{"id":5468,"job_id":11034,"problem_id":1,"lane_id":null,"type":"measure","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 235: Mertens' second theorem with error 2(1 + 4 log 4)/log x, kernel-checked in Lean against #2597's frozen package from three of its lemmas; no fifth input needed\n\n## Result\n\nThe Lean kernel checks L1, L2 and L3 in this project. The project imports #2597's package from `research/lean-replay/inputs/` without editing it.\n\n- **L1.** |A(t) − log t| ≤ C1 = 1 + 4 log 4 < 6.55 for every real t ≥ 1. The integer version holds for every N ≥ 1, and the one-sided bounds are −1 − 4 log 4 < E(t) ≤ log 4.\n- **L2.** The finite Abel identity holds on [2, x] for every real x ≥ 2.\n- **L3.** A(x)/log x → 1, and E(t)/(t log²t) is integrable on (2, ∞). For every real x ≥ 2, |S(x) − log log x − M| ≤ 2·C1/log x, so S(x) − log log x → M. Here M = 1 − log log 2 + ∫_2^∞ E(t)/(t log²t) dt. The closed form of M is not formalized.\n\n`lake env lean Axioms.lean` printed the following (copied from `logs/axioms.txt`):\n\n```\n'Mertens.L1_nat' depends on axioms: [propext, Classical.choice, Quot.sound]\n'Mertens.L1_nat_abs' depends on axioms: [propext, Classical.choice, Quot.sound]\n'Mertens.L1_real' depends on axioms: [propext, Classical.choice, Quot.sound]\n'Mertens.L1' depends on axioms: [propext, Classical.choice, Quot.sound]\n'Mertens.C1_lt' depends on axioms: [propext, Classical.choice, Quot.sound]\n'Mertens.L2' depends on axioms: [propext, Classical.choice, Quot.sound]\n'Mertens.L3_A_div_log' depends on axioms: [propext, Classical.choice, Quot.sound]\n'Mertens.integrableOn_integrand' depends on axioms: [propext, Classical.choice, Quot.sound]\n'Mertens.L3_error' depends on axioms: [propext, Classical.choice, Quot.sound]\n'Mertens.L3_tendsto' depends on axioms: [propext, Classical.choice, Quot.sound]\n'Mertens.L3' depends on axioms: [propext, Classical.choice, Quot.sound]\nexit=0\n```\n\n## Definitions\n\nEvery prime sum uses the inclusive real cutoff of `PrimeLog.theta`, so `PrimeLog.theta x = primeSum (fun p => Real.log p) x` holds by `rfl` (`Mertens/Basic.lean`).\n\n```lean\ndef primeSum (f : ℕ → ℝ) (x : ℝ) : ℝ := ∑ p ∈ Ioc 0 ⌊x⌋₊ with p.Prime, f p\ndef A (x : ℝ) : ℝ := primeSum (fun p => Real.log p / p) x\ndef E (x : ℝ) : ℝ := A x - Real.log x\ndef S (x : ℝ) : ℝ := primeSum (fun p => 1 / (p : ℝ)) x\ndef C1 : ℝ := 1 + 4 * Real.log 4\ndef M : ℝ := 1 - Real.log (Real.log 2) + ∫ t in Set.Ioi 2, E t / (t * Real.log t ^ 2)\n```\n\n## Theorem statements\n\n`Mertens/L1.lean`:\n\n```lean\ntheorem L1_nat (N : ℕ) (hN : 1 ≤ N) :\n    -1 - 4 * Real.log 4 < A N - Real.log N ∧ A N - Real.log N ≤ Real.log 4\ntheorem L1_real (t : ℝ) (ht : 1 ≤ t) : -1 - 4 * Real.log 4 < E t ∧ E t ≤ Real.log 4\ntheorem C1_lt : C1 < 6.55\ntheorem L1 (t : ℝ) (ht : 1 ≤ t) : |E t| ≤ C1\ntheorem L1_nat_abs (N : ℕ) (hN : 1 ≤ N) : |A N - Real.log N| ≤ C1\n```\n\n`Mertens/L2.lean`:\n\n```lean\ntheorem L2 (x : ℝ) (hx : 2 ≤ x) :\n    S x = A x / Real.log x + (∫ t in (2 : ℝ)..x, E t / (t * Real.log t ^ 2)) +\n      Real.log (Real.log x) - Real.log (Real.log 2)\n```\n\n`Mertens/L3.lean`:\n\n```lean\ntheorem integrableOn_integrand {a : ℝ} (ha : 2 ≤ a) :\n    IntegrableOn (fun t => E t / (t * Real.log t ^ 2)) (Set.Ioi a)\ntheorem L3_error (x : ℝ) (hx : 2 ≤ x) :\n    |S x - Real.log (Real.log x) - M| ≤ 2 * C1 / Real.log x\ntheorem L3_A_div_log : Tendsto (fun x => A x / Real.log x) atTop (𝓝 1)\ntheorem L3_tendsto : Tendsto (fun x => S x - Real.log (Real.log x)) atTop (𝓝 M)\ntheorem L3 :\n    Tendsto (fun x => A x / Real.log x) atTop (𝓝 1) ∧\n      IntegrableOn (fun t => E t / (t * Real.log t ^ 2)) (Set.Ioi 2) ∧\n      (∀ x : ℝ, 2 ≤ x → |S x - Real.log (Real.log x) - M| ≤ 2 * C1 / Real.log x) ∧\n      Tendsto (fun x => S x - Real.log (Real.log x)) atTop (𝓝 M)\n```\n\n## Proof route\n\n- **L1, upper half.** `PrimeFactorial.log_factorial_weighted` writes log N! as a von Mangoldt sum. The primes form a sub-sum, and N/p < ⌊N/p⌋ + 1, so N·A(N) ≤ log N! + θ(N). `FactorialLogBounds.log_factorial_bounds` and `PrimeLog.theta_le_log4_mul_x` then give A(N) − log N ≤ log 4.\n- **L1, lower half.** The proof reindexes the von Mangoldt sum over pairs (p, k) and bounds each prime's tail by a geometric series. The primes are then grouped by dyadic block, using `Nat.log 2 p`, which gives ∑_{p ≤ N} log p/(p(p−1)) ≤ 4 log 4 (`dyadic_sum_le`). Together these give log N! ≤ N·(A(N) + 4 log 4), and the lower bound of `log_factorial_bounds` gives −1 − 4 log 4 + 1/N ≤ A(N) − log N.\n- **L1, real t.** A(t) = A(⌊t⌋), and log t − log⌊t⌋ < 1/⌊t⌋ absorbs the 1/N slack.\n- **L2.** The proof is Abel summation by induction over the integer pieces [n, n+1]. On each piece, A agrees with A(n) except at the endpoint n + 1, a null set. The antiderivative of 1/(t log²t) is −1/log t, and that of 1/(t log t) is log log t.\n- **L3.** L1 bounds |E(t)|/(t log²t) by C1/(t log²t). The improper integral of the weight from a to ∞ equals 1/log a. Splitting ∫_{(2,∞)} at x and applying L2 gives the error bound. Both limits then follow by squeezing against C/log x.\n\nOnly three package lemmas are used: `PrimeLog.theta_le_log4_mul_x`, `PrimeFactorial.log_factorial_weighted` and `FactorialLogBounds.log_factorial_bounds`. The fourth, `PrimePowerTail.primePsi_theta_tail_bound`, is not used, and no Mertens module imports `PrimePowerTail`. It is still compiled, because the `KkPackage` library lists all 38 package modules as roots.\n\n## Build\n\nFrom a clean checkout, in `research/lean-mertens/`:\n\n```sh\nexport ELAN_HOME=\"$PWD/.elan\" PATH=\"$PWD/.elan/bin:$PATH\" LEAN_NUM_THREADS=4\ncurl -sSfL https://raw.githubusercontent.com/leanprover/elan/v4.2.4/elan-init.sh -o elan-init.sh\necho 'a620ff1641616222c8d37c54845492004bb84d6877cdbc944dd65c1aa685bf53  elan-init.sh' | shasum -a 256 -c\nsh elan-init.sh -y --no-modify-path --default-toolchain none\nelan toolchain install leanprover/lean4:v4.35.0-rc3\nnice -n 10 lake exe cache get\nnice -n 10 lake build\nnice -n 10 lake env lean Axioms.lean\n```\n\n`lakefile.toml` has two libraries. `KkPackage` compiles the package in place, with `srcDir = \"../lean-replay/inputs\"` and the 38 package modules as roots. `Mertens` holds `Mertens/*.lean`. `.elan/`, `.lake/`, `out/` and `scratch/` are gitignored.\n\n- **Clean-checkout run.** The tree was `git archive HEAD` of `research/lean-mertens` and `research/lean-replay/inputs` at 29bc6d9, unpacked into an empty directory. The build ran from 2026-10-22T23:01:13Z to 23:03:45Z UTC. It printed `Build completed successfully (3580 jobs).` and exited 0. Wall time was 152.45 s, and `/usr/bin/time -l` reported a maximum resident set size of 3,403,923,456 bytes (3.4 GB). `build.log` holds the tail.\n- **Worktree run.** After `.lake/build` was removed, the worktree build took 161.04 s with a maximum RSS of 3,402,645,504 bytes, also 3580 jobs.\n- `lake exe cache get` found the existing Mathlib cache complete (\"No files to download\") and took 1:25.89 wall time.\n- Every build ran under `nice -n 10` with `LEAN_NUM_THREADS=4`, one Lean process at a time.\n\nThe Mertens modules emit linter warnings and no errors. The warnings say that the Mathlib imports are designed for the module system, and that one `simp` argument (`Nat.lt_succ_iff`) in the package file `PrimePowerTail.lean` is unused.\n\n## Pins\n\nPackage files, sha256, unchanged from r4926:\n\n| File | sha256 |\n|---|---|\n| `PrimeLog.lean` | `8ef54f9d6fb8d484e2f9822ccd5ad063d93cf7141e8f1dc8c0bbc6d41022bd90` |\n| `FactorialLogBounds.lean` | `0b396c1731097a09b0a154b3c54c2b102699e880b5c51aae984ae58f88e564ea` |\n| `PrimeFactorial.lean` | `515c254617ee171c49108aa3509c8e7e3b98bdb0a7ff7d4703db97fd9bd8ecc4` |\n| `PrimePowerTail.lean` | `f496214b282b394abb5f0b5ded6c45a06eb17dab92095c56023e0728b5c4f2c8` |\n\nToolchain:\n\n- `leanprover/lean4:v4.35.0-rc3`, `Lean (version 4.35.0-rc3, arm64-apple-darwin24.6.0, commit 470d5ce1400764999581fd26d5d72b00d990b0f4, Release)`.\n- The release tarball has sha256 `967fb9ca8d0c0441c4a72e5d8140208260d0ab6ccf7e7dcf0cc63c54ffbff95c`.\n- The installed `bin/lean` has sha256 `d1d9c26a539f54fe839d2aa7760bf1ac1dd76a30b26e2c9684bebf2900038fcf`, identical to the copy inside the tarball.\n- The installed `bin/lake` has sha256 `d2d63c80ca10bd2cc9557d11d59221b95c6eefb078c9652a11ef887ab8a068a5`.\n- elan 4.2.4 (227caca13 2026-08-25) is installed into `.elan/`, which is gitignored. The `elan-init.sh` used has sha256 `a620ff1641616222c8d37c54845492004bb84d6877cdbc944dd65c1aa685bf53`, the same file as elan's `v4.2.4` tag (checked 2026-10-22).\n- Mathlib commit `331d5244f0d3aad530d9ab00ded135b4c7691502`. `lake-manifest.json` copies `../lean-replay/project/lake-manifest.json` and changes only the package name.\n\n## Additional checks\n\n- **Replay.** `lake env leanchecker -v Mertens.Basic Mertens.L1 Mertens.L2 Mertens.L3 Mertens` replayed all five modules and exited 0 after 57.25 s real (`logs/leanchecker.txt`). `leanchecker` runs the same C++ kernel as the build, so this is a second replay of the environment, not an independent kernel. `leanchecker --fresh` was not run.\n- **Controls.** `controls/run_controls.sh` copies `Mertens/L1.lean`, weakens one statement, and compiles the copy (`logs/controls.txt`). All three edited copies fail to compile:\n  - `drop-P-step4` drops the 4 log 4 term from log N! ≤ N·(A(N) + 4 log 4). Exit 1, 2 errors.\n  - `drop-P-lower` claims −1 + 1/N ≤ A(N) − log N. Exit 1, 3 errors.\n  - `dyadic-zero` replaces 4 log 4 by 0 in `dyadic_sum_le`. Exit 1, 2 errors.\n\n  A rejected control shows only that the given proof does not carry the stronger statement. It does not show the stronger statement is false.\n- **Acceptance greps.** On the committed files, `grep -rnE 'sorry|^axiom|native_decide' research/lean-mertens --include='*.lean'` prints nothing. The brief's grep for home-directory paths and private addresses also prints nothing. Its pattern is not repeated here, because this file would then match it.\n\n## Scope\n\n- L1 is proved for real t ≥ 1, which covers the brief's integer N ≥ 1 and the range t ≥ 2 that L2 and L3 use. The constant is the dyadic route's C1 = 1 + 4 log 4 (`C1_lt : C1 < 6.55`). The upper half is sharper: E(t) ≤ log 4.\n- The L3 error constant 2·C1 comes from this proof: C1/log x from the A(x)/log x term and C1/log x from the tail integral. It is not optimized.\n- M is defined by its integral formula only. Its closed form (Meissel–Mertens, γ + ∑_p (log(1 − 1/p) + 1/p)) is not formalized, and no numerical value of M is claimed.\n- Out of scope and not done: filing, any change to the package, any other route.\n\n## Files\n\n`RESULT.md` (this report), `Mertens/Basic.lean` (definitions), `Mertens/L1.lean`, `Mertens/L2.lean`, `Mertens/L3.lean`, `Mertens.lean` (root), `Axioms.lean` (#print axioms), `lakefile.toml`, `lake-manifest.json`, `lean-toolchain.txt` (the project's `lean-toolchain` file, renamed for upload; restore the name before building), `controls/run_controls.sh` (the three weakened copies of L1), `logs/` (axioms, controls, leanchecker), `build.log` (tail of the clean-checkout build). Flattened names: `Mertens/L1.lean` is `Mertens__L1.lean`. The package itself is #2597's, unchanged, at the sha256 values in Pins; `lakefile.toml` expects it at `../lean-replay/inputs`.","patch":null,"cpu_hours":0.5,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-22T23:11:32.950Z","repo_url":null,"commit":null,"cites":{"returns":[2597,4926,5463]},"tokens":{"log":"claude-code","input":4,"models":{"claude-opus-5-5":535},"output":535,"source":"claude-jsonl","entries":2,"cache_read":279638,"cache_write":1171,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"With #2597's package at `../lean-replay/inputs` and `lean-toolchain.txt` renamed back to `lean-toolchain`: install elan from the v4.2.4 tag (sha256 a620ff16... in RESULT.md), `elan toolchain install leanprover/lean4:v4.35.0-rc3`, `lake exe cache get`, `lake build` (152 s wall, 3.4 GB peak RSS on 4 threads, 3580 jobs), then `lake env lean Axioms.lean`.","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":"2026-10-22T23:11:32.950Z","department_id":"dept_206a3ecdf0ff5c212553942f","run_id":"run_99104b66381542cb4c86eb50","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":"Route 235's sealed step asks to compile the L1-L3 Mertens module inside #2597's frozen package (lean4:v4.35.0-rc3, Mathlib 331d5244, no lake update, no foreign import): L1 bounds A(t) - log t, L2 is the finite Abel identity on [2, x], L3 gives A(x)/log x -> 1 and S(x) - log log x -> M with M = 1 - log log 2 + int_2^inf E(t)/(t log^2 t) dt. Does it compile with no sorry, does #print axioms report exactly propext, Classical.choice, Quot.sound, and does the kernel reject the negative controls?\n\nWhy this step: #2597's accepted package leaves Eq. 8 (the prime-harmonic asymptotic, constant M plus o(1)) as the open obligation O1. A kernel-checked proof against the package's own lemmas closes it, and the lemma count settles whether the fifth input the route listed is needed (#5463 says it is not).\n\nStop when: The return is accepted with the Lean sources, the clean-checkout build log, the #print axioms output, the leanchecker replay and the three rejected controls; or it is refused with a reason that needs new work.","review_deferred":false,"in_triage":true,"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":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/5468/transcript","files":[{"sha256":"a17cd4d37d1166425e75a81a14d5170617c72e9b132acd3753051760e9672eef","name":"Axioms.lean","bytes":355},{"sha256":"84234300d832eb3025819b2d16bbfdef0a220374004913641e67cf106d8db0ff","name":"Mertens.lean","bytes":75},{"sha256":"a43eef9050978d737778077614590303e4bba13258e2e62b8b61f1b2b137a96e","name":"Mertens__Basic.lean","bytes":2536},{"sha256":"432630735d5c31a99a31ae4876d2ff597960ca35cffb1b0e75b125a25b1da513","name":"Mertens__L1.lean","bytes":16108},{"sha256":"251ed19f09fdfb7db3b49538ea56dcd5956a808ac4cd2b496e96ca6d9c843f57","name":"Mertens__L2.lean","bytes":8188},{"sha256":"046c0584657fb90f36dc3b54d576def54e1ae00117a81688645bd2bd66b0610d","name":"Mertens__L3.lean","bytes":6586},{"sha256":"add3da728e13ff66a8cb8e418d62ddaf5d064f791f0ac8e813c160ebaaba27d1","name":"RESULT.md","bytes":10446},{"sha256":"92399274f1238a7230073dd1e31abda5f776113fca6552dbb48c650204254679","name":"build.log","bytes":11768},{"sha256":"c11bc9964ccbcea13881d0b548a38e1b5ad33ab25dbcd4cee4600cdbc4c88d8b","name":"controls__run_controls.sh","bytes":1602},{"sha256":"c3f06a3aea4bfc028c3b3cec275057ee0501c2300181e6916547b07ab032b922","name":"lake-manifest.json","bytes":3232},{"sha256":"fe9455da1e5152322dd21f320e8426cc66a0ca18da8bb7119b9baf7254da9c26","name":"lakefile.toml","bytes":1244},{"sha256":"bc84812c94489d1e3e191baa1dc10d5eb684382d7fe9d2a5ab085e72c1c67e47","name":"lean-toolchain.txt","bytes":29},{"sha256":"d70ab981a5541d0a485b562918b81c26b3bf105ea241e95fffe9aacb7714190e","name":"logs__axioms.txt","bytes":863},{"sha256":"d077371ec8fbdb9857258096b4af852915e8481e20d47d066d218fb0cce8d642","name":"logs__controls.txt","bytes":690},{"sha256":"0de3930e34f085c729e45fa0b2d66b770dc4cef4bc9ed170671daf09b18de793","name":"logs__leanchecker.txt","bytes":144}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"ca4b675cc3698a06571195c5c4623dccfe119429152829d1378e16953bc84c56","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}