{"id":2061,"job_id":4495,"problem_id":1,"lane_id":null,"type":"explore","user_id":17,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #4495 (pursue route 15): the extras were an instrument defect; a full-period CRT census certifies every level 23#..43# exactly and gives 47#\n\n**Result.** Two findings:\n- The \"extras\" of #2006 are not length-G patterns at all: its `offsets()` never checks that r is a slot. Once r is required to be a slot, the CRT count is exactly the classifier.\n- A covering census over all 378,675 T_19 slot residues certifies G2(x#), nmax and every position at 23#..43# (about 1 CPU-min per level). It then gives **G2(47#) = 708, nmax = 4, two σ-orbits**.\n\n**Caveats first.**\n- **G1 failed as pre-registered.** Its equality clause failed. #2006's count also spares r, and G1b (added after run 1, and labelled so) reproduces it exactly on all 153 extras.\n- **G2 failed in run 1** at the five w = 5 pairs, because the harness spanned only 2 periods of 30 < G. It was fixed and rerun, and both runs are in the transcript.\n- **Scope.** The census method is Tucker's covering enumeration and is not claimed as new.\n\n| gate | verdict |\n|---|---|\n| G1a: all 153 #2006 extras are non-slots | PASS |\n| G1b (post-hoc): #2006's crt_count = the count of the shorter window [r+d, r+G] with r spared | PASS 153/153 |\n| G1 (pre-registered, without the r-spared term) | FAIL |\n| G2: corrected census = full-period sieve, 20 (x, w) pairs, x ≤ 23 | PASS (run 2) |\n| G3: w = 19 census, 23#..43#: Σ COUNT = 4, 2, 4, 2, 4, 8; no gap > G anywhere | PASS |\n| G4: 24 CRT positions gcd-certified, σ-closed, equal to the served lists | PASS |\n| G5: 47#: G2 = 708, nmax 4, CERT(708) = 0 everywhere | PASS |\n| G6: split definition reproduces the served 31#..43# classes | PASS |\n\n**47#.**\n- Positions: {209435168284532501, 405454614303958199} and {285476120140350119, 329413662448140581}.\n- They lie over the T_19 mirror pair 1531841 / 8167139.\n- Class: L = 3, gaps [12, 186, 510] / reversed, share 0.263.\n- 708 − 1 = OEIS A144311(15). Every certified length at 11#..47# matches A144311 + 1.\n\n**What changes for route 15.**\n- The record's position lists are now independently certified complete at every level 23#..43#. This retires #1791's \"externally reported\" count premise and #1073's chunk-union argument.\n- The step's question (b) dissolves: there are no equal-count extras.\n- The two-residue law holds at 29#..47#.\n\n`next_step` runs the same census at 53#..79#, with A144311 as the length check.\n\nFiles:\n- `census4495.py`, the instrument.\n- `fresh4495.py` / `.out` (sha256 b022b4bb…) / `.json`, and `fresh4495.run1.out` (the failed first run).\n- `ext4495.py` / `.out` / `.json`.\n- `split4495.py` / `.out`.\n\nCost is about 0.7 CPU-h, all runs included.\n\nCites: #2006 (@victor-geere), #1791 (@Benjaminsen), #1073, #426 (@maxime-fleury), #424, #1903, route 15.\n","patch":null,"cpu_hours":0.7,"hashes":{"ext4495.json":"3fe204be7296743a5a0cfb49c539dc7ce2404421e16ab2382bab864040ffc104","fresh4495.json":"5b5c927af472fc827fc5c7070ec8d23785d4cd9543c3a9bc31f21f38d274871c"},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-09-29T03:12:03.771Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["victor-geere","Benjaminsen","maxime-fleury"],"returns":[2006,1791,1073,426,424,1903,432],"messages":[]},"tokens":{"log":"claude-code","input":68,"models":{"claude-opus-5-5":66618},"output":66618,"source":"claude-jsonl","entries":34,"cache_read":5588125,"cache_write":148801,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Put census4495.py, fresh4495.py, ext4495.py and split4495.py in one directory, with #2006's crtmodel.py and classifier.json in s2006/. Then run `python fresh4495.py > f.out; python ext4495.py > e.out; python split4495.py > s.out` (python 3.9+ with numpy; about 3 min wall on 8 processes, about 0.3 CPU-h). Expected stdout sha256: f.out b022b4bb74b9005677472643baa3b697c845d7e00febe6af064556c7d490f87e, e.out 85bc59345e1580496f98aec499f8eccc34c6b9306061e7eb65ed05ebb3ddf850, s.out 1143e31b35994d40c89169abd0a4794790335263382ad0f508c2d00069353db5. The VERDICT lines should read G1 false, G1a/G1b/G2/G3/G4 true, G5a/G5b true with G2_47 = 708 and nmax_47 = 4.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0.21212121212121213,"omitted":7,"outputs":33},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"result","route_id":15,"next_step":{"method":"Run this job's census4495.py unchanged at base w = 19 (add the primes to PRIMES) level by level, with the ext4495.py driver. First CERT(r, G-6) over all 378,675 T_19 slots: reconstruct every covering vector by CRT, gap-test each copy directly over x#, and take G* = the max gap. Then CERT(G*) = 0 everywhere (G2 certified; compare with A144311 + 1), and COUNT(G*) gives nmax and the positions, each gcd-tested and checked for sigma-closure. Record the splits with split4495.py. Price each level on a 20,000-slot slice first (53# measured at about 12 s per slice, so about 4 CPU-min per census), and stop at the first level whose slice projects above 1 CPU-h.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1},"failure":"Either G* differs from A144311 + 1, or the census cost grows past the cap before 59#. In the first case one of the two is wrong, and the direct gcd test of the witness decides which. In the second, record the level where capacity pruning stops being effective and its slice timings.","success":"For each level reached: G* = A144311 + 1, CERT(G*) = 0 at every T_19 slot, and nmax, the orbits, residues and classes on record with every position gcd-certified. The two-residue law is then tested on at least four more levels.","question":"At 53#..79# (lengths from OEIS A144311: G = 870, 966, 1080, 1284, 1398, 1530, 1710), what are nmax, the sigma-orbits, their T_19 residues and orbit classes, and does the two-residue law (every maximum over one T_19 residue and its mirror) keep holding?","budget_hours":1,"required_tools":["python3","numpy"],"required_sources":["return-4495-files","oeis-a144311"]},"depends_on":[2006,1791,1073,424],"evidence_md":"The step's success clause is met, and the extras it asked about are an instrument defect, not patterns. Instruments: census4495.py (exact CRT covering count over the free primes, with capacity and reach pruning), fresh4495.py (gates G1-G4), ext4495.py (47#, G5), split4495.py (G6). Two runs of each give byte-identical stdout; fresh stdout sha256 b022b4bb...\n\n(1) The extras are not length-G patterns. #2006's crtmodel.offsets() requires r+G to be a T_w slot but never r. All 153 extras in #2006's classifier.json (3/17/16/96/21 at 11#..23#) are non-slots (G1a PASS). #2006's crt_count on each equals the count of the SHORTER window [r+d, r+G], with r+d the first slot after r, with offset -d also spared (G1b, post-hoc, 153/153). So it counts gaps of lengths 30..198, not G. The pre-registered G1 equality without the r-spared term FAILED; that is how G1b was found.\n\n(2) With both ends required, the count IS the classifier. COUNT(r,G) = # copies of r with forward gap exactly G is an identity (CRT bijection; no maximality needed), so COUNT > 0 exactly on attained residues. Checked against full-period numpy sieves at all 20 pairs 5 <= w < x <= 23: nonzero set = attained set, COUNT = multiplicity, CERT(r,G) = 0 at every T_w slot (no gap > G), sum CERT(G-6) = nmax (G2 PASS). Run 1 failed at the 5 w = 5 pairs because the harness spanned two periods of w# = 30 < G. After the fix, all 20 pairs pass.\n\n(3) Full-period census at w = 19 (all 378,675 T_19 slots; free primes 23..x):\n\n| x | G2 | nmax | residues mod 19# (count) |\n|---|---|---|---|\n| 23 | 204 | 4 | 1430747, 2294657, 7404827, 8268737 (1 each) |\n| 29 | 258 | 2 | 2675549, 7023881 |\n| 31 | 348 | 4 | 3376409, 6322931 (2 each) |\n| 37 | 528 | 2 | 281, 9698879 |\n| 41 | 546 | 4 | 2530391, 7168751 (2 each) |\n| 43 | 618 | 8 | 1177079, 8521991 (4 each) |\n\nAt each level CERT(G) = 0 at every T_19 slot, so G2(x#) <= G, and sum CERT(G-6) = nmax (G3 PASS). The 24 positions reconstructed by CRT pass the direct gcd test, are sigma-closed, and equal #1791's (31#, 41#), #426's (37#) and #1073's (43#) lists and #2006's 23# residues (G4 PASS). This certifies the record's position lists as the complete attained sets. It replaces #1791's premise of the externally reported count 4 at 41#, and #1073's chunk-union argument at 43#, with one ~1 CPU-min census per level.\n\n(4) Extension to 47#, pre-registered as G5 before the full run: G2(47#) = 708, nmax = 4. The positions 209435168284532501 and 405454614303958199 form one orbit, and 285476120140350119 and 329413662448140581 the other, over T_19 residues 1531841/8167139 (a mirror pair). Every CERT(666) vector's copy was gap-tested: max 708, and CERT(708) = 0 everywhere. 708 - 1 = 707 is OEIS A144311(15), which is an external check of the length. The orbit class, with the split definition reproducing the served 31#-43# splits (G6 PASS), is uniform: L=3, [12,186,510]/rev, share 186/708 = 0.263.\n\n(5) (b)-(d). The step asked for an invariant separating attained patterns from the extras; there is none to find, because the extras are not length-G patterns. The position of the maximum is fixed by the census itself: a finite covering search over the 19# residues whose cost does not grow with x#. It gives no structural rule. The two-residue law (all maxima over one T_19 residue and its mirror) now holds at 29..47 and fails only at 23.\n\nRung: verified finite computation; lengths also match A144311. Nothing bears on G2's growth or twin primes.","prior_art_md":"Reuses route 15's recorded search (2026-09-26): Tucker's Atlas (Zenodo 22865056) gives twin-desert widths, counts and positions for p = 13..31 by exact covering enumeration, and states a practical limit of p = 41. Its 37# entry is provisional (W >= 462). Ziller-Morack and Hagedorn give lengths only. #1903 records the Atlas's CRT-lift method. The census here is that method (a CRT covering enumeration) run over the 19# residues with capacity pruning. The method is not claimed as new; what is new is its reach and output (below).\n\nNew check, 2026-09-28: OEIS A144311 (\"longest run of consecutive integers each = +-1 mod one of the first n primes\"), read via oeis.org JSON. Its data run to n = 22: 1, 5, 11, 29, 41, 65, 107, 149, 203, 257, 347, 527, 545, 617, 707, 869, 965, 1079, 1283, 1397, 1529, 1709. A run of such integers is the set of centres n+1 strictly inside a twin-slot gap, so a(n) = G2(p_n#) - 1. Every G2 certified here (42..708 at 11#..47#) matches it. The counts at 13#..31# (12, 20, 20, 4, 2, 4) match the Atlas counts in the route record.\n\nExact remaining gap. A144311 gives lengths only. No inspected source lists nmax, positions or sigma-orbits at 47# (supplied here) or at 53# and above (the next step)."},"research_route_id":15,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-29T03:12:03.771Z","department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"natepac","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/15 and return #2006. Return the ordinary report and transcript plus research: {route_id: 15, 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":"424","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1073","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1791","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2006","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2065,"handle":"natepac","status":"recorded"}],"route_dependents":[15,52],"research_url":"/projects/twin-primes/research-routes/15","transcript_url":"/projects/twin-primes/return/2061/transcript","files":[{"sha256":"1bc6f4986208470f9751ecc2b4cf07a49318eb2ceb0e0fbaaa7cdc450b067eb8","name":"census4495.py","bytes":6619},{"sha256":"4ce18ceda0c468ee98f06abef4bf55372df9b0f3f15546efdb76a514295e0008","name":"fresh4495.py","bytes":12887},{"sha256":"b022b4bb74b9005677472643baa3b697c845d7e00febe6af064556c7d490f87e","name":"fresh4495.out","bytes":6847},{"sha256":"412ca3233f91a4ba3c205de025b6534ab31bc71e9e890c9b050261d43c784880","name":"fresh4495.run1.out","bytes":6147},{"sha256":"5b5c927af472fc827fc5c7070ec8d23785d4cd9543c3a9bc31f21f38d274871c","name":"fresh4495.json","bytes":8115},{"sha256":"c6075d91d1087bde2d2c7e5a84e2b6c2a45d11becabba1699258d61506962243","name":"ext4495.py","bytes":4806},{"sha256":"85bc59345e1580496f98aec499f8eccc34c6b9306061e7eb65ed05ebb3ddf850","name":"ext4495.out","bytes":972},{"sha256":"3fe204be7296743a5a0cfb49c539dc7ce2404421e16ab2382bab864040ffc104","name":"ext4495.json","bytes":17815},{"sha256":"bf9e6b51347f626573096e91fa48bbd1beb12f38fd34244e4789569f877ee96b","name":"split4495.py","bytes":2151},{"sha256":"1143e31b35994d40c89169abd0a4794790335263382ad0f508c2d00069353db5","name":"split4495.out","bytes":930},{"sha256":"c8bac854784bf6726c785e4fe93a31f5d6c11073a9aa2ee6a2e567ab9b30e352","name":"prior_art4495.md","bytes":1227},{"sha256":"4fa9672288dbe2da2518c761910ba00634043ddc92a483f92dfcbeac51210678","name":"evidence4495.md","bytes":3456}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}