics.**\n- R_loose(q) is the longest cyclic run of qualifying gaps (#971/#1244).\n- The refined anchor-component spectrum uses refined2588.py's first-slot rule. It is computed by a two-run state machine driven only by the gap codes (x = run of the set {s-2,s}, y = run of {s,s+2}), which is a different algorithm from #1347's numpy look-ahead and from #1800's residue-class one-pass census.\n\n**Gates (all pass).**\n- D = 6 226 553 025.\n- The word sums to P.\n- The max gap is 348 = G2(T31). This matches Tucker's certified twin-desert width W(31) = 348 and #1800.\n- All gaps are multiples of 6, with 55 distinct gaps.\n- sum L*count = 2D = 12 453 106 050 at all 60 primes.\n\n**Controls (all exact).**\n- T29 control (same binary, level-29 mode, 13 s): #1244/#1347 spectra at q = 29 and 31 are reproduced to the unit. R_loose at all 46 primes 29..257 equals #1244's table, and refined L equals #1347's.\n- T31 spectra at q = 37, 41, 43, 47, 53, 59, 61 equal #1800's results-2637.json (e.g. q = 37: {1: 12223428142, 2: 114732724, 3: 70532, 4: 216}).\n- #159's recorded fields at T31 by 37 (max L = 4, longest qualifying run = 3, as read by #971) are reproduced: L = 4, R_loose = 3.\n\n**Results over the 60 primes 31 <= q <= 349 (= G2 + 1; 350 is not prime).**\n\n| R_loose | primes |\n|---|---|\n| 3 | 31, 37, 59, 61 |\n| 2 | 41, 53, 67, 89, 97 |\n| 1 | 43, 47, 71, 73, 79, 83, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 173 |\n| 0 | the other 32 primes, 163 and every prime >= 179 |\n\n| refined L | primes |\n|---|---|\n| 4 | 31, 37 |\n| 3 | 41, 53 |\n| 2 | 24 primes from 43 to 173 |\n| 1 | the other 32 |\n\nFull spectra and loose-run histograms are in t31.json. The spectra at q = 31, 37 are:\n- q = 31: {1: 11913887786, 2: 268582196, 3: 683680, 4: 708}\n- q = 37: {1: 12223428142, 2: 114732724, 3: 70532, 4: 216}\n\n**Structural range.** For the 99 primes 353..1009, no gap in the 55-value gap set is ≡ 0 or ±2 mod q, so R_loose = 0 and L = 1. This is checked from the gap set, consistent with #971's statement for q > G2 + 2.\n\n## 2. What changes\n\n(a) **Route 67's success clause holds at T31.** R_loose(T31,q) <= 3 for every prime in range, so by the proven one-line inequality L <= R_loose + 1 (#1347) the refined correction is at most a four-term sum. The loose support R_loose + 1 is also <= 4, half of the producer's LMAX = 8, so the truncation is idle at T31 in both forms at every fold prime, not only at #159's fold 37. The failure clause (some R_loose >= 7, or a gate/control mismatch) did not fire.\n\n(b) **The ladder is flat:** max R_loose is 2 (T23, #971), 3 (T29, #1244) and 3 (T31, here), and max L is 3, 4, 4. The primes attaining the maximum move: at T29 they are {29, 31}, at T31 {31, 37, 59, 61}.\n\n(c) **Tightness of L <= R_loose + 1.** It is tight at 55 primes and strict at q = 59, 61, 67, 89, 97. At q = 59 and 61, R_loose = 3 comes from exactly two loose runs of length 3 each, while L = 2. So the loose bound over-reads the refined support by one exactly where rare loose runs are not walks inside one 2-set. This is consistent with #1347's observation at T29.\n\n**Rungs.**\n- The census is VERIFIED: an exhaustive finite computation over one full period, with gates plus independent controls.\n- The four-term consequence at T31 is PROVEN, from the census and #1347's one-line inequality.\n\n## Sources\n- #159 (inequality, LMAX = 8; fields quoted via #971's reading of its out/b.log); #161; #162 (D(T31)); #968; #971; #1244; #1347 (refined2588.py sha256 e795b4bc..., frozen T29 values); #1800 (results-2637.json sha256 b64d2e35..., T31 spectra).\n- D. C. Tucker, \"The Atlas of Maximal Gaps: Exact Covering Enumeration for Primorial Sieves\", Zenodo record 22919682 (2026-09-23), file 21_cert_twin_deserts.csv, row p = 31: W(31) = 348. Externally reported and used here only as a gate value.\n\nCost: 0.47 CPU-h (439 s wall, 4 threads, Apple clang -O2), about 0.5 GB RAM.\n\nRequired names from the brief: gcc -> Apple clang cc; return-1244/159/162 were found on the record (values used as frozen literals; #159 via #971's reading). Nothing was rebuilt from those returns except the census itself, which is new code.\n\n24 returns wait for a verdict.\n\nTranscript: removed credentials, private account/session identifiers, absolute paths outside the working folder, and one third-party data file payload (Tucker's CSV, cited above).\n","patch":null,"cpu_hours":0.48,"hashes":{"t31.json":"7efd5477a423af04277d7c7c30e027c582472b1e79cce35d5b518bc528fe8954","checks-2709.out":"363726eb1461f83cac49ca000ce89d5d2aaaf4c55cfc0e4cdf67106efeac1b52"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-26T09:58:20.642Z","repo_url":null,"commit":null,"cites":{"files":["e795b4bc7712993817eed2c0503aa55e4d175ba1446b9b572516fe2f80d66372","b64d2e35e398f2619909b06448c6d9cfda56d65ff8be9118235c2a94ea588e74"],"handles":[],"returns":[159,161,162,968,971,1244,1347,1800],"messages":[]},"tokens":{"log":"claude-code","input":98,"models":{"claude-opus-5-5":47507},"output":47507,"source":"claude-jsonl","entries":49,"cache_read":4349867,"cache_write":126980,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Reproduce (any POSIX machine with a C compiler and python3; about 0.5 CPU-h, 0.5 GB RAM):\n1. Fetch rl31.c and check_2709.py from the server-root file store (/files/<sha256>, sha256 9ead38a0... and bfc0c094..., listed in this return's files). check_2709.py reads the q = 47, 53, 59 controls from ../2637/s31-<q>.json when present (#1800's sweep_T31 in results-2637.json holds the same numbers); without them those three lines are skipped and the checks output hash differs.\n2. cc -O2 -o rl31 rl31.c -lpthread\n3. T29 control: ./rl31 29 257 4 29 > t29.json  -> gates D = 214708725, sum = 6469693230, max gap 258; per_q spectra at 29, 31 equal #1244's; R_loose table equals #1244's (about 13 s wall on 4 threads).\n4. T31: ./rl31 31 349 4 > t31.json   (stdout is deterministic and thread-count independent; progress goes to stderr)\n   expected sha256(t31.json) = 7efd5477a423af04277d7c7c30e027c582472b1e79cce35d5b518bc528fe8954 (439 s wall on 4 threads here)\n5. python3 check_2709.py t31.json  -> 16 PASS lines, \"all_pass\": true; expected output sha256 363726eb1461f83cac49ca000ce89d5d2aaaf4c55cfc0e4cdf67106efeac1b52 when the 2637 control files are present.\nCheapest credible check: step 3 plus step 4 restricted to one prime (./rl31 37 37 1): its q = 37 entry must equal {1: 12223428142, 2: 114732724, 3: 70532, 4: 216} with R_loose = 3.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.07692307692307693,"omitted":4,"outputs":52},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T09:59:57.308Z","file_notes":null,"research":{"outcome":"result","route_id":67,"next_step":{"method":"Extend rl31.c by one streamed fold (T29 in memory, folds by 31 and 37 streamed; about 2.5e11 fold steps). Replace the per-prime loop with a bit-sliced loose-run counter (one 128-bit qualify mask per gap value; B_k = B_{k-1} & mask), so cost is independent of the number of primes. Run refined state machines only at the controls q = 41 (#159: L = 3, R = 3 at T37/41) and q = 43. Gates: D = 217,929,355,875; the word sums to 37#; max gap = measured G2(T37), compared with Tucker's bound.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Some q with R_loose >= 7 (the truncation could bite), R_loose = 4..6 (the four-term bound fails while the truncation stays idle), or a gate or control mismatch.","success":"R_loose(T37, q) <= 3 for every prime in range, and the q = 41 fields equal #159's (L = 3, R_loose = 3).","question":"Does R_loose(T37, q) <= 3 hold for every prime 37 <= q <= G2(T37) + 2 (Tucker: W(37) >= 462, provisional), so the four-term bound and the idle LMAX = 8 truncation persist at the tile of the producer's fold-41 run?","budget_hours":3,"required_tools":["cc","python3"],"required_sources":["return-159","return-1244"]},"depends_on":[159,971,1244,1347],"evidence_md":"T31 (D = 6,226,553,025, P = 31#, G2 = 348 = Tucker's W(31)) streamed in C, exhaustive over one period. R_loose(q) over the 60 primes 31..349: 3 at q = 31, 37, 59, 61; 2 at 41, 53, 67, 89, 97; 1 at 19 primes 43..173; 0 at the other 32. Refined L: 4 at 31, 37; 3 at 41, 53; 2 at 24 primes; 1 at 32. For 353 <= q <= 1009, no gap qualifies (checked from the 55-value gap set). Hence by #1347's inequality the refined correction is at most four terms at every fold prime of T31, and the loose support is <= 4 = LMAX/2, so the LMAX = 8 truncation is idle at T31 in both forms. The route's pre-registered failure (R_loose >= 7 or a control mismatch) did not fire. Controls: the same binary at level 29 reproduces #1244/#1347 exactly (both spectra and all 46 R_loose and L values); the T31 spectra at q = 37, 41, 43, 47, 53, 59, 61 equal #1800's (a different algorithm); #159's T31/37 fields (L = 4, R = 3) are reproduced. Ladder of max R_loose: 2 (T23), 3 (T29), 3 (T31). L <= R+1 is strict at 59, 61, 67, 89, 97. Scope: the number of correction terms at T31 only; no values, exponent, margin, G2 or asymptotic.","prior_art_md":"Search updated 2026-09-26 ~09:45 UTC (run 18). Queries: (1) 'longest run of consecutive admissible twin prime sieve residues modulo prime primorial tile gap sequence runs of gaps congruent 0 or ±2 mod q'; (2) 'twin prime admissible residues primorial consecutive gaps distribution maximal gap 31# twin sieve Jacobsthal function twin analogue'. The new source found is D. C. Tucker, 'The Atlas of Maximal Gaps: Exact Covering Enumeration for Primorial Sieves' (Zenodo 22919682, 2026-09-23). I inspected the record abstract and its 21_cert_twin_deserts.csv. It certifies the maximal twin-desert widths W(p) for 13 <= p <= 31 (W(31) = 348, which equals G2(T31) here) and W(37) >= 462 (maximality provisional). It studies maximal gaps and their positions, not a per-prime run statistic of gap residues. The other results were the Jacobsthal / large-gap line already in the route record (Ford-Green-Konyagin-Tao, arXiv:1408.4505; the Zenodo Poisson/sieve preprints). None defines R_loose or the refined 2-set run statistic on T_x, or bounds the support of a Tail-Count Transport correction. No external coverage and no novelty claim. Remaining gap: T37 (D = 2.18e11) has only #159's quoted fold-41 fields (loose support 4). Its all-prime sweep over 37 <= q <= G2(T37) + 2 (G2(T37) = W(37) >= 462 per Tucker, provisional) is not measured. Access gaps: the Tucker PDF was not read; OEIS was not queried."},"research_route_id":67,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-26T09:58:20.642Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_9a3bec7ea0f2a9712c8cd623","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/67 and return #1347. Return the ordinary report and transcript plus research: {route_id: 67, 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":"159","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"971","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1244","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1347","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/67","transcript_url":"/projects/twin-primes/return/1802/transcript","files":[{"sha256":"9ead38a00356b75cdfcec0100fe47ff7a779e3b9112274669c013d48210fa1d0","name":"rl31.c","bytes":8961},{"sha256":"bfc0c0945989f7926e657c421509c46c9abc7a9939256585a79488f360659561","name":"check_2709.py","bytes":2921},{"sha256":"7efd5477a423af04277d7c7c30e027c582472b1e79cce35d5b518bc528fe8954","name":"t31.json","bytes":7907},{"sha256":"363726eb1461f83cac49ca000ce89d5d2aaaf4c55cfc0e4cdf67106efeac1b52","name":"checks-2709.out","bytes":2352},{"sha256":"007a6b6676908798dc88a811e1312184c58a40e361120f5a3101d3a8c11e68b4","name":"prereg.md","bytes":2100}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}