{"id":6011,"job_id":12113,"problem_id":1,"lane_id":null,"type":"explore","user_id":111,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job 12113: route 239 step check (re-check after a linked change). The step is still open.\n\n**Outcome: promising.** The step is copied unchanged. This was read-and-compare only: nothing was run.\n\nThis re-check reuses step check #6007 (step set by #5554; route 239 `last_return_id` 5554), which found that no recorded return computes the finite-level LP supremum of S/(X·V).\n\n**The only new comparison input is #6010** (route 238, needs_capability, this department). It concerns a Lean prime-counting identity at the pin, with no LP, no S/(X V) and no kappa = 4 bound, so it does not bear on this step.\n\n## Remaining gap\nUnchanged: whether the supremum stays ≤ 12 at kappa = 4 for z ≤ 30.\n\n## Sources\n#6007; #5554; #6010.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-26T09:00:11.138Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5554,6007,6010],"messages":[]},"tokens":{"log":"summary","input":2,"models":{"claude-opus-5-5":1515},"output":1515,"source":"reported","entries":0,"cache_read":764053,"cache_write":1572,"observed_models":[]},"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":"medium","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":"promising","route_id":239,"next_step":{"method":"Exact rational linear programs. For each z in {2,3,5,6,10,30} and each integer density tuple g on primes p<=z with 0<=g(p)<=min(kappa,p-1), maximize S/(X V) over X>0, a_n>=0 with the divisor decomposition r_d = sum_{d|n} a_n - g(d)X/d restricted to |r_d|<=g(d) for every d | P(z); support n <= P(z), weights rational. Report the sup per (kappa,z) for kappa=2,4, the maximizer's z, and the growth of the sup in z. A vertex enumeration over the constraints suffices (the problem is a small LP per g); cross-check one cell per z with scipy.optimize.linprog(method='highs').","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The sup grows with z beyond 12, or a single (g,a,X) with ratio > 12 appears at kappa=4: integrality alone does not bound the constant at finite level, so the annotation must carry Omega_1 with an explicit A=A(kappa) and the kappa=4 bridge at A=5 only.","success":"The sup is bounded by 12 at every z <= 30 for kappa=4, with the maximum attained at z=2 (no growth in z): the integrality field then carries a kappa-only constant at finite level, and the corrected interface can be used without an explicit A field in the annotation (P5 of the report).","question":"For the corrected O5 interface, is the integrality field alone enough at finite level: how large can S/(X V(z)) be for integer-valued multiplicative g with g(p)<=kappa, g(p)0, a_n>=0 with the divisor decomposition r_d = sum_{d|n} a_n - g(d)X/d restricted to |r_d|<=g(d) for every d | P(z); support n <= P(z), weights rational. Report the sup per (kappa,z) for kappa=2,4, the maximizer's z, and the growth of the sup in z. A vertex enumeration over the constraints suffices (the problem is a small LP per g); cross-check one cell per z with scipy.optimize.linprog(method='highs').\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"The sup grows with z beyond 12, or a single (g,a,X) with ratio > 12 appears at kappa=4: integrality alone does not bound the constant at finite level, so the annotation must carry Omega_1 with an explicit A=A(kappa) and the kappa=4 bridge at A=5 only.\",\"success\":\"The sup is bounded by 12 at every z <= 30 for kappa=4, with the maximum attained at z=2 (no growth in z): the integrality field then carries a kappa-only constant at finite level, and the corrected interface can be used without an explicit A field in the annotation (P5 of the report).\",\"question\":\"For the corrected O5 interface, is the integrality field alone enough at finite level: how large can S/(X V(z)) be for integer-valued multiplicative g with g(p)<=kappa, g(p)