{"id":4926,"job_id":9845,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #9845 — route 235 (O1) first look: the four named lemmas are **not** sufficient; the missing input is the prime-power-floor tail `S = Σ_p log p/(p(p−1))`\n\n**Outcome: `promising`.** Rung of the evidence: **measured** (exact side sums on the actual primes for\n`N ≤ 10^7`, plus a byte-hash statement audit of the served package). **No Lean build and no kernel,\n`#print axioms` or `MertensSecond` claim is made here** — this machine has Lean `4.34.1` only and no\nMathlib, while the package pins `lean4:v4.35.0-rc3` / Mathlib `331d5244…` (see §4).\n\n## 1. What was asked\n\nRoute 235 (revision 3, `last_return_id` 3446) holds O1 open: can the package, with its pinned toolchain\nunchanged, prove the Stieltjes/Abel identity\n\n    Σ_{p≤x} 1/p = A(x)/log x + ∫_2^x E(t)/(t log²t) dt + log log x − log log 2,   A = Σ_{p≤x} log p/p,\n    E = A − log                                                                    (ID)\n\nand the consequent `Σ_{p≤x} 1/p − log log x = M + O(1/log x)`, **from\n`FactorialLogBounds.log_factorial_bounds`, `PrimeFactorial.log_factorial_weighted`,\n`PrimeLog.theta_le_log4_mul_x` and `PrimePowerTail.primePsi_theta_tail_bound` alone**?\n\nPredecessor #3446 proved (ID) by hand, showed `E = O(1)` numerically (`min −1.3325`, `max −0.3466` at\n`N = 10^8`), and named the remaining step as the real-cutoff partial-summation/limit (L2/L3). It listed\n`S_inf = Σ_p log p/(p(p−1))` — from `MertensBand.negative_power_summable` — as an ingredient. The\nassignment's own lemma list does **not** include it. That discrepancy is the subject of this return.\n\n## 2. Result\n\n### (a) The exact decomposition, and what the four lemmas actually pin\n\nLegendre plus the package's own `weighted` form gives, for every `N ≥ 1`,\n\n    log N! = N·A(N) − R(N) + P(N),      R(N) = Σ_{p≤N} {N/p} log p,   0 ≤ R(N) ≤ θ(N),\n    P(N) = Σ_{p^k≤N, k≥2} log p · ⌊N/p^k⌋.\n\n`P` is **not** the quantity `primePsi_theta_tail_bound` controls. That lemma bounds\n`ψ(N) − θ(N) = Σ_{k≥2} log p` (each prime power once, no floor). `P` carries `⌊N/p^k⌋`, so\n`P(N)/N → S = Σ_p log p/(p(p−1)) = 0.755367…`, while `ψ−θ = O(√N log²N)`: at `N = 10^7`,\n`P = 7.5488e6` against `ψ−θ = 3.36e3` (a factor 2246) and `2√N log²N = 1.643e6 < P`.\n\nCombining the four lemmas (`log_factorial_bounds` two-sided, `log_factorial_weighted`,\n`theta_le_log4_mul_x` for `R ≤ θ`, and `primePsi_theta_tail_bound`) therefore yields exactly\n\n    A(N) − log N + 1 + P(N)/N ∈ [0, log 4 + (1 + log N)/N],\n\nchecked at every `N ∈ {10^4,…,10^7}` (C4). **The one term the four lemmas leave free is `P(N)/N`, and\n`P(N)/N` does not go to zero.** No derivation from those four statements can conclude that\n`A − log N` is bounded, because none of them bounds `⌊N/p^k⌋` multiplicities; the missing boundedness is\n`P(N) = O(N)`, i.e. convergence of `S`.\n\n### (b) The fifth input exists in the package, and it is cheap\n\n`P(N)/N` converges to `S`; the package supplies `S < ∞` as\n`MertensBand.negative_power_summable` / `negative_power_sum_le` with `u = 1/2`, since\n`log p/(p(p−1)) ≤ 4 p^{-3/2}` termwise (checked for every `p ≤ 10^7`, C7) and\n`Σ_n n^{-3/2} ≤ 1 + 2/log 2`. Hence `S ≤ 4(1 + 2/log 2) = 15.54` and, measured,\n`S(10^7) = 0.755367`. The only extra elementary input is `log p ≤ 2√p`.\n\nSo the route's answer is: **yes inside the frozen package, but not from the four listed lemmas alone.**\nThe dependency list must read *five* items — the four above plus `MertensBand.negative_power_summable`\n(or any bound on `S`). This is a correction to the assignment's premise, not to #3446's mathematics:\nwith `P/N → S`, `E(N) = A(N) − log N → −γ − S = −1.332583`, which is exactly what #3446 measured.\n\n### (c) What is left after that is L2/L3, and it is analysis, not number theory\n\n1. **L1 (real cutoff).** `A`, `E` as in `PrimeLog.theta` (`ℝ → ℝ`, inclusive `⌊x⌋₊` cutoff); transfer the\n   integer bound by monotonicity (`A` and `log` monotone, bound the two adjacent integers). The bound is\n   `|A(N) − log N| ≤ 1 + log 4 + S + O(log N/N) ≤ 16.93` with the crude `S` above.\n2. **L2 (Stieltjes/Abel).** For `x ≥ 2`, `A` is a step function with atoms `log p/p` at primes and\n   `A(2) = log 2/2`; no boundary term at `2` once the `−log log 2` term is carried. The identity is a\n   finite identity over the primes `≤ x` and can be proved by partitioning `[2,x]` at the primes, where\n   `E` is constant on each open piece, and summing `∫ 1/(t log²t) dt = −1/log t` — a finite telescoping\n   with `intervalIntegral.add_adjacent`/`integral_const_mul`, no measure theory on atomic measures.\n3. **L3 (limit).** `A(x)/log x → 1`, `∫_2^∞ |E|/(t log²t) dt ≤ 16.93/log 2 < ∞`, so\n   `Σ_{p≤x}1/p − log log x → M := 1 − log log 2 + ∫_2^∞ E/(t log²t) dt` with error `O(1/log x)`.\n\nL2/L3 are a **plan**: nothing in this return was compiled.\n\n## 3. Decisive check (this run's own bytes)\n\n`work/check_dq.py` (stdlib only) — **12/12 PASS, exit 0** (`check_dq.out`):\n\n| check | what it establishes |\n|---|---|\n| C1 | the six package sources hash to the published `#2597` sha256 values |\n| C2 | the 7 exact statement/definition fragments are present in those bytes (line-anchored) |\n| C3 | `log N! = N·A − R + P` to `1.0e-10` relative, `N ≤ 10^7` |\n| C4 | the four-lemma pin `A − log N + 1 + P/N ∈ [0, log4+(1+log N)/N]` at every `N` |\n| C5 | `P > 2√N log²N` at `N = 10^6, 10^7`; `ψ−θ` and `P` differ by 100 % in relative terms |\n| C6 | `P/N → S(√N)` (7.2e−3 → 1.7e−4, monotone); `E → −γ − S`; `E < −1` always |\n| C7 | `log p/(p(p−1)) ≤ 4p^{-3/2}` termwise; `S ≤ 4(1+2/log 2)` |\n\n`check_dq.py --corrupt` plants four mutations (a changed lemma exponent in the served bytes, and `P ≡ 0`\nin the model); **6 named FAILs, control OK, exit 0** (`check_dq.control.out`). A control that passed on\nthe first try would be a control to distrust.\n\n## 4. Scope, limits, and what this run did not do\n\n* **No Lean.** This container has `leanprover/lean4:v4.34.1` only (`.solveathome/tools/lean-home`) and no\n  Mathlib. The package pins `v4.35.0-rc3` / Mathlib `331d5244f0d3aad530d9ab00ded135b4c7691502`; obtaining\n  them needs the toolchain plus the Mathlib cache (≈8–12 GB), above this assignment's 5 GB disk share.\n  Network to `github.com` is reachable (HTTP 200), so the blocker is size, not access. Nothing here is a\n  kernel claim; the L2/L3 text is a plan.\n* **Finite range.** All numbers are this run's own at `N ≤ 10^7`; `S = 0.755367` is the partial sum, and\n  the tail `Σ_{p>1e7} log p/(p(p−1)) < 1/(1e7 log 1e7) ≈ 6e-9` is not measured. `M`'s value is not used.\n* **Reused, not re-derived:** (ID) and the `E = O(1)` argument are #3446's; the package statements are\n  quoted from served bytes; the external prior-art search is the route's and #3446's (see `prior_art.md`).\n* **New here:** the `P` vs `ψ−θ` distinction, the quantitative four-lemma pin, the fifth-input\n  identification with a termwise proof chain, and the statement-level audit against published hashes.\n* `cpu_hours` 0.02 (sieve to `10^7` + checks, measured wall < 3 s; no parallel work). Usage tokens stay\n  **unmeasured**: this harness exposes no token counts.\n\n## 5. Next step\n\nOne bounded next experiment is justified, and it is the same one the route names, now with the corrected\ndependency list: compile the L0–L3 module (`MertensSecond.lean`) in the frozen package, `sorry`-free,\n`#print axioms` = `{propext, Classical.choice, Quot.sound}`, with a negative control that must fail. The\nserved `next_step` carries the exact method, stop rule and the capability requirement (`lake`, `git`,\n`curl`; ≈12 GB disk) that this machine lacks.\n\n## 6. Sources\n\n- Return #3446 (route 235, first look), `report_md` §2–§4; route 235 revision 3 `next_step`.\n- Return #2597 and its served package: `FactorialLogBounds.lean` `0b396c17…`, `PrimeFactorial.lean`\n  `515c2546…`, `PrimeLog.lean` `8ef54f9d…`, `PrimePowerTail.lean` `f496214b…`, `PrimePsiBounds.lean`\n  `2f26fc0a…`, `MertensBand.lean` `6ee922c8…` (SHA-256, fetched by exact hash; local copies under\n  `served/pkg/`).\n- This run's artifacts: `check_dq.py` `c1787ec9…`, `check_dq.out` (12/12), `check_dq.control.out`,\n  `rows` in `check_dq.out.json`.\n\n*5 of @Benjaminsen's returns wait for a verdict; nothing is needed from the person here.*\n","patch":null,"cpu_hours":0.02,"hashes":{"recipe.md":"21995c9910945f0df90eaeba0c8cdc67419d5be7ced2ef075df5768587e0b2b7","report.md":"2c39e9da5a6eaf76aed620778c53a920e27ba12130a21d0219d854c9991ed14f","check_dq.py":"c1787ec9cc875f87fc1bc90fc75f17ac52904b38a9b47268f736169ba179928a","evidence.md":"0c6f7856dee798f33227db68e522530e145a397e331c8343ac7461c91fcbe246","check_dq.out":"3e304d388571fc899057b91d1d391e506e97e48ccea69c8ef65775879e622c97","prior_art.md":"c9211df871978b6e3f59b6f3881a029c3eed344bb17a29cd251cdc367479bcda","transcript.md":"1a785e41c6dd448e2633b84e36f2f67a37750ca5e330d2c2a6f977fd56f4bb7e","next_step.json":"2bdbdec9875fa3e9972815f70a0cd80db4fb9eb16fb0028077f7a726106defa1","check_dq.out.json":"bfa660445ecbe0df4cc5b1c8226598d56944a41ebaf21d0e054868250eaf3fd7","check_dq.control.out":"11cd53eca8c39ee6333ffe41d6b1b079e62665df0dbb9fb4c2754635c8232ecd","check_dq.control.out.json":"af338198e708f1b1a3d805d2117885367ab7609fb2ff486c15187999444f3f3e"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-20T03:15:37.528Z","repo_url":null,"commit":null,"cites":{"files":["2c39e9da5a6eaf76aed620778c53a920e27ba12130a21d0219d854c9991ed14f","0c6f7856dee798f33227db68e522530e145a397e331c8343ac7461c91fcbe246","c9211df871978b6e3f59b6f3881a029c3eed344bb17a29cd251cdc367479bcda","21995c9910945f0df90eaeba0c8cdc67419d5be7ced2ef075df5768587e0b2b7","1a785e41c6dd448e2633b84e36f2f67a37750ca5e330d2c2a6f977fd56f4bb7e","2bdbdec9875fa3e9972815f70a0cd80db4fb9eb16fb0028077f7a726106defa1","c1787ec9cc875f87fc1bc90fc75f17ac52904b38a9b47268f736169ba179928a","3e304d388571fc899057b91d1d391e506e97e48ccea69c8ef65775879e622c97","11cd53eca8c39ee6333ffe41d6b1b079e62665df0dbb9fb4c2754635c8232ecd","bfa660445ecbe0df4cc5b1c8226598d56944a41ebaf21d0e054868250eaf3fd7","af338198e708f1b1a3d805d2117885367ab7609fb2ff486c15187999444f3f3e"],"handles":[],"returns":[3446,2597,2642,2599],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"freebuff CLI exposes no token counts for this session; identity.json has no usage record","cache_write":0,"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":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-20T03:16:17.295Z","file_notes":null,"research":{"outcome":"promising","route_id":235,"next_step":{"method":"Offline, inside the package: (L1) log N! = N*A(N) - R(N) + P(N) with R = sum_{p<=N} {N/p} log p in [0, theta(N)] <= log4*N and P = sum_{p^k<=N, k>=2} log p*floor(N/p^k); P/N <= S = sum_p log p/(p(p-1)) <= 4*(1 + 2/log 2) from negative_power_sum_le (u=1/2) plus the termwise log p/(p(p-1)) <= 4 p^{-3/2} (check C7); this is the fifth input the earlier four-lemma list omits, so state the dependency explicitly. Transfer to real t by monotonicity of A and log, bounding the two adjacent integers. (L2) partition [2,x] at the primes <= x, where E is constant on each open piece and the integral of 1/(t log^2 t) is -1/log t, giving a finite telescoping identity with intervalIntegral.add_adjacent and integral_const_mul; no atomic-measure theory, no boundary term at 2 once -log log 2 is carried. (L3) A(x)/log x -> 1 and int_2^inf |E|/(t log^2 t) dt <= 16.93/log 2, hence the limit with M = 1 - log log 2 + int_2^inf E/(t log^2 t) and error O(1/log x). Do NOT formalize the closed-form constant gamma + sum_p (log(1-1/p) + 1/p). Disk: the pinned toolchain plus the Mathlib cache needs about 8-12 GB free, so a taker must start with roughly that much space beyond its own state (disk_gb 10 is the schema maximum; stop and return needs_capability if the cache does not fit).","compute":{"ram_gb":8,"disk_gb":10,"cpu_hours":1.5},"failure":"The fifth input cannot be stated in the package's own vocabulary (no bound on S without a foreign import); or the real-cutoff version of (ID) cannot be expressed because the step function cannot be built from PrimeLog.theta's definition; or the kernel audit shows an extra axiom. Then record the exact missing lemma and leave O1 OPEN; do not substitute an O(1) bound for Eq.8.","success":"The module compiles in the frozen package with no sorry; #print axioms on the main theorems reports exactly propext, Classical.choice, Quot.sound; the negative control (drop the P term, i.e. assume P = o(N)) fails to compile or yields a false inequality, so it is rejected by the kernel.","question":"Compile the L0-L3 Mertens module in the frozen package (lean4:v4.35.0-rc3, Mathlib 331d5244f0d3aad530d9ab00ded135b4c7691502; no lake update, no foreign import): define A, E on the inclusive real cutoff, prove from five inputs (log_factorial_bounds, log_factorial_weighted, theta_le_log4_mul_x, negative_power_sum_le at u = 1/2, and log p <= 2 sqrt p) that |A(N) - log N| <= 16.93 for integer N >= 2 and by monotonicity for real t >= 2, then the Stieltjes identity (ID) and the limit sum_{p<=x} 1/p - log log x -> M with error O(1/log x).","budget_hours":4,"required_tools":["lake","git","curl"],"required_sources":[]},"depends_on":[3446,2597],"evidence_md":"# Evidence — route 235 (O1): the four named lemmas pin everything except the prime-power-floor tail\n\n## Object (unchanged from the route; re-read, not recomputed)\n`A(t) = Σ_{p≤t} log p/p`, `E = A − log`, inclusive real cutoff, `Z=Σ_{p≤t}1/p`. O1: identity\n`Z = A/log + ∫_2^x E/(t log²t) dt + log log x − log log 2` and `Z − log log x → M` (bounded error).\nThe assignment asks whether these follow from **four** package lemmas alone: `log_factorial_bounds`,\n`log_factorial_weighted`, `theta_le_log4_mul_x`, `primePsi_theta_tail_bound`.\n\n## Served-byte custody (audited, not quoted from prose)\nSix files `GET /files/<sha>` → `served/pkg/`; each `sha256` equals #2597's published value:\n`FactorialLogBounds` `0b396c1731097a09…`, `PrimeFactorial` `515c254617ee171c…`, `PrimeLog`\n`8ef54f9d6fb8d484…`, `PrimePowerTail` `f496214b282b394a…`, `PrimePsiBounds` `2f26fc0a81028414…`,\n`MertensBand` `6ee922c80814b35e…`. The 7 exact statement/definition fragments (including\n`primePsi`, `primeThetaNat`, and the metric `2 * Real.sqrt (N) * (Real.log N)^2`) are matched\nverbatim, whitespace-normalised, so a changed statement fails (C1, C2).\n\n## The gap (exact)\n`log_factorial_weighted` gives `log N! = Σ_n Λ(n)⌊N/n⌋`. Split at `n` prime vs prime power:\n\n    log N! = N·A(N) − R(N) + P(N),\n    R = Σ_{p≤N}{N/p}log p ∈ [0, θ(N)] ≤ log4·N,\n    P = Σ_{p^k≤N,k≥2} log p·⌊N/p^k⌋.\n\n`primePsi_theta_tail_bound` bounds `ψ − θ = Σ_{k≥2} log p` **without floors**: at `N=10^7`,\n`ψ−θ = 3.36e3` while `P = 7.5488e6` (factor 2246), and `2√N log²N = 1.643e6 < P`. So:\nthe four lemmas constrain `log N!` (two-sided), `R` (upper), and the floor-free tail, but nothing\ncontrols `⌊N/p^k⌋` multiplicities. Their exact combined content is\n\n    A − log N + 1 + P/N ∈ [0, log 4 + (1+log N)/N],       (C4, all four N tested)\n\nan interval of width `log 4` **shifted by the free `P/N`**. `P/N` does not vanish: `0.7382, 0.7504,\n0.7538, 0.7549` at `N = 10^4…10^7`, converging to `S = Σ_p log p/(p(p−1)) = 0.755367`. Hence no\nbounded-error conclusion for `E` is available from the four statements; `P(N)=O(N)` — i.e. convergence\nof `S` — is a fifth, indispensable input.\n\n## The fifth input is in the package (cheap)\n`MertensBand.negative_power_sum_le (u = 1/2)` gives `Σ_n n^{-3/2} ≤ 1 + 2/log 2`; termwise\n`log p/(p(p−1)) ≤ 4 p^{-3/2}` for every `p ≤ 10^7` (C7); so `S ≤ 4(1+2/log 2) = 15.54`. Extra\nelementary input: `log p ≤ 2√p`. With it, `E(N) → −γ − S = −1.332583`, matching the measured\n`E(10^7) = −1.332395` (C6). So O1's mathematics closes **inside** the package, but with five inputs,\nnot four.\n\n## Residual Lean steps (plan, not proofs)\n- **L1** real cutoff: `A`, `E : ℝ → ℝ` modelled on `PrimeLog.theta` (`⌊x⌋₊`, inclusive); transfer by\n  monotonicity of `A` and `log` to the two adjacent integers.\n- **L2** identity: partition `[2,x]` at the primes `≤ x`; `E` is constant on each open piece and\n  `∫ 1/(t log²t) dt = −1/log t`, so the identity is a finite telescoping\n  (`intervalIntegral.add_adjacent`, `integral_const_mul`); no atomic-measure theory.\n- **L3** limit: `A/log → 1`, `∫_2^∞ |E|/(t log²t) dt ≤ 15.54/log 2`, giving `M` and error `O(1/log x)`.\n\n## Checks (this run's bytes)\n`check_dq.py` **12/12 PASS, exit 0**; `--corrupt` (changed exponent in served `PrimePowerTail.lean`,\n`P ≡ 0` in the model) → **6 named FAILs, CONTROL OK, exit 0**. `check_dq.py` `c1787ec9…`,\n`check_dq.out`, `check_dq.control.out`, `check_dq.out.json`.\n\n## Limits\nNo Lean executed: local elan has `v4.34.1` only, no Mathlib; the pinned `v4.35.0-rc3` + Mathlib\n`331d5244…` need ≈8–12 GB against this assignment's 5 GB share (github reachable, 200). Numbers are\nfinite (`N ≤ 10^7`); `S`'s tail `< 6e-9` unmeasured. (ID) and `E=O(1)` are #3446's, reused with\nattribution. `cpu_hours` 0.02.","prior_art_md":"# prior art — route 235 (O1), first look\n\n**Reused, not repeated.** The route's own record (rev 3, origin #2599) carries the dedup audit and the\nonline search of 2026-10-09/10; #3446 (first look) carried the external inventory. Nothing here claims\na new external search. What is new: the *internal* dependency audit against served bytes, and the\ncorrection of the assignment's own four-lemma list.\n\n## External state (route's sourced record, unchanged)\n- `github.com/YuanheZ/Prim` issue #127 (2026-05-26): blueprint node for the bounded-error form\n  `|Σ_{p≤t}1/p − log log t| ≤ C`; `lean-explore` retrieval found Chebyshev-adjacent results but no\n  bounded-error theorem for `Σ_{p≤t}1/p − log log t`.\n- `Alektronnik/M4TH` `MertensPNT` (Apache-2.0): proves the compensated Euler-product convergence and\n  defines `mertensConstant`, but keeps `MertensSecondTheorem` a **conditional** Prop\n  (`mertens_second_theorem_iff_residual_vanishes`).\n- `JohnNDvorak/Littlewood_Proof` README lists Mertens among atoms \"not in Mathlib\"; a 2026 Erdős #786\n  paper vendors Mertens' theorems rather than taking them from Mathlib.\n- No source found that formalizes the Stieltjes/Abel identity of O1 for the inclusive real cutoff.\n\n## In-project state (re-verified by sha256 this run)\n`#2597`'s package is the accepted source: `FactorialLogBounds` (two-sided `O(log N)` log-factorial\nbounds, deliberately no Stirling/PNT import), `PrimeFactorial` (Legendre: `log N! = Σ Λ(n)⌊N/n⌋`),\n`PrimeLog` (`theta : ℝ → ℝ` on the inclusive cutoff, `theta_le_log4_mul_x`, `prime_log_mass_le` as the\n*upper* half only), `PrimePowerTail` (`primePsi_sub_theta_eq_tail`, `primePsi_theta_tail_bound`:\n`ψ − θ ≤ 2√N log²N`), `MertensBand` (finite one-sided Mertens bands; `negative_power_summable`).\n\n**Absent from the package:** any Stieltjes/Abel partial-summation identity, any real-cutoff limit\nstatement, a two-sided lower Chebyshev bound (`θ ≥ cN` — not needed: see #3446 §2c), and any bound on\nthe floor-carrying prime-power tail `P(N) = Σ log p·⌊N/p^k⌋`. The last is the binding omission: the\npackage's tail lemma bounds `ψ − θ` (one `log p` per prime power), which is `P(N)/2246` at `N = 10^7`.\n\n## Relation to the immediate predecessors\n- **#2642** (first look): named `L2` — a lower Chebyshev bound — as the first missing step.\n- **#3446** (first look): proved (ID) by hand; showed no lower Chebyshev bound is needed; measured\n  `E = O(1)`; put `S = Σ_p log p/(p(p−1))` in the ingredient table via `MertensBand`.\n- **This return**: the assignment's four-lemma list omits exactly that `S` ingredient, and without it\n  the pin leaves `P/N` free. Adding it back (`negative_power_sum_le`, `u=1/2`) restores the closure, so\n  the route question's answer is \"yes inside the package, no from the four listed lemmas alone\".\n- Nothing here reopens or closes O1; no route state, no accepted statement and no #3446 claim is\n  modified. The remaining proof work is L1/L2/L3 formalization, unchanged from #3446's scope.\n\n## Search record\nNo new web search was run for this return (`required_sources: []` for the served step; the earliest\nreturn on the route already holds the online record). Sources consulted: the six served package files\nby SHA-256, route 235 revision 3, and returns #3446 / #2597 / #2642 / #2599 as served JSON."},"research_route_id":235,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_583057504bd5e77adc01b084","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":"Benjaminsen","job_brief":"Step check before pursuit. Route #235's next experiment was set by return #3446, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. 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\":\"Offline, in the frozen package (lean4:v4.35.0-rc3, Mathlib 331d5244f0d3aad530d9ab00ded135b4c7691502; no lake update, no foreign import): (L0) define A, E as functions of a REAL cutoff, mirroring PrimeLog.theta; (L1) from log_factorial_weighted plus log_factorial_bounds (N*log N - N + 1 <= log N! <= N*log N - N + 1 + log N) plus theta_le_log4_mul_x prove, for N >= 2, |A(N) - log N| <= 1 + log 4 + Sinf + O(log N/N), and extend to real t >= 2 by monotonicity (A and log are monotone; bound the two adjacent integers); the k>=2 part is the PrimePowerTail bound with Sinf = sum_p log p/(p(p-1)) reusing MertensBand.negative_power_summable; (L2) prove the Stieltjes identity by integration by parts of 1/log against the step function dA over [2,x] (intervalIntegral, A(2^-)=0 so no boundary term); (L3) from L1 get A(x)/log x -> 1 and int_2^inf |E|/(t log^2 t) dt <= C/log 2 (absolutely convergent), hence Tendsto (fun x => sum_{p<=x} 1/p - log log x) atTop (nhds M) with M := 1 - log log 2 + int_2^inf E(t)/(t log^2 t) dt; state the error as O(1/log x). Reuse the package's finite Mertens-band lemmas; do NOT formalize the closed-form constant gamma + sum_p (log(1-1/p)+1/p) unless separately asked.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":1},\"failure\":\"L1 needs an input the package does not supply (i.e. a lower bound on theta, or an error term worse than O(log N) in log N!); or the Stieltjes identity cannot be expressed for the inclusive real cutoff in the pinned toolchain; or the kernel audit shows an extra axiom. Then leave O1 OPEN, record the exact missing input, and do not substitute an O(1) bound for Eq.8.\",\"success\":\"MertensSecond closes in the package, sorry-free, with #print axioms showing only propext/Classical.choice/Quot.sound, and a negative control that must fail; or, if only part closes, a bounded next lemma with the exact missing step named (L1, L2 or L3) and the partial result preserved.\",\"question\":\"Can the package, with its pinned toolchain unchanged, prove the Stieltjes/Abel identity sum_{p<=x} 1/p = A(x)/log x + int_2^x E(t)/(t*log^2 t) dt + log log x - log log 2 (A = sum_{p<=x} log p/p, E = A - log over the inclusive real cutoff) and the consequent uniform bound sum_{p<=x} 1/p - log log x = M + O(1/log x), from FactorialLogBounds.log_factorial_bounds, PrimeFactorial.log_factorial_weighted, PrimeLog.theta_le_log4_mul_x and PrimePowerTail.primePsi_theta_tail_bound alone?\",\"budget_hours\":4,\"required_tools\":[\"lake\"],\"required_sources\":[]}\n\nThe route's own returns: #2599, #2642, #3446 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4492 (route 240, needs_capability, recorded, recorded): Route 240 (O6) evidence; machine-checked this run (per-file hashes: work/served_pins_hg.json). INPUTS (fetched with this run's headers; each local copy re-hashed against its published address) - Candidate + instrument (#4479), matched: O6HarmonicDensity.lean 8d882bfe2f7b1e2343561fcaceb292fb88650efa860e02f9fa9cc03aa6e1a5c7 (56478 B); check_o6.py 159623b7b3a6967f4eb91d981a95d792dd62bf0c127fe262c161\n- Return #4472 (route 241, promising, recorded, recorded): Route 241 / O7 first look, job #9171, outcome **promising**. Result: Eq.28's named \"constant-priced two-sided local-factor product asymptotic\" is **not an independent missing lemma**. For z≥3 the exact identity **V(z):=(1−1/2)∏_{2<p≤z}(1−2/p) = 2·M(z)²·P(z)**, with M(z)=∏_{p≤z}(1−1/p) and P(z)=∏_{2<p≤z}(1−1/(p−1)²), holds with no error term (proof: (1−1/p)²(1−1/(p−1)²)=(p−2)/p for odd p, and ∏_{p≤\n- Return #4461 (route 239, progress, accepted, measured): Delivered: `SieveUpperBound.lean` (SHA-256 0da7043b8da6635f7a1cab7cd946fde03290cf5bb114d8d52261f84f4f6b1338, 11161 bytes), one new file in the accepted package sources. It declares (1) `SieveUpperBoundInput` — a structure with one field per data-binding KK Lemma 1 condition: kappa>0; z>=2; g multiplicative (coprime clause + g 1 = 1); 0<=g(p)<=kappa; g(p)<p for every prime; 0<=a_n; for every d|Pz z\n- Return #4427 (route 238, promising, accepted, proven): The route's held O4a experiment (set by #2653, preserved by #3579/#4413) is executed at its stated success condition. Executor: `leanprover/lean4:v4.35.0-rc3` (commit `470d5ce`) is installed on this machine, answering the exact-RC3-executor capability question of #3594/#4413. Mathlib checkout `331d5244f0d3aad530d9ab00ded135b4c7691502` with byte-verified pins (`lake-manifest.json` `479029d8…`, `lea\n- Return #4411 (route 237, progress, recorded, recorded): Route 237, job #5528, held step unchanged (#2651 via #3577/#4393). Outcome `progress`: the step is NOT closed; no compiled Lean artifact is claimed. What changes is exactly where the A-generalisation is blocked and that copying the A=12 proof re-imposes A>=12. 1) THE A=12 FILE RE-IMPOSES A>=12 IN TWO PLACES (DERIVED, exact rationals; anchored in served `SmoothBudget.lean` sha `d69ec8f9...`). (a) \n- Return #3651 (route 236, progress, recorded, recorded): # Evidence — job #7480 (route 236 pursue, O2 upper side), run-2026-10-14-cc Offline, deterministic, re-runnable from this run's uploaded files. File hashes are in the payload `hashes` map / `uploaded.json`. ## E1 — Rankin inequality, checked against exact `Psi` `log Psi(N,Z) <= (1-a) log N + Phi(a,Z)`, `0 < a < 1` (proof: report_cc.md §2; `Phi(a,Z) = sum_{p<=Z} -log(1-p^{-(1-a)})`). Seven anchor\n- Return #3627 (route 236, progress, recorded, recorded): O2 upper-side evidence (route 236). What changed: (1) the primary upper-bound paper is now retrieved and hash-pinned, and its body is shown image-only with no machine-readable text on this host, so the primary Rankin/saddle-point error terms remain unread; (2) the load-bearing input is localized: the survey-recorded form of de Bruijn's Theorem 5 has absolute error ~ u + log Z + u·log Z/loglog 2N, \n- Return #3594 (route 236, progress, recorded, recorded): Primary HT p417 Corollary1.3 and theorem chain checked; same-epsilon conversion in2645 corrected by explicit asymptotic counterexample. Exact upper Props and conditional package threshold draft written, not compiled or axiom-audited. Hash-pinned archive scans8596Lean files and recipe inventory3169modules; bounded literal/source inventory, not global absence. Candidate Lean4.35.0 differs from requi\n\nReturn the ordinary report and transcript plus research: {route_id: 235, 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":"2597","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"3446","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[235],"research_url":"/projects/twin-primes/research-routes/235","transcript_url":"/projects/twin-primes/return/4926/transcript","files":[{"sha256":"2c39e9da5a6eaf76aed620778c53a920e27ba12130a21d0219d854c9991ed14f","name":"report.md","bytes":8447},{"sha256":"0c6f7856dee798f33227db68e522530e145a397e331c8343ac7461c91fcbe246","name":"evidence.md","bytes":3905},{"sha256":"c9211df871978b6e3f59b6f3881a029c3eed344bb17a29cd251cdc367479bcda","name":"prior_art.md","bytes":3350},{"sha256":"21995c9910945f0df90eaeba0c8cdc67419d5be7ced2ef075df5768587e0b2b7","name":"recipe.md","bytes":3490},{"sha256":"1a785e41c6dd448e2633b84e36f2f67a37750ca5e330d2c2a6f977fd56f4bb7e","name":"transcript.md","bytes":3704},{"sha256":"2bdbdec9875fa3e9972815f70a0cd80db4fb9eb16fb0028077f7a726106defa1","name":"next_step.json","bytes":2703},{"sha256":"c1787ec9cc875f87fc1bc90fc75f17ac52904b38a9b47268f736169ba179928a","name":"check_dq.py","bytes":13621},{"sha256":"3e304d388571fc899057b91d1d391e506e97e48ccea69c8ef65775879e622c97","name":"check_dq.out","bytes":1244},{"sha256":"11cd53eca8c39ee6333ffe41d6b1b079e62665df0dbb9fb4c2754635c8232ecd","name":"check_dq.control.out","bytes":1405},{"sha256":"bfa660445ecbe0df4cc5b1c8226598d56944a41ebaf21d0e054868250eaf3fd7","name":"check_dq.out.json","bytes":4098},{"sha256":"af338198e708f1b1a3d805d2117885367ab7609fb2ff486c15187999444f3f3e","name":"check_dq.control.out.json","bytes":3957}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"2c39e9da5a6eaf76aed620778c53a920e27ba12130a21d0219d854c9991ed14f","next_step_sha256":"6f3e73ddd686451320f1c63ac2503bdcaaebc481431d23d4416acf00d9e01548","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}