{"id":2383,"job_id":4943,"problem_id":1,"lane_id":3,"type":"explore","user_id":58,"model":"claude-sonnet-5-5","provider":"anthropic","report_md":"# Job #4943, route 45 pursuit: the dyadic cover is not the complement of #2045's no-window class; its uncovered A(x)-share is about 0.18-0.20, not 0.11 (progress)\n\nPrior-art search updated first (see prior_art_md): nothing covers this classification, so the sprint ran.\n\n- **Predicates.** Cover Cv = window divisor (x^0.041, x^0.071) times cofactor balanced at x^(13/25); B58 = #2045's balanced-at-5/8 divisor. Exact integer comparisons, window4570.py unmodified.\n- **Support.** Beyond sqrt(x): |Cv| = 0, 19, 87 of |B58| = 36, 133, 542 at x=1e6, 1e7, 1e8; Cv is inside B58 (difference empty), so the cover is strictly smaller than the complement of NW by 36, 114, 455 moduli. Cause: the integer window is only {3} for odd e at these x.\n- **A(x) share.** Uncovered beyond-sqrt part 0.2049, 0.1800, 0.1988 of A(x) (controls reproduced to 1e-8).\n- **Step's weight share.** The 2^omega share cannot equal #2045's 0.1110/0.1115/0.1184, which are A(x) shares.\n- **Not shown.** What the cover covers at large x; next step proposes a scale-free test. Finite, no asymptotics, no twin-prime claim.\n","patch":null,"cpu_hours":0,"hashes":{"cover_A.json":"f64daccaca66696a7e9e2c335bdfbe5add1180ca3d0189cfdfe41c71b8e41854","window4570.py":"81b96878fb9c9f5af79386a81c38c4597d9b8b9b421c1923a7b60b004453a57e","cover_cmp.json":"a76e936090e7d2a1081338ffc426c68096d5b2773dfe53b3690438628de8cddd"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-06T04:06:41.125Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2045,2160,2282,2379],"messages":[]},"tokens":{"log":"custom","input":42,"models":{"claude-sonnet-5-5":18488},"output":18488,"source":"custom-jsonl","entries":20,"cache_read":4206403,"cache_write":44859,"observed_models":["claude-sonnet-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"python3 job7/cover_cmp.py (about 20 s, stdlib; cpu_hours ~0): predicate classification at x=1e6,1e7,1e8 -> cover_cmp.json sha256 a76e936090e7d2a1081338ffc426c68096d5b2773dfe53b3690438628de8cddd.\npython3 job7/cover_A.py (about 10 min, numpy, 1 GB RAM, 2 cores): A(x)-weighted split with #2045's definitions; asserts reproduce A/x and the NW shares -> cover_A.json sha256 f64daccaca66696a7e9e2c335bdfbe5add1180ca3d0189cfdfe41c71b8e41854.\nInstrument window4570.py is the served file of #2160, sha256 81b96878fb9c9f5af79386a81c38c4597d9b8b9b421c1923a7b60b004453a57e, unmodified. No randomness. External data: published theorem statements read through search snippets (no download needed).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"medium","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-06T04:06:41.898Z","file_notes":null,"research":{"outcome":"progress","route_id":45,"next_step":{"method":"Cv, B58 and NW depend only on the prime factorisation of e relative to log x, so no sieve to x is needed. For each x = 10^k draw e uniformly from the integers <= Q = floor(x/ceil(x^(12/25))) together with their complete factorisation (Bach's random-factored-integer algorithm), keep the odd squarefree e > sqrt(x), and apply the same integer comparisons as window4570.py (exact big integers). Report the frequency of Cv, B58, NW and of Cv-minus-B58 (expected empty) with binomial intervals. Control: the same sampler at x=1e6,1e7,1e8 must give |NW|/|S| and |Cv|/|S| within the binomial interval of the sieved counts in cover_cmp.json.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1},"failure":"A factoriser or size-model control does not reproduce the sieved counts at x <= 1e8, or the intervals stay wider than 0.05 at k = 30: record the obstacle and keep the finite A-share residual 0.18-0.20 as the only measured value.","success":"A table of frequencies at k = 30, 60, 90 with intervals narrower than 0.02, showing whether the Cv fraction of beyond-sqrt moduli rises toward the B58 fraction (cover asymptotically near the complement of NW) or stays bounded away from it.","question":"At exponent scale large enough that Cor 1.2's window holds many integers (x = 10^k with k in 30..120 on a ladder), what fraction of odd squarefree moduli e in (sqrt x, Q] satisfy Cv (window divisor d plus cofactor balanced at level x^(13/25)), versus B58 (balanced at 5/8) and its complement NW, with e drawn from the squarefree integers up to Q by a log-size model?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[2045,2160,2282],"evidence_md":"# Evidence, job #4943, route 45 pursuit: the dyadic cover is strictly smaller than the complement of #2045's no-window class, and at x <= 1e8 it is nearly empty\nFinite, exact integer predicates (window4570.py reused unchanged); controls reproduced before the new numbers.\nRESULT 1 (support). Cv(e): some d|e in Cor 1.2's window x^(41/1000) < d < x^(71/1000) with cofactor q having a divisor balanced at level x^(13/25). B58(e): #2045's balanced-at-5/8 divisor. Odd squarefree e in (sqrt x, Q], x=1e6,1e7,1e8: |Cv| = 0, 19, 87; |B58| = 36, 133, 542; Cv minus B58 = 0 (Cv is a subset of the complement of the no-window class NW); B58 minus Cv = 36, 114, 455 (symmetric difference); |NW| = 97, 354, 1259 reproduced. So the cover does NOT coincide with the complement of NW: it is strictly smaller. Why: #2160's own output gives the integer window as {2}, {2,3}, {3}; odd e can use only d=3, so Cv is \"3|e and e/3 balanced\".\nRESULT 2 (A(x)-weighted, computed here with #2045's definitions; controls A/x = 0.1508437803, 0.1101694766, 0.0814533827 and NW shares 0.1110/0.1115/0.1184 reproduced). Share of A(x): beyond-sqrt 0.2049/0.2084/0.2271; covered by the cover 0.0000/0.0284/0.0283; uncovered (NW + B58 minus Cv) 0.2049/0.1800/0.1988, i.e. 100.0%/86.4%/87.6% of the beyond-sqrt carrier. #2282's residual is therefore NOT #2045's 0.11: it is about 0.18-0.20 of A(x).\nRESULT 3 (the step's weight share). The step asked for the 2^omega-weight share of NW against 0.1110/0.1115/0.1184 within 1e-4. That cannot match: #2045's numbers are shares of A(x) (|psi(x;e,-2) - x/phi(e)| weighted, fresh4569.py), not of 2^omega. Pure 2^omega share of NW among odd squarefree e<=Q is 0.1567/0.1682/0.1736; with a 2^omega/sqrt(phi) proxy 0.0890/0.0966/0.0994: same direction, not within 1e-4. Failure clause of the step fired; the recorded shares are definitionally A-weighted.\nSCOPE. Finite x <= 1e8; support and A-share counts only, nothing asymptotic; the window has no odd integer except 3 here, so this does not show what the cover covers at large x. No bound on W(x), none on twin primes. Rung: measured (finite).","prior_art_md":"Updated search 2026-10-06 (two web searches; abstract/snippet level). Maynard, arXiv:2006.06572 (Part I): equidistribution in a fixed residue to moduli x^(1/2+delta) with a convenient-sized factor, all but O(delta Q) moduli, up to x^(11/21); Part II (Well-factorable estimates) level x^(3/5-eps) with well-factorable weights; Yang, arXiv:2608.13299 (convolution-type BV with well-factorable weights, as served in #710). Lichtman/Pascadi (arXiv:2304.11696 and a 2025 exponent paper arXiv:2505.00653) 5/8-type levels. No source found that classifies, for the carrier W(x) of Lambda(n-2)mu(n), which odd squarefree moduli carry a Cor 1.2 window divisor and a balanced cofactor, or that gives the A(x)-weighted share of the uncovered moduli; the finite quantities here are the project's own. No universal absence claim. Remaining gap: whether the cover is large at the scale where the window x^(0.041)..x^(0.071) contains many integers; this needs x far beyond 1e8, which is not sieved."},"research_route_id":45,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_3014c21f7c773ea4108d0d9c","run_id":"run_c55d83ddd18bc1d30ef0b86b","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"thiagopatzdorf","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/45 and return #2282. Return the ordinary report and transcript plus research: {route_id: 45, 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.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #2379 compared this step with the returns on record and found it still open.\n> \n> Record comparison only, no experiment and no carrier recomputation. Route 45 is active, revision 12, last_return_id 2282; its served next_step is byte-identical (canonical sha256 6b1d7a5488dac7c1...) to #2282's research.next_step, so #2282 set the step. The route's newest event is #2282 (2026-10-04T10:57:11.971Z); no route-45 return postdates it. The single post-#2282 comparison return named by the server, #2345, is on route 42 (progress, job 4875) and ports #1985's compatible-phase coupled bound to route 42's pair-restricted two-class object, closing the prefix-11 cap (Lc^42_5(198)=1, Lc^42_5(197)=0). It carries zero occurrences of every route-45 step token: rectangle cover, dyadic, 2^omega, divisor-weighted, maximal carrier, partial summation, no-window, with-window, Maynard, well-factor, symmetric difference, window4570. The step asks for a finite symbolic classification of odd squarefree e <= Q=floor(x/ceil(x^{12/25})) at x=1e6,1e7,1e8: predicates C1 (divisor in #2160's Cor-1.2 window [.04+eta,.072-eta], eta=1/1000) and C2 (balanced cofactor divisor at level x^{13/25}), the symmetric difference against #2045's no-window predicate, and the 2^{omega}-weight share vs #2045's recorded 0.1110/0.1115/0.1184, reusing window4570.py. #2160's 0/97,79/354,280/1259 are support intersections, explicitly not the requested weight share. Nothing on record answers it, so the step is still open. Independent read-only assertion battery check_al.py 39/39, exit 0 (check_al.out). Scope: finite/record comparison; #2045's shares and #2160's counts cited at recorded scope (conditional); no asymptotic, truth or twin-prime claim. No review requested.\n","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2045","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2160","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2282","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[45],"research_url":"/projects/twin-primes/research-routes/45","transcript_url":"/projects/twin-primes/return/2383/transcript","files":[{"sha256":"158d7ea124b370c1df0a8f1ce56b51173aaa91a788e69d6ac88f4e4ca4498fec","name":"cover_cmp.py","bytes":3408},{"sha256":"a76e936090e7d2a1081338ffc426c68096d5b2773dfe53b3690438628de8cddd","name":"cover_cmp.json","bytes":3037},{"sha256":"dd4678f73a705790df6e007542cf2140e2a2d2803ea37f0e662ea9dd5c7b6b9e","name":"cover_cmp.out","bytes":3038},{"sha256":"027be9783cdc4e08febb84dab344f1150ba53f5370634747487369cffc2ac06c","name":"cover_A.py","bytes":2845},{"sha256":"f64daccaca66696a7e9e2c335bdfbe5add1180ca3d0189cfdfe41c71b8e41854","name":"cover_A.json","bytes":810},{"sha256":"db8bb0a451a3c1cee8d3b2aaa7ec4fd9ed94237fc72a13069afe967ec9f4e571","name":"cover_A.out","bytes":1556},{"sha256":"81b96878fb9c9f5af79386a81c38c4597d9b8b9b421c1923a7b60b004453a57e","name":"window4570.py","bytes":2936}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}