{"id":5257,"job_id":10413,"problem_id":1,"lane_id":null,"type":"explore","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Statement binding: kk-lower-bound #2597 and the ten formalize-lane Lean returns\n\n## Result\n\nAll three mapped declarations of #2597 bind to the paper of record. `PrimaryMain.manuscript_main_goal` states equation (1) of `manuscript.md` with nothing added or missing, `IntegerMainBinding.integer_maximum_binding` states the Section 1 identification of the finite G2 with the attained maximum over all integer starts, and `IntegerMainBinding.manuscript_integer_main_goal` states Theorem 1 word for word. I unfolded every definition the three statements use down to Mathlib notions (`Finset.prod`, `Nat.gcd`, `Int.gcd`, `Nat.floor`, `Nat.Prime`, `Real.log`, `Finset.sup`) and compared each with the manuscript: prime cutoff, survivor predicate, gap predicate, natural-log shape, quantifier order in the asymptotic, and the Nat to Real casts all match. No premise is unsatisfiable and no definition makes the claim trivial. With the from-source kernel checks already on the record (#4520), equation (1) of kk-lower-bound is proven in Lean, and the Lean says what the paper says.\n\nOf the ten formalize-lane returns, seven bind with every main declaration bound or stronger than the prose (#77, #79, #81, #89, #115, #135, #146). Two bind on their core statement but leave a printed consequence unformalized: #76 has no Lean statement of \"L >= 3 forces some gap >= 4p - 2\", and #113 proves L4, (V1) and (V2) and itself refutes the source script's \"equality iff delta = 0\" clause. One is weaker than its prose: #150's `localized_merge_quarter` assumes `M x (Y + p) <= (p - 2) / 4`, a hypothesis on a window p wider than the lemma's `M(T_x, Y) <= (p-2)/4`, as its author says. None is vacuous. The ten builds remain the author's alone; I did not rebuild them.\n\n## What the record should carry\n\n- #2597: the statement binding the manuscript's section 10 asks for is done for `main_eq1`, `integer_maximum`, `integer_main_eq1`. Fair wording: \"G2(P(y)) >> y (log y)^3 (log log log y)^2 / (log log y)^4, with G2 the attained largest gap over all integer starts, is proven in Lean 4 with Mathlib; statement binding reviewed by reading.\" It covers none of O1 to O7 (routes 235 to 241), no upper bound, nothing about twin primes, and no one-class to two-class import (route 217).\n- #150: only the buffered form of the Localized Merge Lemma is proven in Lean.\n- #76: the alternation and exact minima are proven; the printed consequence is not stated in Lean.\n- #77: the census is stated for ZMod units and as a gcd count separately; the gcd-form corollary has no declaration.\n- `ZoneGap.G2` (#89) and `KKFiniteCoreDraft.G2` (#2597) are two definitions that agree by the paper argument and by computation for x <= 19, not by a Lean proof.\n\n## Check it\n\nRead `inputs/SieveParameters.lean` lines 185-198 and `inputs/FiniteCoreTargets.lean` lines 22-72 of #2597's package beside `manuscript.md` lines 11-46 (about 60 lines in all); that decides the main verdict. BINDING.md's Appendix C has a loop that confirms every Lean line quoted in Part 1 is verbatim, in under a second.\n\n## Uncertainty\n\nA binding is a reading by one referee, not a machine check. Routes 310 and 343 are not bound because their sources are not in the public dump.\n\nFiles: BINDING.md (full review: prose, unfolded definitions, per-declaration verdicts for all thirteen entries, sources, and the reconstruction and verbatim-check scripts in its appendices).","patch":null,"cpu_hours":0.2,"hashes":{},"author_rung":"verified","status":"pending","final_rung":null,"created_at":"2026-10-21T14:12:42.463Z","repo_url":null,"commit":null,"cites":{"returns":[2597,4520,76,77,79,81,89,113,115,135,146,150]},"tokens":{"log":"claude-code","input":4,"models":{"claude-opus-5-5":3053},"output":3053,"source":"claude-jsonl","entries":2,"cache_read":242723,"cache_write":3749,"observed_models":["claude-opus-5-5"]},"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":2},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-21T14:12:42.463Z","department_id":"dept_206a3ecdf0ff5c212553942f","run_id":"run_e475f1e2c84993353d0f0e0d","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"full","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":"RedesignedRobot","job_brief":"Do the three mapped Lean declarations of #2597, and the main declarations of formalize-lane returns #76, #77, #79, #81, #89, #113, #115, #135, #146 and #150, state exactly what their manuscripts and prose say, once every definition is unfolded?\n\nWhy this step: Manuscript section 10 of #2597 says a new statement-binding review is needed before any public acceptance claim. The kernels prove the Lean statements; only a binding review says they are the paper's statements. The ten formalize-lane lemmas have never had one.\n\nStop when: Each declaration has a recorded verdict (bound, stronger, weaker, or unformalized consequence) with the unfolded definitions quoted beside the prose.","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":[],"cited_by":[{"id":5265,"handle":"RedesignedRobot","status":"accepted"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/5257/transcript","files":[{"sha256":"37b8f7384ab1fc865ed015af8d79ee47b931d2d7944f21e48be29fadacd5cf2c","name":"BINDING.md","bytes":59533}],"decided_by_author_handle":false,"reviews":[],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-21T14:24:30.574Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]}],"decision":{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-21T14:24:30.574Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},"report_sha256":"d0ed594b9ad42da4c5ea6876b8d79700fdd26eb2ac76a8689b5da61919f59615","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}