{"id":6000,"job_id":12102,"problem_id":1,"lane_id":32,"type":"explore","user_id":111,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job 12102: route 293 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 #5994 (same step, set by #5604; route 293 `last_return_id` 5604). #5994 found that no recorded return computes S2(R(37#)), validates a lag-2 fusion functional, or reaches rho_1(R(41#)) or c(41#).\n\n**The new input since #5994 is #5999** (route 365, pending), the only return in this job's comparison list. It diagonalises route 365's constellation lift T(p) = pI + K into Euler products for the twin-wheel gap shares c_L(∞). It contains no S2 statistic, no lag-2 fusion functional, no R-word at 37# or 41#, and no c(41#) for route 293's carrier bridge. It does not bear on this step.\n\n## Remaining gap\nUnchanged from #5994: whether the fusion functional closes at lag 2 and yields rho_1(R(41#)) and c(41#) without a direct sweep.\n\n## Sources\n#5994 (this department's earlier check); #5604; #5999.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-26T08:48:44.678Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5604,5994,5999],"messages":[]},"tokens":{"log":"summary","input":2,"models":{"claude-opus-5-5":1885},"output":1885,"source":"reported","entries":0,"cache_read":674661,"cache_write":2350,"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":293,"next_step":{"method":"Extend the isolated+chain fusion functional to S2 (lag 2): write the isolated-deletion change of sum_i g_i g_{i+2} from the six-gap neighbourhood of the deleted point and the chain change as the corresponding window merge; validate it by predicting the next rung's S2 digit for digit against direct enumeration at 11#->13# through 23#->29#; then compute S2(R(37#)) and the q=41 chain corrections in one streaming pass over the R(31#) word, using max_i(g_{i-1}+g_i) of that word as max_gap(R(37#)) for the q=41 chain test (2*41 = 82), and evaluate S1(R(41#)) = 41*S1(R(37#)) + (2*S2(R(37#)) - S1(R(37#))) + chains, then rho_1(R(41#)) and c(41#).","compute":{"ram_gb":4,"disk_gb":2,"cpu_hours":2},"failure":"If the lag-2 chain correction cannot be validated digit for digit against a directly enumerable rung, keep rho_1(R(37#)) = -0.13839731 as the frontier and leave 41# to a direct enumeration on a machine with more memory and CPU hours than this one.","success":"The lag-2 functional reproduces a directly enumerated S2 digit for digit at a rung above its calibration, and the resulting rho_1(R(41#)) gives c(41#) = (1 - rho_1(A(41#))/rho_1(R(41#)))*ln 41 exactly, deciding whether c keeps falling below the 37# value 2.081844.","question":"Does the single-class fusion functional close at lag 2, so that rho_1(R(41#)) and c(41#) follow from the R(31#) word without a 3.04e14 direct sweep?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[5604],"evidence_md":"Re-check after a linked change; reuses step check #5994 (step set by #5604, route 293 last_return_id 5604, open). The only new comparison return, #5999 (route 365, constellation-lift diagonalisation), contains no S2 statistic, lag-2 fusion functional, R-word at 37#/41#, or c(41#); the step remains unanswered."},"research_route_id":293,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_70682f6fbf6580733b92340d","run_id":"run_916812d086720b070e62de98","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"sarenayu","job_brief":"Step check before pursuit. Route #293's next experiment was set by return #5604, and a linked assessment, correction, status, finding or association changed; the previous comparison no longer covers all inputs. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Extend the isolated+chain fusion functional to S2 (lag 2): write the isolated-deletion change of sum_i g_i g_{i+2} from the six-gap neighbourhood of the deleted point and the chain change as the corresponding window merge; validate it by predicting the next rung's S2 digit for digit against direct enumeration at 11#->13# through 23#->29#; then compute S2(R(37#)) and the q=41 chain corrections in one streaming pass over the R(31#) word, using max_i(g_{i-1}+g_i) of that word as max_gap(R(37#)) for the q=41 chain test (2*41 = 82), and evaluate S1(R(41#)) = 41*S1(R(37#)) + (2*S2(R(37#)) - S1(R(37#))) + chains, then rho_1(R(41#)) and c(41#).\",\"compute\":{\"ram_gb\":4,\"disk_gb\":2,\"cpu_hours\":2},\"failure\":\"If the lag-2 chain correction cannot be validated digit for digit against a directly enumerable rung, keep rho_1(R(37#)) = -0.13839731 as the frontier and leave 41# to a direct enumeration on a machine with more memory and CPU hours than this one.\",\"success\":\"The lag-2 functional reproduces a directly enumerated S2 digit for digit at a rung above its calibration, and the resulting rho_1(R(41#)) gives c(41#) = (1 - rho_1(A(41#))/rho_1(R(41#)))*ln 41 exactly, deciding whether c keeps falling below the 37# value 2.081844.\",\"question\":\"Does the single-class fusion functional close at lag 2, so that rho_1(R(41#)) and c(41#) follow from the R(31#) word without a 3.04e14 direct sweep?\",\"budget_hours\":2,\"required_tools\":[],\"required_sources\":[]}\n\nEarlier step check #5994: reuse its conclusions. Compare only the new candidates listed below and references needed to assess them; do not survey the whole project again.\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5999 (route 365, result, pending): Given Theorem 4, each lift step is T(p) = pI + K with K constant upper triangular and distinct diagonal -4..-13, so all steps commute and v(x#) = sum_j c_j prod_{23<q<=x}(q-j) with exact rational x#-independent c_j = P_j v(23#) (<= 2 m_max modes). A^(1) feeds nothing, so H^(1) has no j=4 term; c_L(inf) = 2 h5/(s_L a4 prod_{5<=q<=23}(1-1/q)) * prod_{q>23}(1-4/((q-1)(q-4))), which for L=11 is Theore\n\nReturn the ordinary report and transcript plus research: {route_id: 293, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"5604","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[{"id":6004,"handle":"sarenayu","status":"recorded"}],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/293","transcript_url":"/projects/twin-primes/return/6000/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"f11aae7499b05cb1b073edbe6735ef6c03f8ff6fdd3f3b4029ab9fcd2c7b785e","next_step_sha256":"9d318858da11b7bfc46ce34ec5d9bf1c223d2d8e30f18a3232737923fea2e21f","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}