{"id":1930,"job_id":4310,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 82, job 4310: singular-series models vs the exact Lambda_1(T_x, p) brackets\n\n**Caveat first.** The models are heuristic, and none meets the step's success clause. The step's first-order (Lemke Oliver-Soundararajan-type) term does not explain the exact brackets of #1925. The rung of the comparisons is measured: deterministic model evaluations against #1925's exact integer brackets.\n\nCaveat first: all three models are heuristic, and none meets the step's success clause. What is measured is how far each misses #1925's exact brackets. The answer to the step is no: the first-order (LO-S-type) singular-series term does not explain the brackets.\n\nModels (no fitted parameter; F(H) = prod_(5<=q<=x) (q - nu_q(H))/(q-2) exactly, delta = prod (q-2)/q):\nM1: h_a ~ F(0,a)(1-delta)^(a/6-1), N(a,b) ~ F(0,a,a+b)(1-delta)^((a+b)/6-2). It tends to L_p as x grows.\nM2 (the step's first-order interior term): every interior candidate is empty with its exact conditional probability p_i = F(H+i)/F(H), treated as independent.\nM3: M2 times exp(sum_(i<j) (p_ij - p_i p_j)/((1-p_i)(1-p_j))), the second cumulant of the interior-empty log.\nGap values run to 2400 (M1/M2) and 1200 (M3). The tail is < 1e-13, and M3 is unchanged at NMAX 160 vs 200.\n\nStep criteria, over the 99 (p, x >= 53) cells, p = 5..31:\n- within 0.01 of the bracket: M1 2, M2 1, M3 61. Max miss 0.306 / 0.182 / 0.070. The success clause fails for all three.\n- T41 -> T101 sign: all three are right at p = 7 and p = 23 (exact +0.017..+0.021 and -0.065..-0.029; M3 +0.022 and -0.061). The p=29/31 miss at T101 is 0.022 / 0.037 / 0 (< 0.05). The failure clause does not fire.\n- p=29 T101: [0.5263, 0.5882] M1 0.6102 M2 0.6252 M3 0.5582. p=13 T53: [0.4530, 0.4536] M1 0.5811 M2 0.6246 M3 0.4937.\n\nMechanism (measured): M1 and M2 put rho_1(60) at 0.45-0.50 at every tile, which is about L(60) = 0.4625 (#1918). So the first cumulant carries none of the finite-x deficit. rho_1(60) exact / c1 / c2: T23 0.0387 / 0.4624 / 0.2397; T31 0.1800 / 0.4754 / 0.3179; T41 0.2574 / 0.4761 / 0.3538; T53 0.3247 / 0.4790 / 0.3874; T71 0.3843 / 0.4971 / 0.4257; T101 0.4199 / 0.4962 / 0.4446. The second cumulant brings h_60 within 1% of exact at every tile, but N(60,60) is still 6.3x too large at T23 and 1.06x at T101. The deficit below L_p lives in higher-order joint emptiness of the interior candidates (how the small primes cover a long empty stretch). The first-order LO-S-type term cannot see it.\n\nOut-of-sample (prereg4310.md, sha in hashes, written before any p >= 37 bracket was computed): M3 at p = 37, 41, 43, 47 against brackets with half-width <= 0.05. Pre-registered FAIL: p = 37 misses by 0.123 (T53), 0.110 (T59), 0.095 (T61) ... 0.015 (T83), and is inside from T89. p = 41/43/47 are inside every bracket, but those brackets are wide. M3 is >= the lower end in 23/23 cells. The overshoot grows with the fold's smallest kill gap (72 at p = 37, 60 at p = 29/31; #1889).\n\nCorrection to #1925's text (same served folds4301.json): the p = 23 peak among the computed tiles is T47 [0.9971, 0.9976], not T41 (0.9925). The p = 7 minimum is also at T47 [0.88350, 0.88353], below T53 [0.88354, 0.88361]. M1 and M3 put the p = 23 peak at T47. M3 puts the p = 7 minimum at T41.\n\nNothing here bounds G2, beta2 or twin primes.\n\n## What was run\n- `model4310.c` (M1, M2) at 18 tiles T23-T101, gap values to 2400: 3 s wall on 4 processes. `m3_4310.c` (M3) at the same tiles, to 1200: 28 s wall. Everything was run under `run-limited`. Output reproduces byte for byte on rerun here (T53, T101).\n- `cmp4310.py` (step criteria), `diag4310.py` (histogram vs order decomposition; hybrid exact-histogram x model pair ratio), `cum4310.py` (rho_1(60) by cumulant order), `oos4310.py` (pre-registered p = 37..47 check).\n- No new exact tile. iecell.c refuses x > 101 (D >= 2^126), so the out-of-sample test used new folds on the existing tiles.\n\n## Sources\n- #1925 files c23..c101.json (iecell, A = 360), folds4301.json, ana4301.py (bracket), fetched as served / this department's own copies.\n- #1918 L(60) = 0.462542; #1889 g* table.\n- Literature: see prior_art_md.\n\n42 returns wait for a verdict.\n\nTranscript: scrubbed. Removed: credentials, local absolute paths outside the working folder, session and account identifiers, and lines not belonging to this assignment.","patch":null,"cpu_hours":0.06,"hashes":{"m53.json":"85d3d909282ee0e74cdff07879aa7947f6d4a6393c38f5bbaaa99ad08aea2633","m101.json":"dce6764aac69cccece356a77183a5f84db944893e9e9be63d93840518440ad98","m3_53.json":"09b7e1b8a189d44c67663bed6c878a6dbaf73fc653f849ebf673b4b205c9db16","m3_101.json":"265d4b1448eabaca293660d7d2031692270d7de25f3728472b16c27b80bd051b","cmp4310.json":"f6e68daff6e3a2684c2e77d57d74ab9bcc36fa6121a3082c01c12cfc576bdc1b","cum4310.json":"9803f22a42d829982f7824a693f4409b2f830e98641da876d592f191a7ce034e","oos4310.json":"0165d7ca5b4d09770d193d580d79b9c1826327734c3f6fd4d04f8b54da7a78c2","diag4310.json":"7a653e20efaecb2b37840df80f30ef1ffef939b247d7a6b59f39061caaaa9715","prereg4310.md":"1338db4f80f0fd14c8bdc4db270be92f73ad9a1a5f3c2b8c1ffc07b0798ff7e0"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-09-27T02:45:15.832Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1925,1918,1889],"messages":[]},"tokens":{"log":"claude-code","input":106,"models":{"claude-opus-5-5":61215},"output":61215,"source":"claude-jsonl","entries":53,"cache_read":6209718,"cache_write":179498,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# inputs: #1925's c<x>.json for x in 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 (in ../4301/), folds4301.json and ana4301.py from #1925 (<project base>/files/<sha>)\ncc -O2 -o model4310 model4310.c -lm; cc -O2 -o m3_4310 m3_4310.c -lm\n./runall.sh    # m<x>.json, NMAX 400, ~9 CPU-s total\n./runm3.sh     # m3_<x>.json, NMAX 200, ~95 CPU-s total\npython3 cmp4310.py ../4301/folds4301.json > cmp4310.json   # summary: d3 n_within_0.01_x>=53 = 61 of 99\npython3 diag4310.py > diag4310.json; python3 cum4310.py > cum4310.json\npython3 oos4310.py > oos4310.json    # verdict FAIL true, max_dist 0.123494 (p=37, T53)\n# hashes are from macOS/Apple clang libm; another libm may move the 6th decimal of m*.json; compare Lambda values to 1e-5, then the verdicts.\n# cheapest check: ./m3_4310 53 200 and ./m3_4310 29 200 plus oos4310.py (under 10 s).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.07017543859649122,"omitted":4,"outputs":57},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-27T02:46:53.331Z","file_notes":null,"research":{"outcome":"result","route_id":82,"next_step":{"method":"Extend m3_4310.c (this return): for each twin-admissible residue class r mod y# (y = 11, then 13), the interior candidates killed by q <= y are removed exactly, and p_i, p_ij are computed from the primes y < q <= x only; sum the class-conditional h_a, N(a,b) over r. Compare with the #1925 brackets (folds4301.json, ana4301.bracket on c<x>.json) with cmp4310.py/oos4310.py. Write the prediction table into a pre-registration before comparing. Label the model heuristic; build on #1925 and this return.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"At y = 13 the p = 37 T53 miss stays above 0.05, or the hybrid is further from the bracket than M3 at more than 10% of those cells.","success":"At y = 13 the hybrid is inside or within 0.02 of every bracket with half-width <= 0.05 for p = 5..37 and x = 53..101, including p = 37 at T53.","question":"Does an exact-small-prime hybrid (residues mod y# enumerated exactly for primes q <= y, the M3 second-cumulant interior expansion only over y < q <= x) remove M3's overshoot of the exact Lambda_1(T_x,p) brackets, in particular p = 37 at T53-T73 and p = 29/31 at T31-T61?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[1925],"evidence_md":"Caveat first: all three models are heuristic, and none meets the step's success clause. What is measured is how far each misses #1925's exact brackets. The answer to the step is no: the first-order (LO-S-type) singular-series term does not explain the brackets.\n\nModels (no fitted parameter; F(H) = prod_(5<=q<=x) (q - nu_q(H))/(q-2) exactly, delta = prod (q-2)/q):\nM1: h_a ~ F(0,a)(1-delta)^(a/6-1), N(a,b) ~ F(0,a,a+b)(1-delta)^((a+b)/6-2). It tends to L_p as x grows.\nM2 (the step's first-order interior term): every interior candidate is empty with its exact conditional probability p_i = F(H+i)/F(H), treated as independent.\nM3: M2 times exp(sum_(i<j) (p_ij - p_i p_j)/((1-p_i)(1-p_j))), the second cumulant of the interior-empty log.\nGap values run to 2400 (M1/M2) and 1200 (M3). The tail is < 1e-13, and M3 is unchanged at NMAX 160 vs 200.\n\nStep criteria, over the 99 (p, x >= 53) cells, p = 5..31:\n- within 0.01 of the bracket: M1 2, M2 1, M3 61. Max miss 0.306 / 0.182 / 0.070. The success clause fails for all three.\n- T41 -> T101 sign: all three are right at p = 7 and p = 23 (exact +0.017..+0.021 and -0.065..-0.029; M3 +0.022 and -0.061). The p=29/31 miss at T101 is 0.022 / 0.037 / 0 (< 0.05). The failure clause does not fire.\n- p=29 T101: [0.5263, 0.5882] M1 0.6102 M2 0.6252 M3 0.5582. p=13 T53: [0.4530, 0.4536] M1 0.5811 M2 0.6246 M3 0.4937.\n\nMechanism (measured): M1 and M2 put rho_1(60) at 0.45-0.50 at every tile, which is about L(60) = 0.4625 (#1918). So the first cumulant carries none of the finite-x deficit. rho_1(60) exact / c1 / c2: T23 0.0387 / 0.4624 / 0.2397; T31 0.1800 / 0.4754 / 0.3179; T41 0.2574 / 0.4761 / 0.3538; T53 0.3247 / 0.4790 / 0.3874; T71 0.3843 / 0.4971 / 0.4257; T101 0.4199 / 0.4962 / 0.4446. The second cumulant brings h_60 within 1% of exact at every tile, but N(60,60) is still 6.3x too large at T23 and 1.06x at T101. The deficit below L_p lives in higher-order joint emptiness of the interior candidates (how the small primes cover a long empty stretch). The first-order LO-S-type term cannot see it.\n\nOut-of-sample (prereg4310.md, sha in hashes, written before any p >= 37 bracket was computed): M3 at p = 37, 41, 43, 47 against brackets with half-width <= 0.05. Pre-registered FAIL: p = 37 misses by 0.123 (T53), 0.110 (T59), 0.095 (T61) ... 0.015 (T83), and is inside from T89. p = 41/43/47 are inside every bracket, but those brackets are wide. M3 is >= the lower end in 23/23 cells. The overshoot grows with the fold's smallest kill gap (72 at p = 37, 60 at p = 29/31; #1889).\n\nCorrection to #1925's text (same served folds4301.json): the p = 23 peak among the computed tiles is T47 [0.9971, 0.9976], not T41 (0.9925). The p = 7 minimum is also at T47 [0.88350, 0.88353], below T53 [0.88354, 0.88361]. M1 and M3 put the p = 23 peak at T47. M3 puts the p = 7 minimum at T41.\n\nNothing here bounds G2, beta2 or twin primes.","prior_art_md":"Search 2026-09-27 (job 4310), updating #1925/#1918's record (LO-S 1603.03720, Holt 2608.26384, Lau 2409.12819, David-Devin-Nam-Schlitt 2105.05048).\nQueries: web \"Lemke Oliver Soundararajan second order term consecutive residues sieve model singular series interior empty correction 2026\"; web \"consecutive gaps residue classes reduced residue primorial sieve twin bias Hardy-Littlewood Poisson correction\"; arXiv API abs:\"consecutive primes\" AND abs:residue AND abs:bias (by date; the API answered with no entries).\nInspected (listing/abstract level): LO-S (PNAS 2016) derives the second-order bias from Hardy-Littlewood with the interior-empty factor expanded to first order. The same first-order structure is our M2. DDNS 2105.05048 (Math. Ann.) does the same for sums of two squares. \"Sums of singular series along arithmetic progressions and with smooth weights\" (IJNT 2025, doi 10.1142/S1793042125500046) gives the averages of singular series in residue classes that such first-order terms need (listing only). A GitHub report (decompwlj) mentions an \"interior-credit law\" beside LO-S (search snippet only; not inspected).\nAccess gaps: no full text read. The IJNT paper is paywalled here.\nExact remaining gap: no located source treats the twin tile T_x, the fold ratio Lambda_1(T_x,p), or cumulant orders of the interior-empty probability beyond the first. This job shows the first order does not explain the finite-x data. The search is bounded; it is not an absence claim."},"research_route_id":82,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-27T02:45:15.832Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_6df1f3248ba720aaf001bcf3","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 #1925. 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; what to do, never when or how fast; 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":"1925","status":"pending","final_rung":null,"canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/82","transcript_url":"/projects/twin-primes/return/1930/transcript","files":[{"sha256":"aa9443ee950d0eb9c42f82e82abff1446b517c5779b1bee9d152a939b5fbd37e","name":"model4310.c","bytes":5040},{"sha256":"c0e78714ff7e6635f860eeb7e0e9f0380d80757c655504a577228fe0bce9d7dc","name":"m3_4310.c","bytes":3485},{"sha256":"621f0e40c94ff07c5b1cd3c2af01555336043ea6affbcb232b438552a0cbd4e5","name":"runall.sh","bytes":232},{"sha256":"81245084d8fdb0943f5d435c848e893fd0d42b65ab4a571dd94e5a42fd2a2c58","name":"runm3.sh","bytes":235},{"sha256":"6c2a895d230a778383e83e21f9e88954c703561b406180a1c2ee7cf9dbaec868","name":"cmp4310.py","bytes":2378},{"sha256":"1e204dacdcbe6107b979c4a29563d56ffa817d1a18e0b59774c8842dc845df27","name":"diag4310.py","bytes":1818},{"sha256":"0f82d2f6a9ed81b3163f64ecbbb453fb9be504357dc21fcb5465985479a9f9f4","name":"cum4310.py","bytes":1797},{"sha256":"9ecaf2f01090d9f12a9615b9535e05718da7cbd78e8f64d69d2d638ad811d87b","name":"oos4310.py","bytes":1285},{"sha256":"1338db4f80f0fd14c8bdc4db270be92f73ad9a1a5f3c2b8c1ffc07b0798ff7e0","name":"prereg4310.md","bytes":1221},{"sha256":"f6e68daff6e3a2684c2e77d57d74ab9bcc36fa6121a3082c01c12cfc576bdc1b","name":"cmp4310.json","bytes":29919},{"sha256":"7a653e20efaecb2b37840df80f30ef1ffef939b247d7a6b59f39061caaaa9715","name":"diag4310.json","bytes":8331},{"sha256":"9803f22a42d829982f7824a693f4409b2f830e98641da876d592f191a7ce034e","name":"cum4310.json","bytes":1302},{"sha256":"0165d7ca5b4d09770d193d580d79b9c1826327734c3f6fd4d04f8b54da7a78c2","name":"oos4310.json","bytes":6820}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}