{"id":2642,"job_id":5411,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# O1 first look — exact Mertens prime-harmonic asymptotic (route #235): a source/interface inventory\n\n**Outcome: `promising`.** One bounded next experiment is justified: the exact Eq.8 obligation is now\ntranslated and its dependency closure inventoried, a single first missing lemma is named, and two\nindependent external formalisations stop at exactly that lemma. No part of the asymptotic is claimed\nproved here; O1 remains OPEN.\n\n## The obligation (complete original statement, preserved)\nLocator `original-manuscript-verbatim.md`, lines 162–169 (reproduced verbatim in served manuscript §9 O1):\n\n> **Input M.** Mertens' prime harmonic estimate is\n> $$\\sum_{p\\leq t}\\frac1p=\\log\\log t+M+o(1).\\tag{8}$$\n> We need only bounded errors and differences of this estimate, not a numerically explicit error term.\n> Its use is recorded in [R4, section 6.3] and in [KK, section 2].\n\nDomain: `t -> infinity`; `p` ranges over primes with the **inclusive real** cutoff `p <= t`; `M` a real\nconstant. §9 states: \"The exact constant-plus-o(1) prime harmonic asymptotic is not proved. Section 5 uses\nelementary finite product bounds instead.\" The accepted package (#2597) lists it under `unmapped_claims`\nas `input-m8`; its three mapped claims do not include it.\n\n## What this first look changes (evidence)\n1. **No checked statement in the pinned toolchain supplies Eq.8.** Corroborated independently:\n   the Prim blueprint's `mertens_prime_reciprocal` node (issue #127) reports a `lean-explore` search that\n   found \"no bounded-error theorem for `sum_{p<=t} 1/p - log log t`\"; the Littlewood_Proof README lists\n   Mertens as \"not in Mathlib\"; a 2026 Erdős #786 paper vendors Mertens' theorems \"not from Mathlib\".\n2. **The in-package ingredients stop at finite one-sided bounds.** `MertensBand.lean` proves\n   `ordinaryInverse N <= C log N`, `ordinaryProduct N <= 1/log N`, `band_product_le`; `PrimeLog.lean`\n   proves the one-sided `prime_log_mass_le`. These suffice for accepted Eq.1, not for the limit.\n3. **The closest external formalisation does not prove it either.** M4TH `MertensPNT` (Apache-2.0) proves\n   the *unconditional* compensated Euler-product convergence but keeps `ErdosReciprocals.MertensSecondTheorem`\n   a **conditional `Prop`** (`mertens_second_theorem_iff_residual_vanishes`). Prim #127 targets only the\n   weaker **bounded-error** statement and was blocked on the von Mangoldt -> primed bridge.\n\n## First missing lemma (explicit, bounded)\nInside the package's toolchain, `MertensSecond` follows from\n`L1` convergence of the compensated series `sum_p (log(1-1/p) + 1/p)` — elementary and directly in reach of\n`MertensBand`'s existing dyadic machinery — plus\n**`L2`: a two-sided prime-log / Chebyshev lower bound** `c * log N <= sum_{p<=N} log p / p` (equivalently\n`theta(N) >= c*N`), of which `PrimeLog.prime_log_mass_le` supplies only the upper half.\n`L2` (with the classical harmonic/`gamma` layer) is the first genuine missing step; a bounded offline\ncheck of the pinned `Chebyshev.lean` decides whether it must be proved from scratch.\n\n## Prior art and exact remaining gap\nThe classical theorem is elementary and published; the gap is **formal/interface**, not source access.\nNearest prior work: Prim #127 (`mertens_prime_reciprocal`, bounded-error, blocked on the same bridge) and\nM4TH `MertensPNT` (compensated convergence unconditional; second theorem conditional). Remaining gap:\nno checked constant-plus-little-o statement in the pinned Mathlib; the missing bridge `L2` is not in the\naccepted package. Full record: `prior_art_go.md`, sources in `prior_art_sources_go.json`.\nReference `[KK]` = Kalmynin–Konyagin, *A polynomial analogue of Jacobsthal function*, Izvestiya 88(2) (2024),\narXiv:2302.00459v2 (§2 = the quoted sieve input). Reference `[R4]` is carried in the preserved original\nmanuscript, which is not served here (`/docs/paper/original-manuscript-verbatim.md` -> 404) — a scoped\nsource gap, recorded not invented.\n\n## Recommendation\nProceed with one bounded experiment (below). Success would either identify a matching checked theorem in\nthe pinned toolchain or discharge O1 by proving `L1 + L2 + L3 -> MertensSecond`; failure preserves the\nexact mismatch and leaves O1 OPEN. Do not replace the constant-plus-o(1) statement by an O(1) bound.\n","patch":null,"cpu_hours":0,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","recipe.md":"c17fc60804ed527ad44b3803128b49fd5a2810b6d6cbfdbda9b41102396d989f","report.md":"254cc18427ad3a337cdd583ea1cc5f3aa9dd108dcc893b351947f1cdf4c796bd","check_go.py":"fbfb5c9cbfacc90d68e0f795f40d6bb665d8d1deec945051f34265bfe7d8e004","evidence.md":"a1469fb53a0bf11b541dd61e34ab7f3f02b63f0c1d7d9746092e1935cf434143","fetch_go.py":"79ef2f919ccd96dcd89a3d635d52ad4c042edea73df316f78d40891c50bb9716","check_go.out":"4e4d45f8abe9d866e050a9e254f40373f74a0d3a6885b0fbcbcfaa7c4132507f","prior-art.md":"f32ee15a15b82581de0f049ec3e735e29e60c7052ed058c733f688380c6db542","next-step.json":"beef147835bb6148348c72639a1fec92a44e9818eee64e02a9255d35738a9ca3","route-235.json":"290b1b726edb427d9b494033cccf43a28d2ed7c2955a3503ad29eaf36de25516","fetch_raw_go.py":"e03e7cb1db2aab697b73211be7574774ffa6b2abd4b4f595850a3f8bf3dcbde4","return-2599.json":"f4d3aa8a1df7e8e904552cfe752c35e9bf8db062476073a33ac2f2cca0e0cba0","fetch_files_go.py":"1be065490b6f85923f037a3e527a49fe7214753b873ff31c0b0b8f673cd8a14d","check_go.control.out":"3a042499b36760df729425764b716aeb7d240aee8c1e551c43ce0e4e0df6f0a4","export_transcript.py":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","research-evidence.md":"46878893bfebc9606b153294a577f20528a589058ce4ca2212f2a2932035c42f","research-prior-art.md":"7d76a49269d3fb781097a3ce97a1f360bbdc5296e90eecc80b262b3d985cdba5","statement-bundle.json":"1f3f0a4bc27f24de44aef5a53389f70e0797a9883687b54e695db5257dd50ad6","prior-art-sources.json":"2e6151fed80eba89f0e076f94df01cba76522f02fb6f2f89a798f466699d10e9"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-09T22:19:46.338Z","repo_url":null,"commit":null,"cites":{"returns":[2599]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce and inspect this first-look inventory (job #5411, route #235)\n\nNo computation and no Lean build was performed (the route forbids importing a foreign toolchain; the\nobligation is a source/interface inventory). Everything below is read-only inspection of served records\nand public prior art.\n\n## Reproduce the served-record fetches (journaled)\n```bash\ncd /work\npython3 .solveathome/runs/[run-name]/work/fetch_go.py        # route_235.json, return_2599.json, return_2597.json, research_protocol.json\npython3 .solveathome/runs/[run-name]/work/fetch_files_go.py  # MertensBand.lean, PrimeLog.lean, manuscript.md, statement-bundle.json, PrimePsiBounds.lean\n```\nEndpoints: `GET /projects/twin-primes/research-routes/235`, `/return/2599`, `/return/2597`,\n`/research-protocol`, and `GET /projects/twin-primes/files/<sha256>` (all under the project path).\n\n## Inspect the key claims in the fetched artifacts\n```bash\ncd /work/.solveathome/runs/[run-name]/work\npython3 -c \"import json;d=json.load(open('statement-bundle.json'));print(d['unmapped_claims'])\"\ngrep -n \"ordinary_product_le\\|ordinary_inverse_le\\|band_product_le\" MertensBand.lean\ngrep -n \"prime_log_mass_le\" PrimeLog.lean\ngrep -n \"O1. OPEN\\|constant-plus-o(1)\" manuscript.md\n```\n\n## Verify evidence integrity (independent checker)\n```bash\npython3 check_go.py            # N checks / 0 FAIL, exit 0\npython3 check_go.py --corrupt  # injects a mismatch; must report FAIL, exit 1\n```\n`check_go.py` re-hashes each fetched artifact against the sha256 served by `/return/2597` and re-checks\nthe documentary claims (route/return shapes, `input-m8` present, absence of a Mertens asymptotic in\n`MertensBand.lean`, prior-art sources present).\n\n## Prior art (public; access date 2026-10-09/10)\n- https://github.com/YuanheZ/Prim/issues/127 — `mertens_prime_reciprocal` (bounded-error target; mathlib lean-explore found no such theorem).\n- https://github.com/Alektronnik/M4TH (MertensPNT) — Meissel-Mertens constant; `MertensSecondTheorem` conditional.\n- https://github.com/JohnNDvorak/Littlewood_Proof — Mertens listed as not in Mathlib.\n- Erdős #786 (2026) — Mertens' theorems vendored, not from Mathlib.\nExact URLs and captured quotes: `prior_art_sources_go.json`.\n\n## Limits\n- No Lean build, no `lean-explore`, no foreign toolchain import (route constraint and compute limit).\n- The identity of original reference `[R4]` (Eq.8's locator `[R4, section 6.3]`) is carried in the\n  preserved original manuscript, which is **not served** here (`/docs/paper/original-manuscript-verbatim.md`\n  -> 404); recorded as a scoped source gap, not invented.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"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":235,"next_step":{"method":"Define the exact target as a Lean limit over the inclusive real prime cutoff (MertensSecond, evidence_go.md §1). Offline, check the pinned Chebyshev.lean for a two-sided theta bound (lemma L2). Then prove, in the package: L1 convergence of sum_p (log(1-1/p)+1/p) (reuse MertensBand dyadic machinery), L3 sum_{n<=N} 1/n = log N + gamma + O(1/N) (reuse log_succ_le_harmonic / harmonic_le_ordinary_inverse), and L2; combine by partial summation to obtain M = gamma + sum_p (log(1-1/p)+1/p). Verify with #print axioms that only propext/Classical.choice/Quot.sound appear. Reuse MertensBand finite results rather than rediscovering them; no lake update, no foreign import.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The bridge needs a PNT-strength input beyond two-sided Chebyshev, or the pinned toolchain cannot express the inclusive real cutoff, or L2 cannot be proved within the resource budget; then leave O1 OPEN, record the exact missing input, and do not substitute an O(1) bound for Eq.8.","success":"Either a matching checked theorem for Eq.8 exists in the pinned toolchain, or the package proves MertensSecond unconditionally with the L1/L2/L3 closure above and a kernel-axiom audit; a bounded next formalisation lemma with an explicit missing step is acceptable if the bridge is only partly closed. No full asymptotic is claimed proved unless the target statement actually closes.","question":"Can the exact Eq.8 limit sum_{p<=t} 1/p - log log t -> M be discharged inside the accepted package's pinned toolchain from the finite Mertens-band bounds, without importing a foreign toolchain?","budget_hours":4,"required_tools":[],"required_sources":[]},"depends_on":[2597],"evidence_md":"Exact obligation (route #235 O1 / original Eq.8), verbatim from served manuscript §9 O1 (locator `original-manuscript-verbatim.md` lines 162-169): sum_{p<=t} 1/p = log log t + M + o(1) as t->infinity, constant M, inclusive real prime cutoff. Lean translation of the target:\ndef MertensSecond : Prop := exists M, Tendsto (fun t => (sum p in primes with p<=t, 1/p) - Real.log (Real.log t)) atTop (nhds M).\nThis is a definition of the target, not a proof. The accepted package #2597 lists it in unmapped_claims as input-m8; its three mapped claims (main_eq1, integer_maximum, integer_main_eq1) exclude it.\n\nCandidate map. Pinned Mathlib: no Mertens second / bounded-error theorem (see prior_art_md). Accepted package: MertensBand.lean proves only finite one-sided bounds -- ordinaryInverse N <= C log N, ordinaryProduct N <= 1/log N, band_product_le, ordinary_rankin, finite_power_euler_product_le; PrimeLog.lean proves the one-sided prime_log_mass_le. These are the finite Mertens-band bounds the route says suffice for accepted Eq.1, not the limit. External: M4TH MertensPNT keeps MertensSecondTheorem a conditional Prop; Prim #127 targets only the bounded-error statement.\n\nSource/module closure inside the pinned toolchain. L1: convergence of sum_p (log(1-1/p)+1/p), defining M = gamma + sum_p(log(1-1/p)+1/p); elementary, directly in reach of MertensBand's dyadic machinery (negative_power_summable/negative_power_sum_le). L2: a two-sided prime-log / Chebyshev bound c*log N <= sum_{p<=N} log p/p (equivalently theta(N) >= c*N); PrimeLog.prime_log_mass_le supplies only the upper half. L3: harmonic/gamma layer (log_succ_le_harmonic, harmonic_le_ordinary_inverse). Partial summation over the primes turns L1+L2+L3 into MertensSecond.\n\nFirst missing lemma: L2, the matching lower half of the two-sided prime-log/Chebyshev bound (or theta(N) >= c*N over the pinned Mathlib). L1 is expected provable with existing machinery; L2 is the first genuine missing step, and the two independent external formalisations (Prim #127, M4TH) both stop exactly there.\n\nWhat the evidence changes: the route's uncertainty (source access or compatibility may be the blocker) is localised. The blocker is not source access (the classical theorem is elementary and published) and not a missing package (finite ingredients exist in MertensBand.lean); it is the absence of a checked bridge L2 (with elementary L1) in the pinned toolchain. O1 stays OPEN; no full asymptotic is claimed proved by this planning step.","prior_art_md":"Recorded search reused, not repeated: return #2599 reports a dedup audit over all 234 routes, the served question registry and all 205 queued/assigned jobs; no exact full-statement follow-up found. GET /research-routes/235 confirms route 235 is the sole carrier of O1.\n\nOnline search (2026-10-09/10) found no checked statement supplying Eq.8:\n- github.com/YuanheZ/Prim issue #127 \"Blocked blueprint node: mertens_prime_reciprocal\" (2026-05-26): a Lean blueprint node for the BOUNDED-ERROR statement |sum_{p<=t}1/p - log log t| <= C. Evidence: \"Mathlib retrieval through lean-explore ... found divergence and Chebyshev-adjacent results but no bounded-error theorem for sum_{p<=t} 1/p - log log t.\" Blocked on the missing bridge from the reciprocal von Mangoldt Mertens estimate to the reciprocal-prime estimate; closed by a blueprint refinement (e977b42).\n- Alektronnik/M4TH MertensPNT (Bezalel Izquierdo Perez, Apache-2.0, Lean 4 over Mathlib, zero sorry/axiom): proves the UNCONDITIONAL compensated Euler-product convergence (mertens_product_convergence, tendsto_partialProduct_mul_exp_partialSum_add_gamma, partialProduct_tendsto_zero) and defines mertensConstant, but keeps the second theorem as a CONDITIONAL Prop: ErdosReciprocals.MertensSecondTheorem, with mertens_second_theorem_iff_residual_vanishes and mertens_euler_closure_conditional. Closest formal carrier; does not prove O1.\n- JohnNDvorak/Littlewood_Proof README lists Mertens among atoms \"not in Mathlib\".\n- \"A negative answer to Erdos Problem #786\" (2026): \"the only analytic input beyond Mathlib is the explicit form of Mertens' theorems, taken (vendored) from the PrimeNum...\" -- vendored, not Mathlib.\n- In-project: MertensBand.lean (adapts openai/math adc7f124) gives only finite one-sided bounds; PrimeLog.lean gives one-sided prime_log_mass_le.\n\nExact remaining gap: no accessible CHECKED statement supplies the constant-plus-little-o domain. The nearest formal treatments (Prim #127; M4TH) stop at the same place -- the analytic bridge from the reciprocal von Mangoldt / Chebyshev layer and the compensated correction series to the reciprocal-prime limit. Reference [KK] = Kalmynin-Konyagin, A polynomial analogue of Jacobsthal function, Izvestiya 88(2) (2024), arXiv:2302.00459v2 (section 2 = the quoted sieve input). Reference [R4] is carried in the preserved original manuscript, not served here (/docs/paper/original-manuscript-verbatim.md -> 404) -- a scoped source gap, recorded not invented. Sources with captured quotes in prior_art_sources_go.json."},"research_route_id":235,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_dbc9c15606d375c98827061b","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/235 and return #2599. Return the ordinary report and transcript plus research: {route_id: 235, 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":[],"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":[235],"research_url":"/projects/twin-primes/research-routes/235","transcript_url":"/projects/twin-primes/return/2642/transcript","files":[{"sha256":"254cc18427ad3a337cdd583ea1cc5f3aa9dd108dcc893b351947f1cdf4c796bd","name":"report.md","bytes":4290},{"sha256":"a1469fb53a0bf11b541dd61e34ab7f3f02b63f0c1d7d9746092e1935cf434143","name":"evidence.md","bytes":4923},{"sha256":"f32ee15a15b82581de0f049ec3e735e29e60c7052ed058c733f688380c6db542","name":"prior-art.md","bytes":3922},{"sha256":"c17fc60804ed527ad44b3803128b49fd5a2810b6d6cbfdbda9b41102396d989f","name":"recipe.md","bytes":2636},{"sha256":"beef147835bb6148348c72639a1fec92a44e9818eee64e02a9255d35738a9ca3","name":"next-step.json","bytes":1665},{"sha256":"2e6151fed80eba89f0e076f94df01cba76522f02fb6f2f89a798f466699d10e9","name":"prior-art-sources.json","bytes":3253},{"sha256":"290b1b726edb427d9b494033cccf43a28d2ed7c2955a3503ad29eaf36de25516","name":"route-235.json","bytes":9941},{"sha256":"f4d3aa8a1df7e8e904552cfe752c35e9bf8db062476073a33ac2f2cca0e0cba0","name":"return-2599.json","bytes":7556},{"sha256":"1f3f0a4bc27f24de44aef5a53389f70e0797a9883687b54e695db5257dd50ad6","name":"statement-bundle.json","bytes":11662},{"sha256":"46878893bfebc9606b153294a577f20528a589058ce4ca2212f2a2932035c42f","name":"research-evidence.md","bytes":2493},{"sha256":"7d76a49269d3fb781097a3ce97a1f360bbdc5296e90eecc80b262b3d985cdba5","name":"research-prior-art.md","bytes":2532},{"sha256":"fbfb5c9cbfacc90d68e0f795f40d6bb665d8d1deec945051f34265bfe7d8e004","name":"check_go.py","bytes":6174},{"sha256":"4e4d45f8abe9d866e050a9e254f40373f74a0d3a6885b0fbcbcfaa7c4132507f","name":"check_go.out","bytes":1452},{"sha256":"3a042499b36760df729425764b716aeb7d240aee8c1e551c43ce0e4e0df6f0a4","name":"check_go.control.out","bytes":1497},{"sha256":"79ef2f919ccd96dcd89a3d635d52ad4c042edea73df316f78d40891c50bb9716","name":"fetch_go.py","bytes":1048},{"sha256":"1be065490b6f85923f037a3e527a49fe7214753b873ff31c0b0b8f673cd8a14d","name":"fetch_files_go.py","bytes":1460},{"sha256":"e03e7cb1db2aab697b73211be7574774ffa6b2abd4b4f595850a3f8bf3dcbde4","name":"fetch_raw_go.py","bytes":1878},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","name":"export_transcript.py","bytes":10230}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}