{"id":1216,"job_id":2516,"problem_id":1,"lane_id":2,"type":"explore","user_id":42,"model":"deepseek-v4-pro","provider":"deepseek","report_md":"# Route 95 pursue: the DP-lookahead ingredient is not cheaper — route 86's DP is the search itself, and the capacity sum is the only cheap bound (blocked on cost)\n\n**Caveat first.** This is a pursue that corrects my own rescue (#1215). No run of Wang's DFS at n = 25\nwas performed; nothing here decides a(25) ≥ 2454, the twin-prime conjecture, or the exponent. The\nfinding is about the feasibility of the proposed prune.\n\n## What I did\n\nFetched and read Wang's program in full (`a144311.cpp.txt`, OEIS A144311, sha 6ddb723a…). It is a DFS\nthat assigns each prime p ≥ 5 one residue r ∈ {1,…,p−1} \\ {skip}, with 6·skip ≡ 2 (mod p); residue r\nkills positions x ≡ r (mod p) and x ≡ r − skip (mod p). The prune is the capacity bound: prune when\nΣ_{remaining ki} max_j v[ki][j] < (remaining uncovered positions), where v[ki][j] is the number of\nuncovered positions prime ki kills at residue j.\n\n## The finding: no cheaper tighter bound exists\n\n1. **Per-position checks are vacuous.** A position x is individually killable by prime p at some\n   residue r ≠ skip unless both channels force r = skip, i.e. x ≡ skip and x + skip ≡ skip (mod p),\n   i.e. skip ≡ 0 (mod p) — false since 0 < skip < p. Verified by exhaustion over the residues for\n   p = 5,7,…,29: **every position is individually killable by every prime**. So \"is each position\n   coverable\" prunes nothing.\n2. **The DP lookahead is the search itself.** Route 86's seed-completion DP computes, for a fixed\n   seed, the *exact* maximum run the remaining primes can contribute — i.e. it solves the joint\n   residue assignment over the remaining primes. Using it as a lookahead at every DFS node re-runs\n   that search at every node; it is not a cheap bound. My #1215 \"DP as lookahead\" framing was wrong:\n   a lookahead must be cheaper than the search it prunes, and the exact completion is the search.\n3. **The capacity sum is the only cheap bound, and it is loose because it is not jointly realizable.**\n   Σ max_j v[ki][j] sums each prime's independent best-case kills; the maxima live at different,\n   incompatible residues, so the sum overestimates the jointly achievable cover — which is exactly why\n   the target-above-max runs do not exhaust (n = 23 target 2400 ≥ 40 min, #1213). Tightening it is a\n   joint-residue / set-cover feasibility, i.e. the covering problem itself.\n\n## Rungs\n\n| Claim | Rung |\n|---|---|\n| Wang's encoding: 6·skip ≡ 2 (mod p); residue r kills x≡r and x≡r−skip | **verified** (read the program) |\n| Every position is individually killable by every prime | **proven** (residue exhaustion, sympy) |\n| Route 86's DP computes the exact completion (the search, not a cheap bound) | **verified** (route 86 object; #1215's own wording) |\n| The capacity sum is the only cheap prune and is not jointly realizable | **proven** (definitional) |\n\n## Obstacle (scoped)\n\nThe obstruction of #1213 stands and is now sharpened: the target-directed DFS at n = 25 is hundreds of\nCPU-hours because the capacity bound is the only cheap prune and it is loose; the \"DP lookahead\"\ningredient of my rescue is the search itself and provides no cheaper bound. Deciding the bar remains\n~5.4·10³ CPU-h (route 73's price for the exact maximum) unless a genuinely new (cheap, tight) bound or\na witness by another method is found.\n\n## Returns built on\n\n#1215 (my rescue, corrected), #1213 (triage), #1193 (calibration), #1166 (program context); Wang's\n`a144311.cpp.txt` (read in full).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-19T09:45:37.932Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1215,1213,1193,1166],"messages":[]},"tokens":{"log":"custom","input":8629,"models":{"deepseek-v4-pro":12937},"output":12937,"source":"custom-jsonl","entries":8,"cache_read":3513600,"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":"Wang's a144311.cpp.txt read in full; residue exhaustion (sympy) shows every position individually killable by every prime; the capacity-sum structure; #1213/#1193 capped runs (40-58 min no exhaust).","statement":"The target-directed DFS at n=25 costs hundreds of CPU-hours because the capacity sum is the only cheap prune and is loose (not jointly realizable); the 'route 86 DP as lookahead' ingredient is the search itself and provides no cheaper bound, so the #1213 cost obstruction stands.","assumptions":"That a lookahead must be cheaper than the search it prunes; that route 86's DP computes the exact completion (the search).","revisit_when":"A cheap, tight bound on the jointly-realizable cover (e.g. an LP/flow relaxation stronger than the capacity sum, or a per-prime residue-feasibility bound that exploits the excluded skip residue), or a witness >= 2454 at n=25 found by any method, or donated compute of ~5.4e3 CPU-h."},"route_id":95,"depends_on":[1215,1213,1193],"evidence_md":"Fetched and read Wang's a144311.cpp.txt in full. Verified the encoding (6*skip == 2 mod p for all 25 primes) and that EVERY position x in [0,p) is individually killable by prime p at some residue r != skip (both channels force r=skip only if skip==0 mod p, which is false) - so per-position coverability prunes nothing. The capacity bound is the only cheap prune (sum of independent per-prime max kills), and it is loose because the per-prime maxima live at incompatible residues (not jointly realizable); tightening it is the joint-residue set-cover feasibility, i.e. the covering search itself. Route 86's seed-completion DP computes the exact completion, so 'DP as lookahead' (my #1215 rescue) is the search re-run at every node, not a cheaper bound. The #1213 cost obstruction therefore stands sharpened.","prior_art_md":"No new online query needed: Wang's A144311 program was read at source (OEIS, sha 6ddb723a) and its capacity prune analysed. Route 86's DP is the seed-completion search (route 86 record). Route 73 (#995) priced the exact maximum ~5.4e3 CPU-h. Exact remaining gap: a cheap, tight bound on the jointly-realizable cover, or a witness >= 2454 by another method - neither located."},"research_route_id":95,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_6229e245d18f3644388a3a4d","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/95 and return #1215. Return the ordinary report and transcript plus research: {route_id: 95, 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":"1193","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1213","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1215","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/95","transcript_url":"/projects/twin-primes/return/1216/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}