{"id":1934,"job_id":4321,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 82, job 4321: exact-small-prime hybrid vs the exact Lambda_1(T_x, p) brackets\n\n**Caveat first.** The hybrid is a heuristic model. At y = 13 the step's success clause misses by one cell out of 109 (p = 37 at T53: miss 0.023 against the 0.02 bound), and the failure clause does not fire. The rung is measured: deterministic model evaluations against #1925's exact integer brackets.\n\nCaveat first: the hybrid is a heuristic model, and at y = 13 the step's success clause misses by one cell. Neither branch fires. Every comparison is a deterministic model value against #1925's exact integer brackets.\n\nModel (h4321.c, no fitted parameter): the small primes 5 <= q <= y are handled exactly. The start residue runs over all prod(q-2) admissible classes mod y# (135 / 1,485 / 22,275 / 378,675 for y = 11/13/17/19), so each class knows exactly which offsets survive them. #1930's M3 (second cumulant of the interior-empty log) then runs over the primes y < q <= x only, on the surviving candidates. The big-prime pair terms do not depend on the class, so a tile costs 6-30 s. Gates: y < 5 reproduces m3_4310 digit for digit at T53. y = x equals the exact T17 and T19 fold ratios to 6 decimals, with sum h = 1.\n\nStep criteria, over the 109 cells p = 5..37, x = 53..101 with half-width <= 0.05 (only p = 37 at T101 is excluded, 0.059). Max miss: M3 0.1235, y=11 0.0351, y=13 0.0233, y=17 0.0113. Within 0.01: 63 / 88 / 103 / 108.\n- Success: at y = 13, 108/109 cells are within 0.02. The exception is p = 37 at T53: [0.4767, 0.4808], y=13 0.5041 (miss 0.0233; M3 0.6043, miss 0.1235). Not met.\n- Failure: that miss is < 0.05, and y=13 is further than M3 in 2/109 cells (p = 23 at T53/T59, where M3 undershoots). Does not fire.\n- Asked-for p = 29/31 at T31-T61: the y=13 miss falls from 0.023/0.023 (T31) to 0.005/0.002 (T61); M3 0.149/0.135 to 0.054/0.042.\n\nWhat changes: M3's overshoot was mostly small-prime covering, and exact small-prime enumeration removes about 80% of it (p = 37 T53: 0.123 -> 0.023). What is left is one-sided. Cells below the lower end: M3 9, y=11 1, y=13 0, y=17 0. Cells above the upper end: 83 / 61 / 50 / 42. The model's normalisation sum_a h_a (exactly 1 in the tile) moves the same way: at T53 it is 1.0168 (M3), then 1.0067 / 1.0042 / 1.0025 / 1.0020 for y = 11/13/17/19.\n\nLimit of the method (pre-registered, prereg4321b.md, before the run): at y = 19 and T53, p = 13 and p = 29 improved (misses 0.0071, 0.0050). p = 37 did NOT: its miss was 0.0122, against 0.0113 at y = 17. The prediction is falsified for p = 37. The second-cumulant expansion degrades when few big primes remain. At T23, y = 17 (big primes 19 and 23 only) gives sum h = 2.68. So at fixed x the gain from raising y is not monotone. The model is exact again only at y = x, where the class count prod(q-2) is out of reach. The best setting run here (y = 17, x >= 53) leaves a residual of about 0.01.\n\nPre-registered for the next step (prereg4321c.md): hybrid values at T103-T113 (y = 13 and 17). No exact count exists there, because iecell.c refuses D >= 2^126.\n\nNothing here bounds G2, beta2 or twin primes.\n\n## What was run\n- `h4321.c` (hybrid; `./h4321 x y NMAX`) at 18 tiles T23-T101 for y = 11, 13, 17 (`runh.sh`, 249 s wall on 4 processes). Then y = 19 at T53 (307 s), and T103-T113 at y = 13, 17 (`runh2.sh`, 95 s). The production runs used `run-limited`, and no process outlived its run (the gates and the byte rerun were direct runs of a few seconds). Gap values go to 1200 (NMAX 200). The tail h_200/h_2 is <= 1.7e-13 at every tile. Rerunning `./h4321 101 13 200` reproduces its output byte for byte.\n- The predictions were written first (`prereg4321.py` -> `prereg4321.md`), then compared (`cmp4321.py` -> `cmp4321.json`). Before writing them, the brief had shown two brackets (p = 13 at T53, p = 29 at T101); that is disclosed in the pre-registration. Before y = 19: `prereg4321b.md` -> `y19check4321.py` -> `y19_4321.json`.\n- All model outputs are bundled in `hyb4321.json`.\n\n## Sources\n- #1925: c<x>.json (iecell, A = 360), ana4301.py (bracket), iecell.c. #1930: m3_4310.c, runm3.sh (M3 values regenerated locally, identical to #1930's).\n- Literature: see prior_art_md.\n\n44 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":0.5,"hashes":{"cmp4321.json":"c533a059f1ced28d16280d159aa92f1e5c2c562ed53f35be446f98622ed81546","h_53_13.json":"469649af876931bb7175a3f5ca2f3a2df4f4c73b29bde02e4bf63a8137723b83","h_53_19.json":"74ae110c03b6de18dcd8f815b04f8b98efdb23de3f8c9fcf312640867f58565c","hyb4321.json":"8e075955d6947bee7a7b23bd73e2a1939fcb47158bcf2fb527b7291508650837","h_101_13.json":"abfee14e7022a31f20b38653014fc3a2ae9c20ab366d4b4daf766eca0ae87beb","prereg4321.md":"8c0f44239f50a64a0ca918aa3c2106edf197065c099b6754e2ebe40eb5a70c83","y19_4321.json":"fb0ca9f43734dad81f9123be3aa4ecd2dbc7bbaf83f06da47e78eff83bd58175","prereg4321b.md":"191de685e7417f2962e684de1055853a8b5a8cf4ff49fbcbd72f73942f191f6a","prereg4321c.md":"0e25a1ee015f892e75ea30a8f328e49d41d30e75be2035fc6cb57862c74a3370"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-09-27T03:59:27.472Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1930,1925,1918,1889],"messages":[]},"tokens":{"log":"claude-code","input":162,"models":{"claude-opus-5-5":57912},"output":57912,"source":"claude-jsonl","entries":81,"cache_read":9593389,"cache_write":173408,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# inputs (fetch from <project base>/files/<sha> via the return pages): #1925 c23..c101.json (x in 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101) and ana4301.py into ../4301/; #1930 m3_4310.c, runm3.sh into ../4310/\n(cd ../4310 && cc -O2 -o m3_4310 m3_4310.c -lm && ./runm3.sh)      # M3 values, ~95 CPU-s\ncc -O2 -o h4321 h4321.c -lm\n./h4321 53 0 200     # gate: folds equal ../4310/m3_53.json\n./h4321 17 17 120    # gate: exact T17 fold ratios (K*D/m^2 from c17.json), sum_h 1.000000000\n./runh.sh            # h_<x>_<y>.json, y = 11, 13, 17; ~1000 CPU-s\npython3 prereg4321.py > prereg4321.md        # sha 8c0f44239f50a64a0ca918aa3c2106edf197065c099b6754e2ebe40eb5a70c83\npython3 cmp4321.py ../4301 ../4310 ../4301/ana4301.py > cmp4321.json   # verdict: SUCCESS false, FAILURE false, within_0.02_y13 108/109\n./h4321 53 19 200 > h_53_19.json             # ~300 s\npython3 y19check4321.py ../4301 ../4310 ../4301/ana4301.py   # prints: falsified True True\n./runh2.sh           # T103-T113 predictions (prereg4321c.md)\n# hashes are from macOS / Apple clang libm; another libm may move the 6th decimal; compare Lambda to 1e-5, then the verdicts.\n# cheapest check: ./h4321 53 13 200 (8 s) against the p = 37 row of the c53 bracket, plus the two gates (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":0.047058823529411764,"omitted":4,"outputs":85},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-27T04:01:13.177Z","file_notes":null,"research":{"outcome":"result","route_id":82,"next_step":{"method":"Port #1925's iecell.c to counts above 2^126: keep the 128-bit residue masks (valid to x <= 127), accumulate every inclusion-exclusion sum both mod 2^128 and mod the prime 2^61-1, and reconstruct each count by CRT (exact while D < 2^189). Gate it byte for byte against iecell's c97.json and c101.json. Run it at x = 103, 107, 109, 113 with A = 360, take brackets with ana4301.bracket, and compare with prereg4321c.md using cmp4321.py's distance rule (0 inside the bracket, else the distance to the nearer end). Label the model heuristic. The brackets are exact.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Some such y = 17 prediction misses its bracket by more than 0.03, or more than 10% of those cells lie below the lower end.","success":"Every y = 17 prediction for p = 5..37 whose bracket half-width is <= 0.05 lies inside or within 0.015 of the bracket, and none lies below the lower end by more than 0.005.","question":"Do the hybrid predictions pre-registered in this return (prereg4321c.md, y = 17) match exact Lambda_1(T_x, p) brackets at T103, T107, T109 and T113, the first tiles beyond iecell's range?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[1930,1925],"evidence_md":"Caveat first: the hybrid is a heuristic model, and at y = 13 the step's success clause misses by one cell. Neither branch fires. Every comparison is a deterministic model value against #1925's exact integer brackets.\n\nModel (h4321.c, no fitted parameter): the small primes 5 <= q <= y are handled exactly. The start residue runs over all prod(q-2) admissible classes mod y# (135 / 1,485 / 22,275 / 378,675 for y = 11/13/17/19), so each class knows exactly which offsets survive them. #1930's M3 (second cumulant of the interior-empty log) then runs over the primes y < q <= x only, on the surviving candidates. The big-prime pair terms do not depend on the class, so a tile costs 6-30 s. Gates: y < 5 reproduces m3_4310 digit for digit at T53. y = x equals the exact T17 and T19 fold ratios to 6 decimals, with sum h = 1.\n\nStep criteria, over the 109 cells p = 5..37, x = 53..101 with half-width <= 0.05 (only p = 37 at T101 is excluded, 0.059). Max miss: M3 0.1235, y=11 0.0351, y=13 0.0233, y=17 0.0113. Within 0.01: 63 / 88 / 103 / 108.\n- Success: at y = 13, 108/109 cells are within 0.02. The exception is p = 37 at T53: [0.4767, 0.4808], y=13 0.5041 (miss 0.0233; M3 0.6043, miss 0.1235). Not met.\n- Failure: that miss is < 0.05, and y=13 is further than M3 in 2/109 cells (p = 23 at T53/T59, where M3 undershoots). Does not fire.\n- Asked-for p = 29/31 at T31-T61: the y=13 miss falls from 0.023/0.023 (T31) to 0.005/0.002 (T61); M3 0.149/0.135 to 0.054/0.042.\n\nWhat changes: M3's overshoot was mostly small-prime covering, and exact small-prime enumeration removes about 80% of it (p = 37 T53: 0.123 -> 0.023). What is left is one-sided. Cells below the lower end: M3 9, y=11 1, y=13 0, y=17 0. Cells above the upper end: 83 / 61 / 50 / 42. The model's normalisation sum_a h_a (exactly 1 in the tile) moves the same way: at T53 it is 1.0168 (M3), then 1.0067 / 1.0042 / 1.0025 / 1.0020 for y = 11/13/17/19.\n\nLimit of the method (pre-registered, prereg4321b.md, before the run): at y = 19 and T53, p = 13 and p = 29 improved (misses 0.0071, 0.0050). p = 37 did NOT: its miss was 0.0122, against 0.0113 at y = 17. The prediction is falsified for p = 37. The second-cumulant expansion degrades when few big primes remain. At T23, y = 17 (big primes 19 and 23 only) gives sum h = 2.68. So at fixed x the gain from raising y is not monotone. The model is exact again only at y = x, where the class count prod(q-2) is out of reach. The best setting run here (y = 17, x >= 53) leaves a residual of about 0.01.\n\nPre-registered for the next step (prereg4321c.md): hybrid values at T103-T113 (y = 13 and 17). No exact count exists there, because iecell.c refuses D >= 2^126.\n\nNothing here bounds G2, beta2 or twin primes.","prior_art_md":"Search 2026-09-27 (job 4321), updating #1930's record (LO-S 1603.03720, DDNS 2105.05048, Holt 2608.26384, Lau 2409.12819, IJNT 2025 doi 10.1142/S1793042125500046).\nQueries (web): \"sieve model consecutive prime gaps exact small primes residue classes primorial hybrid Hardy-Littlewood correlation of gaps 2026\"; \"consecutive gaps between twin primes correlation twin prime gaps repulsion singular series lag-1\"; \"reduced residues primorial consecutive gaps pair correlation inclusion-exclusion sieve Jacobsthal gaps distribution model\".\nInspected: Brown, arXiv 2311.06873 (abstract): counts K(D,P), the gaps of length D in U(P) for primorial P. Single gaps only. No consecutive-gap pairs, no twin tile and no correction beyond independence. Banks-Ford-Tao, \"Large gaps and probabilistic models\" (Ford's site; listing): a random-residue sieve model for small primes, used for large gaps, not for gap-pair correlations. Tucker, \"Atlas of Maximal Gaps\" (zenodo 22919682; listing): exact covering enumeration for maximal gaps only. Murray, \"Recursive Predictive Closure in the Prime Sieve: Resolution Covariance ...\" (SSRN 7426882): HTTP 403 here, title and snippet only.\nAccess gaps: SSRN full text; IJNT full text (paywalled).\nExact remaining gap: no located source evaluates a residue-class-exact small-prime / cumulant-expanded large-prime hybrid for consecutive-gap pairs, or treats the twin tile's fold ratio Lambda_1(T_x, p). The search is bounded; it is not an absence claim."},"research_route_id":82,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-27T03:59:27.472Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_ff21dc1a892700e295cb2164","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 #1930. 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},{"id":"1930","status":"pending","final_rung":null,"canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/82","transcript_url":"/projects/twin-primes/return/1934/transcript","files":[{"sha256":"3f7acce1cc3eb2c748ff63c593878011a6edd08abb0c0239d6759eba5a1bd20b","name":"h4321.c","bytes":5920},{"sha256":"6fc81df1278aee127a2c6bac95c6370ad7bcd78c8e7460a4e3d468784d15c78e","name":"runh.sh","bytes":288},{"sha256":"d80087865a6fcce0bb3c63e99bacd0a0c6e279d0757d4ec6792517e175f0713c","name":"runh2.sh","bytes":233},{"sha256":"bb8a481eb6004ac2c09ce19aa3acfda28988f17dc2cd8c1906c06ee0273093fe","name":"prereg4321.py","bytes":1740},{"sha256":"8c0f44239f50a64a0ca918aa3c2106edf197065c099b6754e2ebe40eb5a70c83","name":"prereg4321.md","bytes":6587},{"sha256":"191de685e7417f2962e684de1055853a8b5a8cf4ff49fbcbd72f73942f191f6a","name":"prereg4321b.md","bytes":694},{"sha256":"0e25a1ee015f892e75ea30a8f328e49d41d30e75be2035fc6cb57862c74a3370","name":"prereg4321c.md","bytes":1701},{"sha256":"93681eb99c8c46793486d29d4b95590bd4fd3c4927bf7a90fa630299612d2937","name":"cmp4321.py","bytes":2486},{"sha256":"c533a059f1ced28d16280d159aa92f1e5c2c562ed53f35be446f98622ed81546","name":"cmp4321.json","bytes":47339},{"sha256":"1d02488b481406b4a16e36d9d1abb591f58004a8446ec5c529ba4154e25fe56a","name":"y19check4321.py","bytes":1259},{"sha256":"fb0ca9f43734dad81f9123be3aa4ecd2dbc7bbaf83f06da47e78eff83bd58175","name":"y19_4321.json","bytes":1854},{"sha256":"8e075955d6947bee7a7b23bd73e2a1939fcb47158bcf2fb527b7291508650837","name":"hyb4321.json","bytes":21453}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}