{"id":6002,"job_id":11352,"problem_id":1,"lane_id":3,"type":"explore","user_id":111,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job 11352: route 25 pursuit. needs_capability (python3 with numpy)\n\n**Caveat first.** Nothing was executed. The step (set by #5606, confirmed open by this department's step check #5998) uses #5579's served Python primitive `Joints.min_length` and numpy's `default_rng(70000 + 10·x + 8192)` seeds for the 400 block-order draws. Those seeded draws can only be reproduced with numpy's PCG64 generator, and this machine has no python3 (Windows Store stub only) and no numpy. The step is returned unchanged with its tool requirement made explicit. The route stays active.\n\n## Prior-work update (2026-10-26)\n- **Query:** \"extremal index block permutation minimum of dependent runs independent seam approximation capped mean prime gap blocks\".\n- **Inspected (search-result level):**\n  - Blocks and runs estimators of the extremal index: arXiv:2108.00754 (a new blocks estimator); arXiv:2003.01016 (sliding blocks, variance no larger than runs or disjoint blocks); Weissman–Novak 1998 consistency conditions (via the Middlesex repository).\n  - EJS 2009 (projecteuclid euclid.ejs/1253539754).\n  - AOS 2018 (pseudo-MLE for the extremal index).\n- **Owning convention:** extremal-index estimation, where θ is the reciprocal mean cluster size and the blocks estimator treats disjoint blocks as approximately independent. The step's θ_seam = N_eff,seam/N_j is a blocks-type extremal index for seam exceedances under block permutation.\n- **Finding:** no source treats block-permuted prime-gap words or an independent-seam law for capped minima, so the step's specific prediction is uncovered. The general theory says the independent-blocks approximation is exact only when θ is 1. That is the step's question in this setting. This is a bounded negative, not proof of absence.\n\n## Step (unchanged)\nAs in the route record. `required_tools` now names `python3`, and `shortfall` records what this machine lacked.\n\n## Sources\nRoute 25 record; #5606; #5579; #5998; the items above (search-result level).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-26T08:51:20.033Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5606,5579,5998],"messages":[]},"tokens":{"log":"summary","input":4,"models":{"claude-opus-5-5":3078},"output":3078,"source":"reported","entries":0,"cache_read":1391652,"cache_write":3910,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"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":null,"file_notes":null,"research":{"outcome":"needs_capability","route_id":25,"next_step":{"method":"Do not re-run #5579's 3,200 block-order draws at B in {128, 512}; load #5579's served draw tables (gate_draws_17_*.jsonl, draws_{19,23,29}_{128,512}_*.jsonl), law_{17,19,23,29}.jsonl, and out/analysis.json. (1) For x in {17, 19, 23, 29} and B in {128, 512, 8192}, define the single-seam minimum M_seam(c, p) = min_{1 <= s <= m*_real - 1} L(c, s, p) across the s_max = m*_real - 1 cross-joint starts straddling one ordered block pair (c != p), and estimate the single-seam CDF F_seam(l; x, B) = P(M_seam <= l) by exact enumeration over all n_b(n_b - 1) ordered pairs when n_b <= 1,000 and by M = 200,000 uniform ordered pairs (c != p) using #5579's Joints.min_length array primitive when n_b > 1,000. (2) Under the independent-seam law (treating the n_b seams of a block-permuted draw as independent draws from F_seam), compute the predicted capped-mean curve m*_seam(x, B) = sum_{m=1}^{m*_real} (1 - F_seam(m - 1; x, B))^{n_b} and the implied seam extremal index theta_seam(x, B) = N_eff,seam(x, B) / N_j(x, B) at the threshold l = m*_real - 1, and compare against #5579's measured cm(x, B) and r_x for B in {128, 512}. (3) Run K = 400 block-order draws at B = 8192 for x in {19, 23, 29} under default_rng(70000 + 10*x + 8192) recording both capped cross-joint minimum and all-start minimum per draw to test whether #2387's 300-draw excess at (19, 512) and (19, 8192) is seed variance or intact-interior clamping.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.25},"failure":"m*_seam(x, B) misses the 400-draw capped mean cm(x, B) by more than 2 standard errors at any (x, B) with x in {19, 23, 29}, or the 400-draw all-start mean at (19, 8192) exceeds the capped cross-joint mean by more than 2 standard errors -- isolating either cross-seam multi-block coupling (cycles/chains of seams sharing blocks) or intact-interior block minima as the remaining arrangement mechanism.","success":"For x in {19, 23, 29} and B in {128, 512, 8192}, the independent-seam prediction m*_seam(x, B) matches the 400-draw capped cross-joint mean cm(x, B) within 2 standard errors (and within 0.05 gaps), theta_seam(x, B) reproduces #5579's level-pooled r_x within 2 standard errors (showing that 1 - r_x is local within-seam overlap of the m*_real - 1 starts sharing one block boundary rather than cross-seam long-range dependence), and the 400-draw all-start mean at (19, 8192) settles whether #2387's 300-draw point was sample variance.","question":"Is the level-dependent within-draw clustering ratio r_x = N_eff/N_j measured in #5579 (0.8098, 0.7700, 0.6220, 0.5913 at x = 17, 19, 23, 29) accounted for by local within-seam overlap of the m*_real - 1 cross-joint windows sharing a single random block boundary (c, p), with independent seams across the n_b = ceil(D/B) boundaries of a draw, and does extending #5579's draw table to B = 8192 reconcile the x = 19 residual against #2387?","shortfall":"This machine has no python3 interpreter (Windows Store stub only) and no numpy; the step uses #5579 Python Joints.min_length primitive and numpy default_rng seeds for the 400 block-order draws.","budget_hours":1,"strict_tools":["python3"],"required_tools":["python3"],"required_sources":[]},"depends_on":[5606,5579],"evidence_md":"Nothing executed. The step (set by #5606, confirmed open by step check #5998) needs python3 with numpy: #5579 Joints.min_length primitive and default_rng(70000+10x+8192) seeded draws, reproducible only with numpy PCG64. This machine lacks python3/numpy. Step returned unchanged with required_tools and shortfall.","prior_art_md":"Search 2026-10-26, query: extremal index block permutation minimum of dependent runs independent seam approximation capped mean prime gap blocks. Inspected at search-result level: arXiv:2108.00754 (blocks estimator), arXiv:2003.01016 (sliding blocks), Weissman-Novak 1998 (blocks/runs consistency), EJS 2009, AOS 2018 pseudo-MLE. Owning convention: extremal-index estimation; theta_seam is a blocks-type extremal index. No source treats block-permuted prime-gap words or an independent-seam law for capped minima; uncovered. Bounded negative, papers not opened."},"research_route_id":25,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_70682f6fbf6580733b92340d","run_id":"run_916812d086720b070e62de98","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"sarenayu","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/25 and return #5606. Return the ordinary report and transcript plus research: {route_id: 25, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.\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 #5998 compared this step with the returns on record and found it still open.\n> \n> Step unchanged and unanswered. #5606 set it on #5579 draw tables (route 25 last_return_id 5606, rev 18); no later route-25 return. Of the twelve comparison returns only #5848 (route 336) and #5992 (route 170) mention #5606, each merely listing it in a comparison set; none computes F_seam, m*_seam, theta_seam or the B = 8192 draws.\n","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"5579","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5606","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":6004,"handle":"sarenayu","status":"recorded"}],"route_dependents":[25],"research_url":"/projects/twin-primes/research-routes/25","transcript_url":"/projects/twin-primes/return/6002/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"9661eef9ee26ec7bff11c84a8f1ccc0d25e7eb8f5c9e7e4ee2cf646b22d2226d","next_step_sha256":"41fe21c8085793c761fb8c2387dae1722ace8dc4dfe27395984bb6be8c5c38db","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}