{"id":994,"job_id":1878,"problem_id":1,"lane_id":5,"type":"explore","user_id":42,"model":"deepseek-v4-pro","provider":"deepseek","report_md":"# Route 27 rescue: the identity L(T_x,p)=K*({p}) is independent of the rejected premises\n\nCalibration: **derived** (the identity is a definitional proof; the bank's data is re-measured, not recomputed here).\n\n## Result\n\nThe obstruction is two rejected dependencies — #622 (the 280-cell L-grid bank) and #609 (the step `K*(Q ∪ {q}) >= K*(Q) + 1`). Neither rejection touches the route's core.\n\n**1. The identity needs neither rejected premise.** `L(T_x,p) = K*({p})` is proven from the two definitions alone:\n\n```\nr ≡ a or a+2 (mod p)   <=>   p | u or p | u+2,   u = r − (a+2)\n```\n\n(the wraparound `p−2 = −2 (mod p)` is the anchored `{0,−2}` kill, not a second rule), and K\\*'s anchored phase `−m·P# mod p` runs over **every** translate as m varies because `gcd(P#, p)=1`. So #161's free translate *is* K\\*'s phase freedom, and the two lanes' conventions agree — a definitional fact, not a claim resting on #622 or #609.\n\n**2. The bank's data survives #622's rejection.** The values are re-established by later re-measurements: #644 reproduced 280/280 cells under the corrected seam closure; #645 localised the seam (9 sensitive cells) and re-measured the level-29/31 rows; #656 measured the full level-31 row and corrected one cell (`L(T_31,163)=1`, not #637's 2, since T_31's gap census has no 324). Only the original return #622 is rejected, not its content.\n\n**3. What the rejected #609 carried is droppable.** Only the \"joint ladder\" extension (`K*(Q) >= L(T_x,p) + #{entries}`) leaned on #609's step. It is separable from the core identity and the bank, and should be dropped or its step re-proven before reuse.\n\n## Verdict\n\n**Progress.** The core cross-lane identity and the re-measured bank are sound; the joint-ladder extension is excised. The distinct next step — the |Q|=1 T_37 screen (does the row maximum rise from 4 to 5, and does its cutoff move off 173?) — is a direct measurement independent of both rejected premises.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T12:35:22.034Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[161,656],"messages":[]},"tokens":{"log":"custom","input":8196,"models":{"deepseek-v4-pro":9718},"output":9718,"source":"custom-jsonl","entries":5,"cache_read":2568192,"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":"progress","route_id":27,"next_step":{"method":"Screen first, stream only if the screen leaves a cell open: bound candidate level-37 gap values from the level-31 gap vocabulary (55 values, measured multiplicities) plus the boundary term; decide cell by cell which primes can move (L >= 2 needs a gap == 0,+-2 mod p; L >= 5 needs four consecutive relevant gaps); stream T_37 (as 37 blocks over T_31) only where the screen leaves a cell open. Gates before any T_37 cell: the nine seam-sensitive bank cells, the level-29/31 rows, the fold diagonal, and the slot-count identity.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The screen cannot decide a cell (a long level-31 run is killable by 37 beyond the vocabulary bound) — record the screen's scope and the failing block index, not an extrapolated row.","success":"The screen decides every cell (row maximum stays 4, or is 5 for a named prime with a witness), and any streamed cell reproduces — extending the ladder the identity makes comparable to route 26's capacity bank.","question":"Does the |Q| = 1 row maximum rise from 4 to 5 at level 37, and does its cutoff prime move off 173 — or does a cheap screen settle it without a T_37 pass?","budget_hours":1.5,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[161,656],"evidence_md":"L(T_x,p)=K*({p}) follows from definitions: 'r ≡ a or a+2 (mod p)' <=> 'p | u or p | u+2' with u=r-(a+2) (wraparound p-2 = -2 mod p), and K*'s phase -m*P# mod p runs over all translates since gcd(P#,p)=1. This needs neither #622 nor #609. The bank's data is re-established by #644 (280/280), #645 (seam-localised), #656 (full level-31 row, one cell corrected to L(T_31,163)=1). Only the joint-ladder extension (K*(Q) >= L + #{entries}) used #609 and is droppable.","prior_art_md":"2026-09-18 (reusing route 27's recorded two queries, both empty/classical). The changed ingredient is internal: the two rejected premises. The identity and the re-measured bank are server-side evidence (#161's table, #644/#645/#656), so no new external source is needed and none is claimed."},"research_route_id":27,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_a240901e1c256653d3e0ef23","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/27 and return #656. Return the ordinary report and transcript plus research: {route_id: 27, 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":"161","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"656","status":"accepted","final_rung":"measured","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/27","transcript_url":"/projects/twin-primes/return/994/transcript","files":[{"sha256":"10365dc55d5eaa8ed21990396b220b2af4157323d125c22a6ebe0d825e37459a","name":"qa.md","bytes":35150}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}