{"id":958,"job_id":1811,"problem_id":1,"lane_id":null,"type":"explore","user_id":42,"model":"deepseek-v4-pro","provider":"deepseek","report_md":"# Route 66 triage: the phase-drift rotation reformulation is a re-description; no rotation bound transfers\n\nCalibration: **heuristic** (the reformulation is exact but imprecise; the non-transfer is a structural assessment, not a disproof of every conceivable bound).\n\n## Result\n\nThe direction's own central uncertainty is resolved against it: the rotation reformulation adds **no quantitative leverage** over the exact CRT covering reduction.\n\n1. **It is a re-description.** The phase-drift `k ↦ (a_q(k))_q = (-k·P mod q)_q` is exactly the per-block phase walk already inside #594's exact block+phase collapse. \"Covered by q\" for a slot r means the phase point lies in the **slot-dependent** set `C_r = ⋃_q {x_q ∈ {r mod q, (r+2) mod q}}` — a moving target, not the fixed region `⋃_q {x_q ∈ {a_q, a_q−2}}` stated in #957. The run is over the **slot sequence**, not over the rotation orbit (which is indexed by blocks). Dynamical language adds no new object.\n\n2. **None of the three toolkits transfers.**\n   - *Sturmian / rotation coding*: bounded by the continued-fraction structure of **irrational rotations of the circle** (Chaika–Constantine arXiv:1802.01370). The phase-drift is a **finite, periodic** rotation on `∏_q Z/qZ` — no continued fractions. Finite-vs-infinite gap.\n   - *Kronecker discrepancy*: bounds the **global** max deviation from uniformity; `K*(s)` is a **local** clustering quantity (longest consecutive run). A discrepancy bound does not give a run-length bound.\n   - *Lonely Runner*: unproven (no theorem), and a **separation** statement — the dual extremal problem, not a run-length bound.\n\n## Verdict\n\n**Blocked.** The direction's stated failure criterion is met: every standard rotation bound is vacuous for `K*(s)`. The proposed next experiment (evaluate rotation bounds against the measured K*(32)=25, K*(34)=29, K*(36)=33) is not justified — the toolkits apply to the wrong object. The reformulation remains a correct but inert re-description.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T22:16:15.458Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[594,957],"messages":[]},"tokens":{"log":"custom","input":6236,"models":{"deepseek-v4-pro":20828},"output":20828,"source":"custom-jsonl","entries":9,"cache_read":3124096,"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":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"Slot-dependent covered set C_r; Sturmian theory restricted to S^1 (continued fractions); discrepancy is a uniformity measure; Lonely Runner unproven. The measured K*(32)=25, K*(34)=29, K*(36)=33 are obtained exactly by the CRT scan, which the reformulation does not improve.","statement":"The phase-drift 'rotation' is the CRT phase walk already exact in #594; the covered set is slot-dependent (C_r), the run is over the slot sequence (not the rotation orbit), and no standard rotation bound — Sturmian run-length (irrational S^1 only), Kronecker discrepancy (global uniformity), or Lonely Runner (unproven separation) — transfers to a non-trivial bound on the finite combinatorial run-length K*(s).","assumptions":"That a finite/profinite rotation on prod_q Z/qZ admits the run-length machinery of irrational circle rotations, or that discrepancy/separation bounds imply a local run-length bound.","revisit_when":"A run-length theory for finite/profinite rotations is developed, or a specific discrepancy-type bound on the two-class covering run is proven (not merely conjectured by duality)."},"route_id":66,"depends_on":[594,957],"evidence_md":"The phase-drift k -> (a_q(k))_q = (-k*P mod q)_q is the CRT phase walk of #594's exact reduction; the covered set for a slot r is C_r = U_q {x_q in {r mod q, (r+2) mod q}} (slot-dependent), not the fixed region of #957. Sturmian run-length theory is for irrational S^1 rotations (continued fractions); the phase-drift is a finite periodic rotation on prod_q Z/qZ (no continued fractions). Kronecker discrepancy is a global uniformity measure, not a local run-length bound. Lonely Runner is unproven and a separation statement. Hence the reformulation adds no bound beyond the exact CRT scan.","prior_art_md":"2026-09-17 update for the changed ingredient (rotation run-length / discrepancy bounds transferring to the two-class covering run). Searched: 'longest run of consecutive hits of a rotation against a target set', 'Sturmian word run length continued fraction', 'Kronecker sequence discrepancy run length', 'covering system two residue classes primorial dynamical reformulation'. Inspected: Chaika-Constantine arXiv:1802.01370 (Sturmian coding of an IRRATIONAL circle rotation); 'On the discrepancy of Halton-Kronecker sequences' (uniformity); Filaseta-Harvey covering congruences (combinatorial). No source gives a run-length bound for a FINITE/profinite rotation, and no source applies rotation dynamics to the primorial two-class covering run. The gap is structural, not just unsearched."},"research_route_id":66,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_f60e0aa013ff494788a759be","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/66 and return #957. Return the ordinary report and transcript plus research: {route_id: 66, 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":"594","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"957","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/66","transcript_url":"/projects/twin-primes/return/958/transcript","files":[{"sha256":"bb485f9d4d2e28b9345a52df4e4e0e280b37cba36124d71002799eb116bbfd97","name":"qa.md","bytes":27073}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}