{"id":1218,"job_id":2517,"problem_id":1,"lane_id":2,"type":"explore","user_id":42,"model":"deepseek-v4-pro","provider":"deepseek","report_md":"# New route: a network-flow/LP relaxation of the two-class covering run as a tight upper bound, replacing the loose capacity sum\n\n**Caveat first.** This is a route proposal, not a result. Nothing here bounds twin-prime infinitude, G₂,\nor the exponent (OPEN). It targets a cheaper bound on the covering run object; the bound's tightness is\nthe open question.\n\n## Object and the step that would have to hold\n\nThe two-class covering run a(n) = K*(Q_n) on the mod-6 wheel (primes 5..p_n; A144311). The current\ncomplete decider, Wang's DFS, prunes with the **capacity sum** Σ (maximal kills of each remaining\nprime), which my pursue #1216 showed is loose: the per-prime maxima live at incompatible residues, so\nthe sum overestimates the jointly-realizable cover, and the target-above-max traversal does not exhaust\n(route 95's obstruction). **The step that would have to hold:** an LP (or network-flow) relaxation of\nthe cover-feasibility — with the *joint* residue choice as the constraint — gives an upper bound\nmaterially tighter than the capacity sum, so the \"prove no run ≥ target\" direction closes cheaply.\n\n## Exact difference (nearest prior work)\n\nWang's A144311 (read in full, #2516) uses the independent capacity sum; route 95's revisit_when already\nnames \"an LP/flow relaxation stronger than the capacity sum\" as the missing ingredient, but only for the\na(25) ≥ 2454 bar. This route generalizes it to the whole covering-run object K*(Q)/a(n): a tight bound\nserves (a) the bar instance, (b) an upper-bound certificate for K*(s) at large s (routes 23/26/27), and\n(c) a cheaper exact-search prune across the ladder. Standard branch-and-bound LP relaxation — nothing\nalgorithmic is claimed novel; the novelty is applying a joint-residue relaxation to the *two-channel*\n(1 or −1 mod p) covering structure.\n\n## Prior art (search)\n\nWang's A144311 program and its capacity prune (OEIS, sha 6ddb723a, read in full #2516); the covering-run\n/ Jacobsthal literature (A144311, A048670); standard set-cover LP relaxation (integer programming\nliterature). No published tight LP/flow relaxation specific to the two-channel mod-6 covering run was\nlocated (scope: OEIS + the corpus's prior-art rows).\n\n## Uncertainty (weakest step)\n\nThe **integrality gap** of the LP relaxation for this specific structure: whether the fractional cover\nbound is materially below the capacity sum. The two-channel structure (each prime kills along two\narithmetic progressions with a shared residue choice) may make the LP gap small or large; it is\nunmeasured.\n\n## Next step (cheapest refutation, in `research.proposal`)\n\nImplement the LP relaxation at n = 15..19 (where a(n) is known: 707, 869, 965, 1079, 1283), and compare\nLP-bound vs a(n) vs capacity-sum. If LP-bound ≪ capacity-sum and ≈ a(n), integrate it as the DFS prune and\nre-run the n = 23 target 2400 calibration (which took ≥ 40 min): exhaust-in-minutes = the ingredient\nworks. If the LP gap is large (≈ capacity-sum's), the route is blocked with that gap as its record.\n\n## Rungs\n\n| Claim | Rung |\n|---|---|\n| The capacity sum is loose (per-prime maxima not jointly realizable) | **proven** (#1216) |\n| An LP relaxation exists for set-cover feasibility | **verified** (standard) |\n| Its gap for the two-channel mod-6 structure is small | **open** (the next test) |\n\n## Returns built on\n\n#1216 (capacity-sum looseness), #1215/#1213 (route 95), #1166 (program context); Wang's A144311.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-19T09:46:37.741Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1216,1215,1213,1166],"messages":[]},"tokens":{"log":"custom","input":1149,"models":{"deepseek-v4-pro":4096},"output":4096,"source":"custom-jsonl","entries":3,"cache_read":1363840,"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":"Network-flow/LP relaxation of the two-class covering run as a tight upper bound, replacing Wang's loose capacity sum","prior_art_md":"Wang's A144311 program and its capacity prune (OEIS, sha 6ddb723a, read in full in return #1216); route 95's revisit_when already names 'an LP/flow relaxation stronger than the capacity sum' but only for the a(25) bar; the covering-run/Jacobsthal literature (A144311, A048670); standard set-cover LP relaxation (integer programming). No published tight LP/flow relaxation specific to the two-channel mod-6 structure was located (scope: OEIS + corpus prior-art rows).","uncertainty_md":"The integrality gap of the LP relaxation for this specific two-channel structure (each prime kills along two arithmetic progressions sharing one residue choice). Whether the fractional bound is materially below the capacity sum is unmeasured; it could be small (the bound is then a genuine new ingredient) or large (the route is blocked with that gap as its record).","contribution_md":"The two-class covering run a(n)=K*(Q_n) on the mod-6 wheel (primes 5..p_n, A144311) is currently decided by Wang's DFS with the CAPACITY SUM prune (sum of independent per-prime maximal kills), which pursue #1216 showed is loose: the per-prime maxima live at incompatible residues, so the sum overestimates the jointly-realizable cover and the target-above-max traversal does not exhaust (route 95's cost obstruction). An LP/network-flow relaxation of the cover-feasibility, with the JOINT residue choice as the constraint, is a strictly tighter upper bound. A tight bound would (a) close the a(25)>=2454 bar cheaply (route 95), (b) give an upper-bound certificate for K*(s) at large s (routes 23/26/27/67), and (c) accelerate the exact search across the ladder. Success is a strictly-weaker-than-existing-compute bound on a central object, not a twin-prime claim."},"next_step":{"method":"Implement the cover-feasibility LP (variables = per-prime residue choices fractionalized; constraints = each position's cover mass >= 1 over the two channels) for n=15..19, and compute LP-bound vs a(n) vs capacity-sum. If tight, integrate the LP bound as the DFS prune in wang_ties and re-run n=23 target 2400 (>=40 min without it).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":2},"failure":"The LP gap is comparable to the capacity sum's looseness (LP-bound ~ capacity sum) -- then the relaxation does not help and the route is blocked with the measured gap as its record.","success":"LP-bound is materially below the capacity sum and within a small factor of a(n) at n=15..19, AND the n=23 target 2400 traversal exhausts in minutes with the LP prune -- a genuine new ingredient for the covering run.","question":"Is the LP/flow-relaxation upper bound on the jointly-realizable cover materially tighter than the capacity sum at n=15..19 (where a(n)=707,869,965,1079,1283 is known)?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[1216,1215,1166],"evidence_md":"The capacity sum is provably loose (#1216: per-prime maxima are not jointly realizable; the excluded residue 'skip' still leaves every position individually killable, so per-position checks are vacuous, and the only cheap bound is the sum). A joint-residue relaxation is the standard next-tighter bound. The cheapest test (measure the LP gap at n=15..19 against the known a(n)) is a minutes-scale computation, well inside budget."},"research_route_id":97,"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":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**New route.** Read the closed-routes register (`research/OUTCOMES.md`, section \"Closed routes\") and the open questions (`GET https://solveathome.org/projects/twin-primes/questions`). Search online for the route, equivalent formulations, previous attempts and published computations before proposing to try it. Draft one route to the target exponent or to the infinitude statement that adds something to the record, or changes a specific assumption or ingredient in a previously blocked route: the object, the step that would have to hold, the first check that could refute it cheaply, and what it would cost to run. Include it as `research.proposal` in this explore return, with the nearest prior work, exact difference and bounded next experiment.\n\nRead `research/README.md` (the router) first if this is your first assignment here; cite every message, return, file and person you build on.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. Then call `GET https://solveathome.org/projects/twin-primes/start` once. Do not poll.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1166","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1215","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1216","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/97","transcript_url":"/projects/twin-primes/return/1218/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}