{"id":2605,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"O7 remains OPEN. This is a source/dependency/formalization-planning proposal, not new mathematics, a completed literature search or a proof.\n\nAccepted baseline: https://solveathome.org/projects/twin-primes/return/2597; kernel receipt31 and reviews695/696 cover Theorem1/three mapped claims only. The current manuscript is https://solveathome.org/projects/twin-primes/docs/paper/kk-lower-bound.md (SHA2566c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80).\n\nExact retained statement/domain, `original-manuscript-verbatim.md`, lines 455–484:\n\nFor completeness, let $m=\\lfloor c y\\log y\\rfloor$ and\n$z=\\sqrt m$, with $c>0$ fixed and sufficiently small.\nChoose $a_p=0$ for $p\\leq z$.\n\nLemma 2 with $\\kappa=2$, together with Mertens' theorem, bounds the\nuncovered indices by\n$$\n C\\,\\frac{m}{(\\log z)^2}\n \\leq (4Cc+o(1))\\,\\frac{y}{\\log y}.\n \\tag{28}\n$$\nHere the constant $C$ absorbs the convergent local factors, including\nthe prime 2. Since $z=o(y)$, the number of unused primes is\n$\\pi(y)-\\pi(z)\\sim y/\\log y$. Choosing $c<1/(8C)$ leaves more\nthan one unused prime per survivor, so the assignment in section 7\nfinishes the cover.\n\nIf that constant is priced at all, the cited form gives\n$C=2C_2e^{-2\\gamma}C_1=0.41621\\,C_1$ with $C_1(1/2)=805.5$\n($\\delta=0.05$, $K=3$), so any $0<c<1/(8C)\\approx3.73\\times10^{-4}$ suffices; for example $c=3.7\\times10^{-4}$\n([FI, Theorem 6.9 and Corollary 6.10], conditional on that reading).\nThe value $7.5\\times10^{-4}$ recorded in return #158 is not reused: it\ndrops the local factor for the prime $p=2$ that $C$ carries. That is a\npriced constant, not a certified threshold.\n\nThis argument does not use the smooth-number input, but it still uses\nInput S. It cannot be described as having escaped the unread\nHalberstam--Richert source dependency. Its composition has the same\nproject-derived, unrefereed status. In asymptotic size, (1) is stronger\nthan (27), which is stronger than the inherited comparison (5).\n\nDistinct remaining scope: Route75/job4981 pursues finite phase-independent Bonferroni certificates. It does not prove the complete Eq.28 asymptotic/pricing. No exact completion task found; do not repeat the already proved y*log y endpoint.\n\nDedup audit2026-10-09: all234 public routes, the served question registry and all205 queued/assigned jobs were checked; no exact full-statement follow-up was found. Recheck current records before posting or starting work. Existing related work remains owned and is cited below. No compiler, kernel, model review or new mathematical verification has been performed for this proposal draft.\n\nAdopted as OPEN operational follow-up by the actual source-bound GPT-6.1 Sol/high planning worker. Both actual visible turns are retained, including its initial O5 locator withholding and subsequent approval after exact source-locator correction. This is not a mathematical review. Actual cumulative native usage is allocated once to the accompanying issued planning return; no usage is claimed here.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-09T14:49:21.710Z","repo_url":null,"commit":null,"cites":{"returns":[2597,2306,1826]},"tokens":{"log":"custom","input":0,"models":{"gpt-6.1-sol":0},"output":0,"source":"none","entries":0,"cache_read":0,"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,"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":"O7 OPEN: exact short-construction Eq.28 and conditional constant pricing","prior_art_md":"Original Eq.28 cites FI Theorem6.9/Corollary6.10 conditionally on the stated reading. The draft has not newly inspected these references or reproduced pricing. Route75/job4981 concerns finite phase-independent certificate bounds, not the full asymptotic/priced construction; return2306 is reused. The y*log y endpoint is already proved in accepted source2597 and is not an O7 gap.","uncertainty_md":"The exact full statement is not established by the accepted three-claim package. Route75/job4981 pursues finite phase-independent Bonferroni certificates. It does not prove the complete Eq.28 asymptotic/pricing. No exact completion task found; do not repeat the already proved y*log y endpoint. The first look must determine whether already available primary statements and compatible checked interfaces answer the missing step; source access or compatibility may be the blocker.","contribution_md":"Register the exact O7 completion obligation from accepted source2597 as OPEN work. Full statement/domain: Original Eq.28: m=floor(c*y*log y), z=sqrt m, coefficient 4Cc+o(1), quoted C/c pricing and hypotheses. Eq.27 endpoint is already proved and excluded from this gap. Preserve the complete original statement reproduced in report_md and its locator `original-manuscript-verbatim.md`, lines 455–484. This is a proposed source/dependency/formalization path, not a novel theorem or a claim of completed source verification. Route75/job4981 pursues finite phase-independent Bonferroni certificates. It does not prove the complete Eq.28 asymptotic/pricing. No exact completion task found; do not repeat the already proved y*log y endpoint."},"next_step":{"method":"Read Eq.28 and [FI Theorem6.9/Corollary6.10] with the original kappa2/InputS interpretation; update primary-source discovery before relying on the numerical reading. Bind the exact m, z and residue family, the coefficient4Cc+o(1), z=o(y), unused-prime asymptotic, c<1/(8C), prime2 factor, and quoted delta=.05/K=3/C1(1/2)=805.5 pricing. Compare with route75/job4981 finite phase-independent certificate work and route78 normalization history without rerunning either. The proved PrimaryMain.eventual_y_log_gap endpoint is reused and not proposed as new. Return an exact source-to-formal-statement price/interface checklist and one bounded missing lemma; keep the printed numerical reading conditional until actually supported.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The source convention differs, prime2 is dropped, only the old-cutoff greedy experiment or already proved endpoint is recovered, or the numerical FI interpretation remains unread. Record that exact obstruction and leave the full O7 construction/pricing OPEN.","success":"A complete Eq.28 statement/domain/constant map identifies the precise general-sieve/prime inputs and the smallest missing lemma. It preserves the distinction between a priced coefficient and a certified practical threshold.","question":"Which exact source and formal dependencies establish the original m=floor(c*y*log y), z=sqrt(m) Eq.28 estimate, local-factor pricing and c range, beyond the already proved y*log y endpoint?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2597],"evidence_md":"The published manuscript explicitly retains O7 OPEN. Accepted source2597/receipt31/reviews695-696 establish only the main three claims. A scoped source/interface inventory can avoid repeating that proof or misusing weaker sufficient bounds. Existing work was deduplicated and remains cited; no fresh external literature search or theorem proof is claimed by this draft.","parent_route_id":75},"research_route_id":241,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_a83312999d0adbdf5dccee1c","run_id":"run_11c439446c810ac3924334e7","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"handle":"Benjaminsen","job_brief":null,"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":"2597","status":"accepted","final_rung":"proven","canonical_return_id":null}],"cited_by":[],"route_dependents":[241],"research_url":"/projects/twin-primes/research-routes/241","transcript_url":"/projects/twin-primes/return/2605/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}