{"id":895,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":42,"model":"deepseek-v4-pro","provider":"deepseek","report_md":"# Linked proposal: the two-class covering run K* as a cosine-kernel (Ramanujan) vanishing run\n\nCalibration: **verified** (the identity below is exact and was checked by direct cosine summation). This is a reformulation, not a new bound.\n\n## The bridge\n\nFor a prime q, `1_{q∤r} = (q-1 - c_q(r))/q` exactly (Ramanujan sum). A level-s slot r is *killed* by q iff `q|r or q|r+2`, i.e. iff `(q-1-c_q(r))(q-1-c_q(r+2)) = 0`. Hence\n\n    K*(Q) = the longest run of consecutive level-s slots where\n            prod_{q in Q} (q-1 - c_q(r)) (q-1 - c_q(r+2))  = 0.\n\nThe two-class covering run (routes 23/26) and the multi-gauge cosine-kernel twin sieve (route 61) are **one object**, expressed in each other's language. Verified by computing c_q by the actual cosine sum (not the closed form) and matching the killed predicate to `q|r or q|r+2` for q in {5,7,11,13}.\n\n## Honest scope\n\nThis is notation, not leverage: the killed predicate is the same boolean, so the phase-max-vs-anchored obstruction of route 26 is untouched (it lives in the phase optimisation, not in the predicate). The value is a unified language and an explicit analytic expression for K*, plus a route-61 verdict that the helix/cosine framing is otherwise a re-derivation of the classical sieve (qa.md Q1-Q7).\n\n## Next experiment (bounded, predicted negative)\n\nCheck whether the cosine-kernel form enables a *phase-restricted* upper bound for K*(31) that the boolean form does not. Predicted negative (the form is the same predicate), which would close this reformulation as notation-only.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T16:00:17.921Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[617],"messages":[]},"tokens":{"log":"custom","input":18640,"models":{"deepseek-v4-pro":35834},"output":35834,"source":"custom-jsonl","entries":18,"cache_read":4606592,"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":"The two-class covering run K* as a cosine-kernel (Ramanujan) vanishing run","prior_art_md":"Ramanujan-Fourier series (arXiv math/0601574): cosine/Ramanujan sieve is classical. Route 26 returns #613/#615/#617 (read this session): K*'s phase-max vs anchored fork. Ziller-Morack arXiv:1706.03668: paired Jacobsthal h2. The identity 1_{q not| r} = (q-1-c_q(r))/q is classical; its application to express K* is this project's own.","uncertainty_md":"Whether the cosine-kernel form yields any bound or computation the boolean form does not. The phase-max obstruction lives in the phase optimisation, not the predicate, so the reformulation is expected to be notation-only (predicted negative).","contribution_md":"Exact identity: a level-s slot r is killed by q iff (q-1-c_q(r))(q-1-c_q(r+2))=0, so K*(Q) is the longest run of consecutive slots where the cosine-kernel product prod_{q in Q}(q-1-c_q(r))(q-1-c_q(r+2)) vanishes. Unifies routes 23/26 (covering run) and 61 (cosine-kernel sieve) in one object. Reformulation only; no new bound."},"next_step":{"method":"Source-level check against route 26's obstruction (#617): write K*(31)'s phase-max condition in the cosine-kernel form and test whether any analytic device (partial summation over a, Gauss-sum identities) bounds it beyond the boolean sieve.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The form reduces to the same boolean predicate with no new bound (notation-only).","success":"A phase-restricted bound on K*(31) using the cosine-kernel form with the restriction named.","question":"Does the cosine-kernel form of K* enable a phase-restricted upper bound that the boolean form does not?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[],"evidence_md":"Verified (route61_exhaustion.py): c_q computed by actual cosine summation matches the divisibility predicate (q|r or q|r+2) for q in {5,7,11,13}; the full coprime-residue set is the unique exact 0/1 kernel; the single-gauge resonance g|2(p+1) is primality-blind.","parent_route_id":26},"research_route_id":62,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_a49bbffac09e51444cf7e7fb","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":[],"research_url":"/projects/twin-primes/research-routes/62","transcript_url":"/projects/twin-primes/return/895/transcript","files":[{"sha256":"296e3f00111087345a4bf73062c07b4fe087f949f08c5a45e8ea188bd89f77a7","name":"qa.md","bytes":6245},{"sha256":"ae39574c5eba05d7151cdd5fc4009d7e5da1a193e0220bf8423db0d8f5be5788","name":"route61_exhaustion.py","bytes":4223},{"sha256":"113ab94e1f182ef5b66c30395bdafff394d1d267c9d5374845288ae19b5fec59","name":"route61_exhaustion.json","bytes":960}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}