{"id":953,"job_id":1802,"problem_id":1,"lane_id":5,"type":"explore","user_id":42,"model":"deepseek-v4-pro","provider":"deepseek","report_md":"# Route 23 pursue: #933's reduction is understood precisely; a per-block phase subtlety is located\n\nCalibration: **heuristic** (implementation reading + small-case checks; the 35x domain arithmetic is exact).\n\n## Result\n\nI read return #933's full proof and implemented/checked the weighted-cover identity it states. Two things are now precise.\n\n**1. The reduction, stated correctly.** #933 section 3's exact statement is: over every phase a (mod 37) and every old-base (31#) cyclic window W fully covered by {37} ∪ Q' under phase a, the maximum over W of the survivor count (slots of W with r mod 37 ∉ {a, a−2}) equals the new K*. The Q' phases b_q are **unrestricted**. My return #951's gloss was wrong in two independent ways — it fixed b_q = 0 and dropped the phase-a maximization — so its small-case \"did not match\" was expected.\n\n**2. A per-block phase subtlety.** The 35× is exact in slot count: N_37 = 217,929,355,875 = 35 × N_31, and translated phase-a copies are exactly the 37 value-blocks [kP,(k+1)P). But a naive single-phase-a implementation fails the ground truth: the validated C engine gives K*(7) = 3, while a single-phase-a survivor-count search gives 2. The missing witness is the cross-block run [17, 29, 41] (parents 17, 29 in block 0 and 41 = 11+30 in block 1): a covering run that crosses a block boundary needs the 37-phase to **change at the boundary** (block t carries a different phase a than block t+1). A single phase a cannot represent a window whose survivors span two blocks. This is the precise implementation requirement the summary did not spell out: the survivor-count objective must carry a per-block (or per-slot) 37-phase, not one global phase.\n\n## Conclusion\n\nThe 35× domain reduction is real and exact in slot count, and the reduction's objective is now precisely specified (unrestricted Q' phases + per-block 37-phase). What is not yet established is whether that objective runs 35× cheaper than the direct 37# scan, because the per-block phase handling adds per-window cost that a naive implementation does not capture. The next step is a native implementation with per-block 37-phases, benchmarked against #938's 12.3/117.8 CPU h pricing. This does not touch #938's finite certificates (the 29-slot witness and 17 L34 negatives) or the already-exact K*(34)=29, K*(36)=33.\n","patch":null,"cpu_hours":0.05,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T21:01:19.434Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[933,938,951],"messages":[]},"tokens":{"log":"custom","input":11302,"models":{"deepseek-v4-pro":48286},"output":48286,"source":"custom-jsonl","entries":19,"cache_read":8031488,"cache_write":0,"observed_models":["deepseek-v4-pro"]},"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":"high","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":"progress","route_id":23,"next_step":{"method":"Implement the reduced search in the native incremental engine: iterate the 37 blocks, carry the per-block 37-phase, run the exact DFS over Q' with unrestricted phases on survivor windows, and record the max survivor count; validate against the C engine ground truth on s=7/s=11/s=13, then benchmark a prefix against #938's 12.3/117.8 CPUh.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The per-block phase handling makes the reduced per-window cost no cheaper than the direct scan (no runtime saving despite the 35x slot count); record the measured factor.","success":"A reduced-domain native scan matching the direct K* on small cases, priced well under 4 CPUh for the full K*(37).","question":"Does a native implementation of #933's survivor-count objective with PER-BLOCK 37-phases (not one global phase) reproduce the direct phase-max K* on small cases and give a runtime saving close to the 35x slot-count factor?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[933,938],"evidence_md":"Read #933's full proof: K*(37) = max over phase a (mod 37) and old-base windows W fully covered by {37} U Q' of the survivor count (slots with r mod 37 not in {a,a-2}); Q' phases unrestricted. Ground truth from the validated C engine: K*(7)=3; a single-phase-a survivor-count search gives 2, missing the cross-block run [17,29,41] whose 37-phase must change at the block boundary. So the objective needs PER-BLOCK 37-phases. N_37=35*N_31 is exact; runtime speedup unmeasured.","prior_art_md":"2026-09-17. #933 (pending) supplies the weighted-cover identity and two-sided proof; #938 (pending) supplies the 37#-base benchmark (12.3/117.8 CPUh) and finite certificates (29-slot witness, 17 L34 negatives); #936/#951 supply the complete-domain repair and the 35x domain arithmetic. No new external literature needed. The precise remaining gap: a native implementation of the survivor-count objective with per-block 37-phases, benchmarked."},"research_route_id":23,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_fb22a1982ed97962998ff647","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","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/23 and return #951. Return the ordinary report and transcript plus research: {route_id: 23, 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>, 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":"933","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"938","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/23","transcript_url":"/projects/twin-primes/return/953/transcript","files":[{"sha256":"0f737dbcb7ca324767a37086f7116b2c4b500f5b00864a854cc169b70038efcf","name":"validate_weighted_cover.py","bytes":2814},{"sha256":"2df50d5b03c6b9c1dd8bfe010791d8034b9496ad45a5fbabfe904a9059903e99","name":"qa.md","bytes":15506}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}