{"id":947,"job_id":1728,"problem_id":1,"lane_id":null,"type":"explore","user_id":42,"model":"deepseek-v4-pro","provider":"deepseek","report_md":"# Route 62 rescue: the cosine/Ramanujan re-expression is exact but yields no new bound\n\nCalibration: **verified** (identity re-checked over q in {5,7,11,13,37,41} by actual cosine sums; short-window fluctuation measured).\n\n## Result (bounded negative)\n\nThe reformulation is valid but notation-only; the obstruction stands and sharpens.\n\n**The identity is exact.** With c_q(r) = Σ_{gcd(a,q)=1} cos(2π a r/q) (= q−1 if q|r else −1), the factor (q−1−c_q(r)) equals 0 iff q|r and q otherwise, so\n  (q−1−c_q(r))·(q−1−c_q(r+2)) = 0  ⇔  q|r or q|r+2.\nVerified for every residue of q ∈ {5,7,11,13,37,41}. The phase-correct kernel F_a on actual positions r_i (with r_(i+D)=r_i+P, the CRT bijection m ↦ (−mP mod q)) expresses the covering predicate exactly. This part of #895 is correct.\n\n**But the Fourier form supplies no short-window bound.** Expanding each indicator 1_{r≡a_q mod q} = (1/q)Σ_{t} e^{2πi t(r−a_q)/q} turns the window cover count into a main term ρ|W| plus non-principal partial sums Σ_{r∈W} e^{2πi ν r/q}. Over a full period those partial sums vanish exactly (Ramanujan complete-sum cancellation), which is what fixes the density ρ. Over a *short* window W of L consecutive slots the partial sums do not vanish and have no uniform bound. Measured at s=11 (P(11)#=2310, Q={13,17,19,23}): the killed density is ρ=0.4, so ρL=8.0 for L=20, but the window killed-count ranges over [3,13] — a deviation of ±5, order L/2. Hence proving \"K*<L ⇔ Σ_j F_a(r_(i+j)) ≥ 1 for all i,a\" requires a uniform lower bound on ρL+E that the complete-sum identity does not provide. That uniform short-window estimate over the irregular admissible set is the substantive missing ingredient (short exponential sums / discrepancy of the level-s slots), not supplied by the reformulation.\n\n## Conclusion\n\nRoute 62 is a correct unification (covering run ↔ cosine-kernel vanishing run) with no new bound. It remains blocked under the reformulation-only scope. The route's genuinely useful output is the connection to the two-class Jacobsthal function (Ziller–Morack h2), which lives in routes 23/26, not in the cosine-kernel form itself. Separately, #896's phase_span.c suffix-defect finding (control-flow: `linear=best` omits a terminal `cur`) is acknowledged and is a fixable evidence-script bug, not a change to this mathematical conclusion.\n","patch":null,"cpu_hours":0.01,"hashes":{"route62_obstruction_verify.json":"bf1d962124267bf394b3dd6b4188cf5f3ae1fde8a7058d7016e4f6c97fe42f92"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T20:17:40.699Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[895,896],"messages":[]},"tokens":{"log":"custom","input":6538,"models":{"deepseek-v4-pro":15052},"output":15052,"source":"custom-jsonl","entries":8,"cache_read":2131712,"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":"(q-1-c_q(r))(q-1-c_q(r+2))=0 iff q|r or q|r+2 for q in {5,7,11,13,37,41}; s=11 rho=0.4, L=20, window killed-count in [3,13] vs rhoL=8.0; #896's CRT phase-correct kernel F_a and complete-sum argument.","statement":"The cosine/Ramanujan re-expression preserves the covering predicate exactly but supplies no uniform short-window estimate: expanding the indicators gives a main term rho*|W| plus partial sums that vanish only over full periods, and the short-window fluctuation is of order |W| (measured s=11: rhoL=8.0 vs window sums [3,13]), so K*<L still needs a uniform bound on rho*L+E that is not provided.","assumptions":"Distinct odd Q-primes coprime to the base primorial P; fixed separation 2; consecutive admissible positions with seams r_(i+D)=r_i+P. Scope is the stated re-expression; does not cover a separately supplied uniform exponential-sum or discrepancy estimate.","revisit_when":"A uniform short-window exponential-sum/discrepancy bound for the level-s admissible set, or a justified phase restriction with a new lower bound for rho*L+E; plus correcting the phase_span.c suffix defect before its scans are used as evidence."},"route_id":62,"depends_on":[895,896],"evidence_md":"Identity re-verified exactly for q in {5,7,11,13,37,41}: (q-1-c_q(r))(q-1-c_q(r+2))=0 iff q|r or q|r+2, with c_q by actual cosine sums. Short-window fluctuation measured at s=11 (P=2310, Q={13,17,19,23}): killed density rho=0.4 gives rho*L=8.0 for L=20, but window killed-counts range over [3,13] (deviation +-5 ~ L/2), so the complete-sum density does not bound the short-window sum; the uniform estimate rho*L+E is still required and is not supplied.","prior_art_md":"Reused route 62's search (DLMF 27.10.2-5, OEIS A144311/A072753, Ziller-Morack 1706.03668 h2, Nguyen 202608.1299 sec3.3-3.4). The elementary exact encoding is known algebra; no new external source changes this bounded negative. The one route to a genuine bound is the two-class Jacobsthal function h2 (Ziller-Morack), already tracked under routes 23/26, not the cosine form."},"research_route_id":62,"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":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/62 and return #896. Return the ordinary report and transcript plus research: {route_id: 62, 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":"895","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"896","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/62","transcript_url":"/projects/twin-primes/return/947/transcript","files":[{"sha256":"c9bac1153578e61e08de7e077b6cba6f48a27c46d54f94b4a076475a8da8ed5d","name":"verify_route62_obstruction.py","bytes":3180},{"sha256":"bf1d962124267bf394b3dd6b4188cf5f3ae1fde8a7058d7016e4f6c97fe42f92","name":"route62_obstruction_verify.json","bytes":719}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}