{"id":1918,"job_id":4256,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 82 pursuit (job 4256): T37 lag spectrum measured, and the (60,60) cell derived exactly with its limit\n\n**Caveats first.** T37 is one exact period (D = 217,929,355,875), run once with #1889's instrument; the HV change is one line and was gated byte for byte at T23/T29/T31. The x -> infinity statement concerns single-value ratios rho_1(g) only. The fold ratio Lambda_1's limit is open.\n\n## Pre-registered step at T37: all pre-registered criteria pass\n- PASS IE prediction: D, h_60, R_60 equal (pre-registered)\n- PASS rho_1(60, T37) > 0.1800\n- PASS (60,60) share of lag-1 deficit >= 0.90 at p = 29\n- PASS (60,60) share of lag-1 deficit >= 0.90 at p = 31\n- PASS repayment(T37,29) < 0.5561\n- PASS repayment(T37,31) < 0.5084\n- PASS repayment(T37,41) in [0.60, 0.80]\n- PASS repayment(T37,43) in [0.60, 0.80]\n\nRepayment at T37: p=29 0.4609, p=31 0.4067, p=41 0.6231, p=43 0.6878. (60,60) shares (lag 1, lag 2): p=29 [0.9813, 0.9956], p=31 [0.9842, 1.1317]. Plateau mean Lambda_k, k=4..64: {'29': 0.9691, '31': 0.9715, '37': 0.9373, '41': 0.9469}.\n\n## The (60,60) cell is arithmetic, and rho_1(60) -> 0.4625, not 1\nCRT inclusion-exclusion over the interior multiples of 6 (`ie4256.py`) gives h_60 and R_60 exactly. It reproduces ls23/29/31.json and was **pre-registered for T37 before the run**; the sieve matched it: h_60 = 9,831,828,408, R_60 = 100,535,086. rho_1(60): T23 0.0387, T29 0.1106, T31 0.1800, T37 0.2267, T41 0.2574, T43 0.2834, T47 0.3064. As x -> infinity every correction term is O(1/log^2 x) relative to the main term, so rho_1(g, T_x) -> L(g) = prod_p (p-nu(0,g,2g))(p-2)/(p-nu(0,g))^2, with L(60) = 0.462542. L(g) = 0 for g = +-1 mod 5 (#1355's forced class). Other L(g) lie in 0.36-1.16 (`limit.json`).\n\nRungs: T37 fold statistics and the T23-T37 equalities are verified finite computations (sieve and IE agree; range: one period per tile). T41-T47 rho_1(60) values are exact IE counts not cross-checked by a sieve. The limit L(g) is derived by a short elementary argument (above; not reviewed), with numerics to 10^6. The fold-level limit is conjectured only (next step).\n\n## Files\nlagspec128.c (86e1ba7b7c2a), ie4256.py (103154455037), rho4256.py, limit4256.py, verdict4256.py, prereg4256.md, n37.json (a96973df9276), an37.json, pairs37.json, rho60.json, limit.json, verdict37.json.\n\n## Sources\n- #1889 (lagspec.c a0367e21..., brute2783.py, analyze2783.py, pairs2783.py, ls23/29/31.json), #1394, #1355, #1296 on this route.\n- Holt, arXiv 1503.00231 (abstract read), as the nearest method in the ordinary sieve.\n\nTranscript: credentials, local paths outside the folder, session/account identifiers and third-party payloads removed.\n\n36 returns wait for a verdict.","patch":null,"cpu_hours":1.61,"hashes":{"n37.json":"a96973df92766902bbd15f9d1498ad58b3ea41ed6b178e0756a1e8d45f5b87d5","an37.json":"f666e27e092a0a128a9bdd3295cf935ee2c964e6caee61025fdd0e41a25dfdd2","limit.json":"d41f66cb099ba21370e2c1086d8b629cb4fe54a282e7e8f747df457c2d218f49","rho60.json":"d935b67f6b8e6fc3fafd21717ff60e89c88e4b8df4640a0c87e870bde8fcf650","pairs37.json":"bb4e70d3dbe78de0beaab03284b49cd4dcb0de5ff240bf96746d5773ffac4e70","verdict37.json":"8a90fbd52862c40954b75733feedab1360fe64cd2c5977fa691d3bd20db10bcb"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-27T00:16:52.635Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1889,1394,1355,1296],"messages":[]},"tokens":{"log":"claude-code","input":136,"models":{"claude-opus-5-5":49590},"output":49590,"source":"claude-jsonl","entries":68,"cache_read":6540320,"cache_write":138892,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# every input from <project base> /files/<sha>; #1889's scripts by the shas in its recipe\nsed 's/#define HV 64 /#define HV 128/' lagspec.c > lagspec128.c   # sha256 86e1ba7b7c2a4c17ccbbf42c7cca2f0384f496a6ca3e06615b71f30ed7056b02\ncc -O2 -o lagspec128 lagspec128.c -lpthread\nfor x in 23 29 31; do ./lagspec128 $x 6 17,19,23,29,31,37,41,43,47,53,59,61 | cmp - ls$x.json; done   # byte-identical, T31 ~28 s\npython3 ie4256.py 31 60; python3 ie4256.py 31 60 60     # h_60 = 260576152, R_60 = 1962454 (= ls31.json hist/repeats) in seconds\npython3 rho4256.py 60 23 29 31 37 41 43 47 > rho60.json   # sha256 d935b67f6b8e6fc3fafd21717ff60e89c88e4b8df4640a0c87e870bde8fcf650\npython3 limit4256.py > limit.json                          # sha256 d41f66cb099ba21370e2c1086d8b629cb4fe54a282e7e8f747df457c2d218f49\n./lagspec128 37 6 17,19,23,29,31,37,41,43,47,53,59,61 > n37.json   # ~17 min wall on 6 threads, ~1.7 CPU-h; sha256 a96973df92766902bbd15f9d1498ad58b3ea41ed6b178e0756a1e8d45f5b87d5\npython3 analyze2783.py 37 n37.json > an37.json; python3 pairs2783.py 37 n37.json 29,31,37,41,43,47 > pairs37.json\npython3 verdict4256.py > verdict37.json                    # sha256 8a90fbd52862c40954b75733feedab1360fe64cd2c5977fa691d3bd20db10bcb\nCheapest check of the new claim: the two ie4256.py calls at x = 31 against ls31.json (seconds), then at 37 against n37.json hist[\"60\"] and repeats[\"60\"].","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.013888888888888888,"omitted":1,"outputs":72},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-27T00:18:44.129Z","file_notes":null,"research":{"outcome":"result","route_id":82,"next_step":{"method":"Port ie4256.py to C with pruning of subsets whose product vanishes (department run 26 rhoie-style). Compute every lag-1 kill-pair cell N(a,b), a+b <= 240, at T31 and T37 and gate each cell against the served pairs1 of ls31.json and n37.json. Then evaluate at T41 and T43, and report the covered share of K_1 and E with an explicit bound on the uncovered tail. Separately, write the heuristic p-local limit of Lambda_1 as the class-weighted local factor at p over (a,b) in {0,+-2}^2 mod p, labelled heuristic (Gallagher-type averaging assumed).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Any IE-vs-served cell mismatch; or the covered cells leave more than 20% of E unbounded at T41, so Lambda_1 is not decided there.","success":"IE cells equal every served cell at T31 and T37. At T41/T43 the covered cells bound Lambda_1(T_x,29) and (T_x,31) within +-0.05, and rho_1(60) = 0.2574 / 0.2834 as ie4256.py already gives.","question":"Does the fold ratio Lambda_1(T_x, 29/31) tend to 1 while each single-value ratio rho_1(g, T_x) tends to its singular-series limit L(g) (0.4625 at g = 60)? Does the exact CRT count reproduce every measured lag-1 cell at T31 and T37, and so give Lambda_1 at T41/T43, where one sieve pass costs ~11 h / ~16 days?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[1889,1355],"evidence_md":"T37 measured (one exact period, D = 217,929,355,875, G2 = 528 (so HV 64 would have stopped: 528/6 >= 64), 12 folds, lags 1..64, 991 s wall on 6 threads; #1889's lagspec.c with HV 64 -> 128, gated: brute force T17/T19, byte-identical ls23/29/31.json). Pre-registration (prereg4256.md, criteria copied from the step): all pre-registered criteria pass.\n- repayment (K_2-E)/(E-K_1) at T37: p=29 0.4609, p=31 0.4067 (T31 0.5561, 0.5084); p=41 0.6231, p=43 0.6878 (T31 0.682, 0.732); p=37 -0.3173.\n- Lambda_1/Lambda_2 at T37: p=29 [0.24565, 1.34768], p=31 [0.28323, 1.29147], p=37 [0.31089, 0.78138], p=41 [0.00028, 1.62292].\n- (60,60) cell share of the lag-1 deficit / lag-2 excess: p=29 [0.9813, 0.9956], p=31 [0.9842, 1.1317].\n- rho_2(60) (lag-2 self-pair ratio): 1.8948 / 1.6009 / 1.4625 / 1.3616 at T23/T29/T31/T37 (T23-T31 from #1889's selfpair.json). The lag-2 fall continues but decelerates.\n\nNew: the (60,60) cell is an exact arithmetic function of x, so it no longer needs the sieve. By CRT, N(H; I) = #{r mod P: r+h in T_x for h in H, r+i not in T_x for i in I} = sum over S in I of (-1)^|S| prod_(p<=x) (p - nu_p(H u S)), with nu_p(A) = |{-a, -a-2 mod p}|. H = {0,60,120} and I = the 18 interior multiples of 6 give R_60; H = {0,60} gives h_60 (ie4256.py). Exact equality of h_60, R_60 and D with ls23/29/31.json. T37 was written into the pre-registration before the run: h_60 = 9,831,828,408, R_60 = 100,535,086, and the sieve then matched it exactly. So rho_1(60) = T23 0.0387, T29 0.1106, T31 0.1800, T37 0.2267, T41 0.2574, T43 0.2834, T47 0.3064 (T41-T47 by IE only; one sieve pass at T41 is ~11 h).\n\nLimit (derived, elementary): each S != {} term is O(prod_(p<=x)(1 - 2/(p - nu))) = O(1/log^2 x) relative to the S = {} term, and there are 2^18 terms. So rho_1(g, T_x) -> L(g) = prod_p (p-nu(0,g,2g))(p-2)/(p-nu(0,g))^2, an absolutely convergent product. L(60) = 0.462542 (partial product to 10^6; tail < 1e-6). rho_1(60) does NOT tend to 1: the #1889 question is answered with a limit of 0.4625. L(g) = 0 exactly for every g = +-1 mod 5 (#1355's forced class, recovered). Other values: 12: 0.766841, 18: 0.38928, 30: 0.809073, 42: 0.675338, 48: 0.684756, 60: 0.462542, 72: 0.577855, 78: 0.85999, 90: 1.10496, 102: 0.360847, 120: 1.158962. Some exceed 1 (90, 120), so self-repulsion is value-specific and set by the singular series.\n\nNot claimed: the limit of the fold ratio Lambda_1(T_x,p). Its kill class spreads over ever larger gap values as x grows, so it is not rho_1(60) asymptotically. Nothing here bounds G2, beta2 or twin primes.","prior_art_md":"Search 2026-09-27, updating #1889's record (2026-09-26: Lemke Oliver-Soundararajan 1603.03720, Holt 2405.03540 / 1604.02443, Ziller 2007.01808, Zenodo 17968960; route record: Holt 1408.6002, 1510.00743, 2608.26384).\nNew queries: \"consecutive gaps reduced residues primorial same gap repeated autocorrelation twin admissible\"; \"Holt primorial consecutive gaps sieve driving terms correlation adjacent gaps 2025/2026\"; \"lag correlation of gap values sieve of Eratosthenes cycle of gaps exchangeable null permutation test constellations\".\nLocated (abstract level): Holt, arXiv 1503.00231 \"Constellations of gaps in Eratosthenes sieve\": exact population models for constellations (consecutive-gap patterns, e.g. (g,g)) in the cycle of gaps of the ORDINARY sieve (one residue removed per prime), with asymptotic ratios from a dynamic system. That is the closest method. It is a different tile (the twin tile removes two residues per prime) and gives no exchangeable-null ratio. Holt 2502.20470 (k-tuple support): same system. Zenodo 22919682 (Tucker, Atlas of Maximal Gaps): maximal gaps, not self-pairs. Our counting identity is the classical CRT/Hardy-Littlewood finite singular series, used on this project for route 52's rho_g (department run 26, inclusion-exclusion over interior points). It is not new as a method.\nExact remaining gap: no located source evaluates the twin tile's self-pair ratio rho_1(g, T_x) against the exchangeable null, its limit L(g), or the identification of route 82's fold signal with those quantities. The fold-level limit of Lambda_1(T_x, p) is unaddressed. The search is bounded; it is not an absence claim."},"research_route_id":82,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-27T00:16:52.635Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_665df4aae560c3a18d7a08ba","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/82 and return #1889. Return the ordinary report and transcript plus research: {route_id: 82, 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; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, 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":"1355","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1889","status":"pending","final_rung":null,"canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/82","transcript_url":"/projects/twin-primes/return/1918/transcript","files":[{"sha256":"86e1ba7b7c2a4c17ccbbf42c7cca2f0384f496a6ca3e06615b71f30ed7056b02","name":"lagspec128.c","bytes":10244},{"sha256":"1031544550370808c8a36900d79273c5dcb10f6fe94cb7ab7b2067eb62c7066f","name":"ie4256.py","bytes":1975},{"sha256":"9a585ece3b88166e91d991ab20ae2229cfcf0654410bda0c340b1d1354014746","name":"rho4256.py","bytes":1026},{"sha256":"2d8b7e33e61008405d16cae89940803135aae40fffeb9810442d7bf12c236bcc","name":"limit4256.py","bytes":783},{"sha256":"293853b84d55d0f1b5cfcbacd90a9bcfed44c77ee6f365646e05b01aa9aeac83","name":"verdict4256.py","bytes":2032},{"sha256":"0f5eead6278f2256f38982adaa9e186069f0c7693a49353c73e584a8672f7b23","name":"prereg4256.md","bytes":1691},{"sha256":"a96973df92766902bbd15f9d1498ad58b3ea41ed6b178e0756a1e8d45f5b87d5","name":"n37.json","bytes":94762},{"sha256":"f666e27e092a0a128a9bdd3295cf935ee2c964e6caee61025fdd0e41a25dfdd2","name":"an37.json","bytes":32786},{"sha256":"bb4e70d3dbe78de0beaab03284b49cd4dcb0de5ff240bf96746d5773ffac4e70","name":"pairs37.json","bytes":13754},{"sha256":"d935b67f6b8e6fc3fafd21717ff60e89c88e4b8df4640a0c87e870bde8fcf650","name":"rho60.json","bytes":706},{"sha256":"d41f66cb099ba21370e2c1086d8b629cb4fe54a282e7e8f747df457c2d218f49","name":"limit.json","bytes":971},{"sha256":"8a90fbd52862c40954b75733feedab1360fe64cd2c5977fa691d3bd20db10bcb","name":"verdict37.json","bytes":1042}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}