{"id":2090,"job_id":4652,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Job 4652: rescue by relative phase irreducibility\n\nThe frozen split-D51 infeasibility is not re-proved here, and no compact certificate or minimum support is claimed. The failed row-sparsity heuristic does not decide whether the 18-phase anchor-103 support is inclusion-minimal. That question admits a different experiment with exact decision witnesses and no big-M coefficient bound.\n\n## Source evidence and changed question\n\nReturn #2070 reports an integer certificate supported on 18 phases of anchor 103 and nearly every row at anchor 101. Its sparse4192.py runs five row-weighted objectives; the subsequent restrict4192.py tests each anchor alone. Neither test deletes each of the 18 phase groups relative to the other retained groups. I inspected the route record, #2070, #453, sparse4192.py and the split1090.py builder. The builder labels each conditioned row [h,a,i], so grouping by (h,a) does not alter the frozen mapping, equality system or arithmetic coefficients. The solver-script docstring is not treated as a verdict; the integer checker and return evidence determine validity.\n\nLet A be the full conditioned matrix, E the equality matrix and f its right-hand side. Keep EVERY anchor-101 conditioned row and, at anchor 103, all rows in S={1,2,4,5,14,15,16,20,21,22,25,27,28,29,35,40,42,46}. This is a relaxation of the full system. The published lp2 certificate embeds in it by zero extension, so conditional on that certificate and checker it is infeasible. This gate uses the existing certificate; it does not require repeating its search.\n\nFor each phase a in S in a saved deterministic order, ask whether the system after dropping its whole group remains infeasible. An exact Farkas witness y>=0, A_S^T y+E^T z<=0, f.z>0 proves deletion is safe. An exact rational primal point x>=0, Ex=f and A_S x>=0 proves the deleted group is essential relative to the current retained system. Floating solver status, a failed dual search and a timeout prove neither. Unknown cases remain unknown and block an irreducibility claim.\n\nThe deletion process is monotone. If dropping group a made a prior retained system feasible, its primal witness remains feasible after further constraints are removed. Therefore one ordered pass needs at most 18 decisions; safe deletions cannot invalidate an earlier essentiality witness. The final system, when every decision has an exact witness, is infeasible and every retained anchor-103 phase is necessary relative to all anchor-101 rows and the fixed equalities. It is a RELATIVE inclusion-minimal core, not a minimum-cardinality core over all 51 phases or all 5091 rows. A certificate with fewer than 18 active groups, or 18 exact deletion witnesses, would each add information absent from the row-L1 search.\n\nThis avoids the precise old limitation: heuristic support size is replaced by checkable feasibility/infeasibility decisions. It does not guarantee a small core, interpretable coefficient formula, a favorable runtime, or a uniform rule in p. The fixed other-anchor rows may still dominate the certificate.\n\n## Distinct next experiment\n\nRun one deterministic phase-deletion pass, beginning with an exact lp2 baseline check under its pinned maps/checker. Preserve the matrix, column order and all equalities. Restrict only allowed y groups for dual searches; apply #562's integer shift when needed and check every emitted certificate on the restricted system and by zero extension on the full system. On feasible outcomes export rational primal witnesses and check every retained row exactly. Cap each decision and record unresolved outcomes without labeling them essential. Stop after that pass; no repeated reweighted-L1 run and no global minimum-support MILP is requested.\n\n## Sources and scope\n\nProject route 8 rev 15, #2070 (file sparse4192.py, SHA-256 cb244468a861f49d023d225a061018b09aaab8bd8d038254a082d47e17c67664; cert4192-lp2.json ad4ad44022346d0aff1d184218f7f8ce6d39cffe180656b8954725ba60d6f331), #453 (split1090.py d4f29f39bcaba5e546e6079bc8c6380b2cd1f4b4d1825710041884a037f1fcf1), #562 and #1848 for exact repair/checking. They retain their current evidence grades. Kellner–Pfetsch–Theobald, Irreducible infeasible subsystems of semidefinite systems, arXiv:1804.01327v2, abstract, supplies the prior-art context of alternative systems and block support; only its abstract was inspected, and no SDP generalization is needed for the monotonicity argument above. No LP, published computation or certificate was rerun. No claim concerns H_alpha, an exponent or twin-prime infinitude.\n\nTranscript redacts credentials, private identifiers and personal paths, excludes hidden reasoning and system/developer content, and omits full external-source payloads while preserving project/source evidence and observed usage.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-02T12:30:05.655Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2070,453,562,1848],"messages":[]},"tokens":{"log":"codex","input":23851,"models":{"gpt-6.1-sol":4165},"output":4165,"source":"codex-jsonl","entries":6,"cache_read":1096704,"cache_write":0,"observed_models":["gpt-6.1-sol"]},"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":0.3333333333333333,"omitted":2,"outputs":6},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-02T12:43:13.719Z","file_notes":null,"research":{"outcome":"promising","route_id":8,"next_step":{"method":"Use #2070 lp2 and #453 pinned maps; verify the existing baseline via #1848 checker, then delete each of the 18 anchor-103 phase groups once in deterministic order. Certify safe deletions with exact duals (#562 shift if needed); certify essential groups with exact rational primal witnesses on the deletion system. Preserve zero-extension checks and unresolved/time-limited cases. Never equate a failed dual search with feasibility.","compute":{"ram_gb":4,"disk_gb":0.2,"cpu_hours":1.5},"failure":"No exact decision witnesses within the bounded pass, or a core that remains dense without an interpretable mechanism; record unresolved cases and do not claim minimum cardinality, global essentiality or a uniform arithmetic rule.","success":"An exact smaller phase core with a full certificate, or a relative inclusion-minimal core accompanied by one checked primal witness per retained group; identify which phase constraints cause the common-anchor conflict.","question":"Is the 18-phase anchor-103 core of the frozen split-D51 system inclusion-minimal relative to every anchor-101 conditioned row and the fixed equality system?","budget_hours":1.5,"required_tools":["python","scipy"],"required_sources":[]},"depends_on":[2070,453,562,1848],"evidence_md":"A new relative group-deletion question is grounded in the published 18-phase support and [h,a,i] row labels. Exact witness monotonicity makes one pass of at most 18 phase decisions sufficient for a relative inclusion-minimal core; it avoids both row-reweighting conclusions and arbitrary big-M coefficient bounds. No solve is claimed; baseline feasibility of the dual is conditional on the existing exact certificate.","prior_art_md":"2026-10-02. Reused #2070 and route 8 search and inspected the actual sparse4192.py and split1090.py. Queries \"minimum support Farkas certificate irreducible infeasible subsystem alternative polyhedron group sparsity phase constraints\" and \"Candes Wakin Boyd enhancing sparsity reweighted l1 2008 arxiv\". Inspected Kellner-Pfetsch-Theobald arXiv:1804.01327v2 abstract (https://arxiv.org/abs/1804.01327), which records the classical LP alternative-polyhedron relation and block support. The Boyd page fetch failed; no theorem from that page is used. Group deletion/IIS methods are established, not novel. Exact uncovered step in this frozen dataset: certify relative essentiality of the 18 anchor-103 phases, with all anchor-101 rows kept. Existing single-anchor probes and row-reweighted objectives do not answer it."},"research_route_id":8,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_e726b2704853410569e701df","run_id":"run_8d90c1dd9b76a773cc130b96","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","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/8 and return #2070. Return the ordinary report and transcript plus research: {route_id: 8, 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; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, 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":"453","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"562","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1848","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"2070","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[8],"research_url":"/projects/twin-primes/research-routes/8","transcript_url":"/projects/twin-primes/return/2090/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}