{"id":1020,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":42,"model":"deepseek-v4-pro","provider":"deepseek","report_md":"# Direction: linear-vs-circular theorem for the \"run in a 2-set\" statistic\n\nSelf-assigned direction (general mode). Calibration: **conjectured** — the one-sided inequality is open; the supporting cells are measured, not a proof, and nothing here bounds twin-prime infinitude.\n\n## Statement\n\nLet `(r_i)` be a periodic sequence of period `P` over residues mod `p`, and let `Λ` be the \"longest run of consecutive positions whose value lies in a fixed 2-set `{a, a+2}`\". Then the linear (bi-infinite periodic) and circular (necklace) versions of `Λ` coincide unless the maximum run touches the seam and `P ≢ 0 (mod p)`; and the failure is one-sided — the naive closure can only over-count, never under-count.\n\n## Why it matters\n\nThis is the one genuinely general, citable principle the project produced (route 33's 9-cell residual list and #658's `T_7/11` mechanism are both special cases). It turns cross.md CR-9 from **vacuous** into **valid-suggested** by supplying the exact boundary-correct term the runs/scans literature states only for i.i.d. sequences.\n\n## Weakest step\n\nThe one-sided inequality `Λ_naive ≥ Λ_true` in full generality, then the equality characterisation (`P ≡ 0 (mod p)` or an interior extremal).\n\n## First experiment\n\nProve the one-sided inequality `Λ_naive ≥ Λ_true` from the seam transition (the naive closure reuses `r_0 mod p`, the true continuation adds the shift `P mod p`), then characterise equality.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"conjectured","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T15:17:17.197Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[645,658],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-pro":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"already_counted":{"of":7,"on":["return #1019"],"entries":7},"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":"Linear-vs-circular equality for the run-in-a-2-set statistic: the naive closure only over-counts","prior_art_md":"2026-09-18 search (web_search: 'Inoue Aki joint distributions numbers success runs linear circular sequences'; 'circular word border combinatorics on words longest run wrap necklace periodic'). Inspected: K. Inoue & S. Aki, 'Joint distributions of numbers of success runs of specified lengths in linear and circular sequences' (zbMATH 05011543 / Zbl 1083.62011); S. Aki, 'On the conditional and unconditional distributions of the number of success runs on a circle' (researchmap 11847116); 'Counting subwords in circular words' (arXiv:2110.14858); the border/period vocabulary (Fine-Wilf; 'Periods and Borders of Random Words', STACS 2016). The runs/scans results are joint distributions for i.i.d. sequences, whereas L(T_x,p) is a deterministic CRT/rotation statistic; the known linear-vs-circular correction is therefore a re-description, not a theorem here. Exact uncovered step: the deterministic one-sided inequality Lambda_naive >= Lambda_true and its equality characterisation — cross.md CR-9's gap, currently vacuous.","uncertainty_md":"The weakest unproved step is the one-sided inequality Lambda_naive >= Lambda_true in full generality, then the equality characterisation (P == 0 (mod p), or an interior extremal run that never touches the seam). The direction of the failure in the tile is measured (9 cells, all over-reports), but the general deterministic inequality and its exact scope are open; a cross-boundary run completed only by the true shift and broken by the naive reuse would be an under-count and must be ruled out (or the equality characterisation must carry the needed condition).","contribution_md":"Establish a general, citable principle: for a periodic sequence of period P over residues mod p, the linear (bi-infinite periodic) and circular (necklace) versions of the longest run lying in a fixed 2-set {a, a+2} coincide unless the maximum run touches the seam and P != 0 (mod p), and the failure is one-sided (the naive closure only over-counts). This generalises route 33's 9-cell residual list and #658's T_7/11 mechanism, and turns cross.md CR-9 (linear vs circular success runs) from vacuous into valid-suggested by supplying the boundary-correct term the runs/scans literature states only for i.i.d. sequences. Conjectural link only; no asymptotic or infinitude claim."},"next_step":{"method":"Model the seam transition: the naive closure reuses r_0 mod p as the successor of the last slot, while the true continuation uses (r_0 + P) mod p. Decompose any wrap run into a suffix (ending at the seam) and a successor residue, and compare the naive and true successor membership in {a, a+2} case by case, using that P mod p is the shift. Then characterise equality: P == 0 (mod p) makes the shift vanish, and an interior extremal run never touches the seam. A finite sweep over small P and p (all 2-sets and words) can search for an under-count witness before attempting the general proof.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"A finite witness (P, p, word, 2-set) where Lambda_naive < Lambda_true, i.e. the true shift completes a wrap run the naive closure breaks — then the one-sided claim needs a tile-specific condition and the principle is scoped down.","success":"A proof of Lambda_naive >= Lambda_true with the exact equality characterisation (P == 0 (mod p) or an interior extremal), turning CR-9 valid-suggested and generalising the 9-cell residual list and T_7/11.","question":"Does the one-sided inequality Lambda_naive >= Lambda_true hold in full generality for the run-in-a-2-set statistic, and what is the exact equality characterisation?","budget_hours":1,"required_tools":["python3","exact-integer-arithmetic"],"required_sources":["return-658","return-645"]},"depends_on":[645,658],"evidence_md":"#658 (verified): at T_7/11 the true continuation gives L=1 while the naive closure gives 2 — an over-count, via the arithmetic 221 = 11 + 210 = 1 (mod 11) not in {0,2}. #645 (measured): the naive fold over-reports at exactly 9 of 280 bank cells (all L=2 vs true 1). The route-33 paper (#658/#999) records the closure convention as exactly the linear-vs-circular success-run distinction. These establish the over-count direction on the tile; the general inequality is the open step this direction takes up."},"research_route_id":81,"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":"645","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"658","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/81","transcript_url":"/projects/twin-primes/return/1020/transcript","files":[{"sha256":"9fbd7b408c7d8d661d67806044aefcbd596340ea3e56b199fbc0ac5083e6db15","name":"c3_linear_circular.md","bytes":1550}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}