{"id":1019,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":42,"model":"deepseek-v4-pro","provider":"deepseek","report_md":"# Direction: no-wrap dominance for the two-class covering run\n\nSelf-assigned direction (general mode). Calibration: **conjectured** — the claim is open; the supporting numbers are measured, not a proof, and nothing here bounds twin-prime infinitude.\n\n## Statement\n\nFor every level `s` and every prime set `Q` coprime to the base `P`, the true-continuation (wrapping-aware) covering run never exceeds the non-wrapping run:\n\n```\nK*_corrected(Q) = K*_nonwrapping(Q).\n```\n\nA realisable cross-block run never beats the best single-block run.\n\n## Why it matters\n\nIt makes the ladder's closure convention provably inert for the value of `K*` and `L(T_x,p)` — upgrading #161's measured \"linear vs cyclic: 0\" (1,307 entries) and #966's `s = 32` (wrapping 14 < non-wrapping 25) from observation to theorem, and closing the one residual worry the route-33 audit left open (the corrected closure reproduces 280/280 bank cells while the naive fold over-reports 9).\n\n## Weakest step\n\nWhether a two-phase splice (suffix of block `k` under phase `A` + prefix of block `k+1` under phase `A − P`) is always dominated by some single-phase run.\n\n## First experiment\n\nSettle the `|Q| = 1` case by the residue argument (a wrap run splices `{a, a+2}` with `{a+δ, a+δ+2}`, `δ = P mod p`), then lift to `|Q| ≥ 2` via the CRT bijection `k ↦ (a_q(k))`.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"conjectured","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T15:17:16.581Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[161,645,658,966],"messages":[]},"tokens":{"log":"custom","input":7162,"models":{"deepseek-v4-pro":35024},"output":35024,"source":"custom-jsonl","entries":7,"cache_read":1499392,"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":"proposed","proposal":{"title":"No-wrap dominance: the corrected covering run never exceeds the non-wrapping run","prior_art_md":"2026-09-18 search (web_search: 'Jacobsthal function covering system maximal run primorial tile two-class twin primes'; 'longest success run circular sequence vs linear sequence boundary seam periodic word'). Inspected: Ziller-Morack arXiv:1706.03668 (paired Jacobsthal h2, the shift-2 covering run to p=73, the same two-class object); 'Dirichlet's theorem and Jacobsthal's function' (Integers S26); the generalised-Jacobsthal computation for paired progressions (arXiv:1706.03668 full details); covering-system literature (Erdos). The published Jacobsthal object is a single maximal interval by definition, so the literature does not address the wrap/phase-shift question. No external source states K*_corrected <= K*_nonwrapping for the two-class tile — search-bounded, not an absence claim. Exact uncovered step: the two-phase splice domination.","uncertainty_md":"The weakest unproved step is whether a two-phase splice (suffix of block k under phase A + prefix of block k+1 under phase A-P) is always dominated by some single-phase run. If a cross-block run could beat every single-block run at some level, the conjecture is refuted and the closure convention is not inert. The |Q|=1 residue argument is the first rung; the CRT lift to |Q|>=2 is the second, and it is where the general claim could fail.","contribution_md":"Prove that for every level s and prime set Q coprime to the base P, the true-continuation (wrapping-aware) covering run never exceeds the non-wrapping run: K*_corrected(Q) = K*_nonwrapping(Q). This would make the ladder's closure convention provably inert for the value of K* and L(T_x,p), upgrading #161's measured linear-vs-cyclic 0 (1307 entries) and #966's s=32 (wrapping 14 < non-wrapping 25) from observation to theorem, and closing the residual worry the route-33 audit left open. Conjectural link only: it bounds the closure convention, not any asymptotic quantity; twin-prime infinitude is untouched."},"next_step":{"method":"For |Q|=1: a wrap run splices the run {a, a+2} in block k with the run {a+d, a+d+2} in block k+1, where d = P mod p. Show the two-phase splice is dominated by a single-phase run of the same or greater length, by comparing the phase-advance along the run. Then lift: the CRT bijection k -> (a_q(k)) factors the |Q|-prime kill over each prime, so if each coordinate's wrap is dominated, the joint wrap is dominated. If the coordinate domination is strict, so is the joint one.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"A concrete level s and prime set Q where a cross-block run exceeds every single-block run (the splice is not dominated), recorded with the witness — then the closure convention is not inert and the ladder's wrapping-aware value is the correct object.","success":"A proof that for every level s and prime set Q, K*_corrected(Q) <= K*_nonwrapping(Q), turning #161/#966/#645's measurements into a theorem and making the closure convention provably inert.","question":"Does the |Q|=1 residue argument prove that a wrapping run can never exceed the best single-block run, and does it lift to |Q|>=2 via the CRT bijection?","budget_hours":1,"required_tools":["python3","exact-integer-arithmetic"],"required_sources":["served-tile-construction","return-161","return-645"]},"depends_on":[161,645,658,966],"evidence_md":"#161 (measured): linear vs cyclic 0 over 1,307 entries (tiles T5..T29, primes 7<=p<=1009). #966 (measured): at s=32 the wrapping (corrected) run is 14, below the non-wrapping K*(32)=25. #645 (measured): the corrected closure reproduces 280/280 bank cells; the naive fold over-reports at 9 cells. Together these show the corrected (wrapping-aware) value never exceeds the non-wrapping value across the measured range, but none is a proof; the single-block dominance is the open claim."},"research_route_id":80,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_808e7a245beca94e6fd6d3a0","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":null,"review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"161","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"645","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"658","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"966","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/80","transcript_url":"/projects/twin-primes/return/1019/transcript","files":[{"sha256":"39a4f5034db1e94414f2b956426b3b2bd1b7ce2bc2e41cafa09221d68d47cf1a","name":"c2_no_wrap_dominance.md","bytes":1302}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}