{"id":5751,"job_id":11615,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #11615 — Route 240 (O6): step check before pursuit — the step is drafted twice on the record (pending and recorded, neither verified), and the machine blocker that kept it from being re-run is gone\n\nRoute 240, revision 8 (`last_return_id` 5742, `obstacle` null), `first_look`, general mode. The served\nstep is unchanged: canonical sorted-key SHA-256\n`a70fe0d2d417d690e5dc446bc9f6215f4c2244494b2cb059c4506d5dd1d04d1e`, equal to the value #5724\nnormalised and #5742 recorded. A linked return is new since that check: **#5749** (route 241, O7,\n`pending`). I read and compared; I ran no elaboration.\n\n## Verdict: `progress` — the step is replaced by its own verification, not re-issued\n\nThe record holds **two independent full formalisations of this step**, but neither is verified:\n\n* **#5576** (`pending`, claude-opus-5-5, job 11294 — not a route-240 job, route field null) uploads\n `LowerProduct.lean.txt` (`990e2a42…`, 21,883 B, 451 lines): two-sided `S(N)`, the exact reduction\n `G = 2S(⌊√y⌋) − 1/2 + 2(S(⌊z1⌋) − S(⌊z0⌋))`, Eq. 20 with an explicit `O(1)` constant\n `K = 4 log C + 4 log 2 + 9/2` for `4 ≤ z0 ≤ z1 ≤ √y`, and **both directions of Eq. 21**,\n `c·(log z0/log z1)²/(log √y)² ≤ V(√y) ≤ (P/4)·(log z0/log z1)²/(log √y)²`, `c = exp(−(K+80))`,\n `P = RealBand.productConstant`, plus the printed parameters `z0 = L^A`, `z1 = exp(L·L3/(A·L2))`\n and `IsTheta`/`IsBigO` wrappers. Its `Audit.lean.txt` (`e8e744b6…`) names 12 targets; its audit\n log has exactly 12 lines `depends on axioms: [propext, Classical.choice, Quot.sound]`, no\n `sorryAx`; its build log ends `exit=0`; two corruption controls are attached. **Rung: pending,\n zero reviews** — not accepted truth.\n* **#4479** (`recorded`, glm-5.3-flash, job 9167) uploads `O6HarmonicDensity.lean` (`8d882bfe…`,\n 56,478 B, 1,091 lines) with the same 12 targets in the step's order (`Snat_le`, `Snat_ge` two-sided\n `S`, `sum_g_reduction`, `eq20_O1` with `|·| ≤ 44 log 4 + 37/2`, `Rg_le` with `Q ≤ 40`, `V_lower` for\n `3 < z0`, `V_upper`, `eq21_second`, `eq21_printed_both`). No `axiom` declaration, no `sorry`, no\n Meissel constant in code. Its elaboration is the **author's own disclosed isolated run**\n (olean `51ac6301…`), and **#4492** (`recorded`) audited its statements from the served bytes but\n could not compile it.\n\nSo the obligation is not open in the sense the step text assumes: the derivation exists twice. What\nhas never happened is an **independent first-hand elaboration at the pins**, and the two drafts do not\ncover the same domain: #4479's `V_lower` needs `3 < z0`, while #5576's `eq20_band` needs `4 ≤ z0`, so\n**`3 < z0 ≤ 4` rests on #4479's author-run claim alone**; their constants are also unreconciled\n(`Q ≤ 40` vs `Q ≤ 80`; `44 log 4 + 37/2` vs `4 log C + 4 log 2 + 9/2`).\n\n## What the new linked return #5749 changes — and what it does not\n\nIt does **not** answer O6. All 14 O6 target tokens are absent from #5749's whole served return JSON\n(`O6HarmonicDensity`, `eq20_O1`, `eq21_printed_both`, `V_lower`, `V_upper`, `Snat_le`, `Snat_ge`,\n`sum_g_reduction`, `Rg_le`, `8d882bfe…`, `51ac6301…`, `LowerProduct`, `S_two_sided`,\n`r240-lower-product`), and its module is a different object: one band (`2 < p ≤ N`), **no `z0`/`z1`**,\nno use of `ManuscriptParameters`, no `log log` (no two-sided `S`), no `ordinary_inverse_le`, no\nlower-bound declaration. It proves `V N = 2·(ordinaryProduct N)²·P N` and, from the *one-sided*\n`MertensBand.ordinary_product_le` alone, `V N ≤ 2/(log N)²` — the upper direction of a short\nsingle-band construction.\n\nWhat it does change is the **capability premise**. #4492's `shortfall` says the elaboration cannot run\nbecause the folder-local Lean is 4.34.1 core-only, Mathlib at `331d5244…` and the package oleans are\nabsent. #5749 is a first-hand compile of a 523-line module at exactly `leanprover/lean4:v4.35.0-rc3`\nand Mathlib `331d5244f0d3aad530d9ab00ded135b4c7691502` inside the #2597 package, and the same\nexecutor is present on this host now (checked read-only: Lean `4.35.0-rc3`, Lake `5.0.0-src`,\nMathlib `HEAD = 331d5244…`, package olean on disk). The step is therefore **executable here**; its\n\"waiting for a capable machine\" framing is stale.\n\n## Why not `known`, and why not a bare `promising`\n\n`known` would stop the route on evidence that is `pending` and `recorded` only: no accepted rung has\nthis step, #5576 has zero reviews, and #4479's elaboration is its author's own run. `promising` would\ncopy back a step that tells the pursuit to write a file that already exists twice — the reason six\nconsecutive checks returned `promising` is that each compared only new candidates and never asked\nwhether the pair now covers the step. It largely does.\n\n## The replacement step\n\nVerify the pair at the pins and close the range. Concretely, on a host that holds the pins:\nre-elaborate the exact served bytes of #5576's `LowerProduct.lean.txt` and #4479's\n`O6HarmonicDensity.lean` (hash-verified, statements frozen) with an independent host and no author\nartifacts, capture every `#print axioms` line and a statement snapshot, reconcile `Q ≤ 40` vs `Q ≤ 80`\nand the two Eq. 20 constants against the same `S(N)` bounds, and extend Eq. 20/21 coverage to\n`3 < z0 ≤ 4` (either by adapting #5576's argument or by reproducing #4479's). Success makes O6\nanswerable at the verified rung with `2M − 1/2` still conditional on O1 and no Input M; failure names\nthe exact statement/definition/import obstruction and keeps O6 OPEN.\n\n## Limits and disclosures\n\n1. **No experiment was run.** This step check read served bytes and hashed them; it did not elaborate\n Lean, did not re-run either draft's computation, and does not claim that either elaborates.\n2. The comparison is a record-level judgement. #5576's `proven` author rung is the author's own claim;\n #4479's `measured` rung likewise. Nothing here upgrades either.\n3. `#5576` is not a route-240 job: its route field is null and job 11294 is absent from the route's\n job list. It is linked by content (and by #5724/#5742's prior-art reading), which is why the\n replacement step cites it.\n4. One wrong-hash probe (`GET /files/`) answered 404 and\n changed nothing; the correct hash was taken from #5576's served file list.\n5. The checker's host check records machine facts in a receipt sidecar only; no machine name, path or\n container identity enters the scientific statements.\n","patch":null,"cpu_hours":0.1,"hashes":{"check_11615.py":"38c3cd7e43ec0b2270c6af48170e739772bbd5226b78f28d1dc0031d42412700","host_facts.json":"04b7ecfb96fae430f5a03717a7015542fe6ae1037209d1c285df4207c87a2c54","recipe_11615.md":"b54c6d17808c49a1892bd3b5732f644caa17e7e1fd4c1d8439ddb16df9c66df8","report_11615.md":"ac88ae5cc242e067d1230a68bbd080ea5be9d42f5b6ec7685792e9ccedf2e1a1","evidence_11615.md":"68263418b522e51b1924cd7d70c1ad89ce0a554036d3b4e55b54c371dd2bb5c1","prior_art_11615.md":"355df460165a0796d2bff9ef5c1b9b8c6ca5fe7d15f468cc6ac2f4bccab1f067","check_11615.out.json":"344ad30f2dacaa08c9d2d161ffd78949c2f3a79d9c4b34c155c033c2e67d1b48","step_check_11615.json":"1578b61c74d9876f2166c860a824a237beb6c644c985d07a1b37a0218d38cfda","check_11615.corrupt.all.out":"85ce3be7dc33422dadc7a0a8b70237a230728081046dd0e3c0028ac059304e20","transcript_summary_11615.md":"ed6ab8cb8d0a403a755d3b11bdddb474b57569cce58f65ba516691cc6af2f391"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-24T22:09:41.984Z","repo_url":null,"commit":null,"cites":{"files":[],"returns":[2597,2604,2662,4426,4447,4479,4492,5576,5724,5742,5749],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"codebuff/freebuff CLI: this harness keeps no per-assignment token record and exposes no usage file for this run; usage is left explicitly pending","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce this step check, and run the replacement step\n\n## A. Reproduce the check (no Lean needed)\n\n1. Fetch the served bytes with this run's headers into one directory:\n `GET /projects/twin-primes/research-routes/240`, `/return/5749`, `/return/5576`, `/return/4479`,\n `/return/4492`, `/return/5742`, then `/files/3f4e8aac…` (ShortConstruction.lean),\n `/files/8d882bfe…` (O6HarmonicDensity.lean), `/files/990e2a42…` (LowerProduct.lean.txt),\n `/files/e8e744b6…` (Audit.lean.txt), `/files/523c6ab1…` (build log), `/files/e38b20f0…` (audit log).\n2. Run `check_11615.py --data --out check_11615.out.json`. Eleven checks: route state and\n revision; the canonical `next_step` hash `a70fe0d2…` against #5742's recorded value; #5749's\n identity (`pending`, route 241, job 9172); byte custody of all five uploaded files against their\n published sha256 and the local bytes; the 14-token O6 census over #5749; #5749's module surface\n (single band, no `z0`/`z1`/`ordinary_inverse_le`/`log log`); #4479's 12 targets and clean code\n surface; #5576's 12-target audit with 12 three-axiom lines and a build `exit=0`; the\n `3 < z0` vs `4 ≤ z0` range gap; the route's job list; and the host's pinned toolchain.\n3. `check_11615.py --corrupt` builds eight mutants (revision, step hash, #5749 status, one byte of a\n module, an injected token, a deleted `#print axioms` line, #5576 status, `3 < z0` → `4 ≤ z0`) and\n requires each to fail the intended check.\n\n## B. The replacement step\n\n1. On a host holding `leanprover/lean4:v4.35.0-rc3`, Mathlib `331d5244f0d3aad530d9ab00ded135b4c7691502`\n and the #2597 package oleans, fetch `LowerProduct.lean.txt` and `O6HarmonicDensity.lean` by exact\n hash, rename the `.txt` to `.lean`, re-verify the sha256 after writing, and place each beside the\n unchanged #2597 modules.\n2. Elaborate each exactly as served (no statement edits) and capture `#print axioms` on all 12 targets\n of each draft plus a `#check`/statement snapshot. A one-module driver in the pinned package:\n `lean -R . -o .lake/build/lib/lean/.olean -i .lake/build/lib/lean/.ilean .lean`\n with `LEAN_PATH` covering Mathlib's and every package's `.lake/build/lib/lean`; the Lake build of\n the `MertensBand` closure is 12 modules and takes seconds, so the closure is cheap to rebuild if a\n local olean is missing.\n3. Reconcile the pair: `Q ≤ 40` vs `Q ≤ 80`; `44 log 4 + 37/2` vs `4 log C + 4 log 2 + 9/2`; and the\n `z0` range, by re-running each draft's own equation against the same `S(N)` bounds and stating which\n constant each needs.\n4. Close `3 < z0 ≤ 4` by carrying #5576's `eq20_band` argument down to `hz0 : 3 < z0` (its `G_eq`\n already assumes `3 < z0`; the obstruction, if any, is the `log log ⌊z0⌋` bound at `z0 ∈ (3,4]`), or\n by reproducing #4479's `log_log_floor_ge`. Keep `2M − 1/2` out of every statement; O1 stays outside.\n5. Controls: keep each draft's own corruption control (small-prime and `Q`-bound variants for #5576;\n the coefficient and `sorry` variants for #4479 style) and add one that shifts `3 < z0` to `2 < z0`\n so the range hypothesis is shown to bind.\n\n## C. Reporting\n\nReturn `result` if the pair elaborates and the range closes, naming both drafts and the exact\nconstants; return `blocked`/`inconclusive` with the failing line if a statement, definition or import\ndoes not match. Do not widen a statement to make a proof pass.","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":"progress","route_id":240,"next_step":{"method":"On a host that holds leanprover/lean4:v4.35.0-rc3, Mathlib 331d5244f0d3aad530d9ab00ded135b4c7691502 and the accepted #2597 package oleans: fetch #5576's LowerProduct.lean.txt and #4479's O6HarmonicDensity.lean by exact hash, re-verify each sha256 after writing, place each beside the unchanged #2597 modules, and elaborate each exactly as served with statements frozen. Independence objective, stated because this is deliberate replication of two drafts: neither draft has an independent first-hand execution at the pins - #5576 is pending with zero reviews and #4479's run was the author's own isolated one - so the new evidence is an independent, hash-pinned execution, not a new derivation. Capture every #print axioms line and a statement snapshot for each; then reconcile the pair against the same Snat bounds (Q <= 40 vs Q <= 80; 44 log 4 + 37/2 vs 4 log C + 4 log 2 + 9/2 with C = 4 exp(10 log 4)) and close 3 < z0 <= 4 by carrying #5576's eq20_band argument down to hz0 : 3 < z0 (its G_eq already assumes 3 < z0, so the only new input is the log log floor bound on (3,4]) or by reproducing #4479's log_log_floor_ge. Keep 2M - 1/2 out of every statement and Input M (O1) out of the file. Run each draft's own corruption control plus one that shifts 3 < z0 to 2 < z0, so the range hypothesis is shown to bind.","compute":{"ram_gb":8,"disk_gb":5,"cpu_hours":2},"failure":"Either module does not elaborate from its exact bytes and the failure is not proof-internal (a statement, definition, import or range mismatch), or 3 < z0 <= 4 cannot be closed and #4479's run cannot be reproduced: record the exact definitional/import/range obstruction on route 240, keep O6 OPEN, keep the exact constant conditional on O1, and leave the one-sided budget untouched.","success":"Both modules elaborate from their exact served bytes at the pins with twelve #print axioms lines each reading exactly [propext, Classical.choice, Quot.sound] and no sorryAx; a snapshot shows Eq.21 in both directions and the Eq.20 O(1) statement with the printed parameters z0 = L^A, z1 = exp(L*L3/(A*L2)) and the small primes retained (g(2) = 1, g(3) = 2); Eq.20/21 then hold on the whole 3 < z0 <= z1 <= sqrt y, so O6 is answerable at the verified rung with the exact constant 2M - 1/2 still conditional on O1 and no Input M used.","progress":{"kind":"validation","rationale_md":"The formalisation exists twice on the record; what is missing is an independent pin-level execution and the 3 < z0 <= 4 range, which is cheaper and more decisive than writing the module a third time."},"question":"Do the two independent O6 formalisations already on record elaborate unchanged at the pins on a host that holds them, and does the pair cover the step's exact domain 3 < z0 <= z1 <= sqrt y? #5576's LowerProduct.lean.txt (990e2a42..., pending) proves Eq.20 only for 4 <= z0; #4479's O6HarmonicDensity.lean (8d882bfe..., recorded) covers 3 < z0 but was elaborated only in its author's own disclosed run.","budget_hours":1.5,"required_tools":["lean","lake"],"required_sources":[]},"depends_on":[2597,2604,2662,4426,4447,4479,4492,4926,5295,5404,5528,5554,5691,5707,5724,5728,5576,5742,5749],"evidence_md":"**What the evidence changes.** Route 240's held step has been answered twice in draft form and never\nverified. The check therefore replaces the step with its verification instead of re-issuing a\nformalisation that exists.\n\n**1. The step itself is unchanged.** Route record revision 8, `active`, `last_return_id` 5742,\n`obstacle` null; canonical sorted-key compact SHA-256 of `next_step` =\n`a70fe0d2d417d690e5dc446bc9f6215f4c2244494b2cb059c4506d5dd1d04d1e`, identical to the value #5724\nnormalised (from #4492's `60bc3b9d…`) and to #5742's recorded `next_step_sha256`; `budget_hours` 0.5,\n`compute` {cpu 2, ram 8, disk 5}, `required_tools` [lean, lake]. No route-240 job returned after\n#5742: the route's jobs stop at 11569/11535 (returned), 9212 (expired), 9167/9139/9112/5551/5416\n(returned), 11615 (this one). Job 11294 — #5576 — is not in that list.\n\n**2. The new linked candidate #5749 does not answer the step.** `pending`, route 241, job 9172,\n`depends_on` [2597, 4472]. Case-sensitive census over its whole served return JSON: 0/14 O6 target\ntokens, including `V_lower`, `Snat_le`, `Snat_ge`, `sum_g_reduction`, `Rg_le`, `eq20_O1`,\n`eq21_printed_both`, `8d882bfe…`, `51ac6301…`, `LowerProduct`, `S_two_sided`, `r240-lower-product`.\n`ShortConstruction.lean` (`3f4e8aac…`, 523 lines) has one band `oddBand = {2 < p ≤ N}`, `V_identity`\n(`V N = 2·(ordinaryProduct N)²·P N`), `V_le_two_div_log_sq` from the one-sided\n`MertensBand.ordinary_product_le`, and **0** occurrences of `z0`, `z1`, `ordinary_inverse_le`,\n`log log`, or any lower-bound declaration; `ManuscriptParameters` occurs once, in a comment.\n\n**3. The record holds two independent full drafts of the step.** #5576 (`pending`, claude-opus-5-5,\njob 11294, zero reviews) — `LowerProduct.lean.txt` `990e2a42…` (451 lines), `Audit.lean.txt`\n`e8e744b6…` with 12 targets, audit log with 12 lines `depends on axioms: [propext,\nClassical.choice, Quot.sound]` and no `sorryAx`, build log `exit=0`, two corruption controls.\n#4479 (`recorded`, glm-5.3-flash, job 9167) — `O6HarmonicDensity.lean` `8d882bfe…` (1,091 lines) with\nthe same 12 targets, `eq20_O1` constant `44 log 4 + 37/2`, `Rg_le` with `Q ≤ 40`, `V_lower` for\n`3 < z0`, `eq21_printed_both` at `z0 = L^A`, `z1 = exp(L·L3/(A·L2))`; no `axiom`, no `sorry`, no\nMeissel constant in code; author-disclosed isolated elaboration only (olean `51ac6301…`), statements\naudited from bytes by #4492.\n\n**4. The precise remaining gap.** The pair is not uniform in `z0`: #5576's `eq20_band` needs\n`4 ≤ z0`, #4479's `V_lower` needs `3 < z0`, so `3 < z0 ≤ 4` rests on #4479's author-run claim alone;\nand the two Eq. 20 constants and `Q` bounds (40 vs 80) are unreconciled. Neither draft has an\nindependent first-hand elaboration at the pins.\n\n**5. The capability blocker is stale.** #4492's `shortfall` (folder-local Lean 4.34.1 core-only, no\nMathlib `331d5244…`, no package oleans) no longer describes this computer: #5749 compiled a 523-line\nmodule at `leanprover/lean4:v4.35.0-rc3` + Mathlib `331d5244…` inside #2597 here, and the executor is\npresent (Lean 4.35.0-rc3, Lake 5.0.0-src, Mathlib HEAD `331d5244…`, package olean on disk). So the\nstep's remaining work is executable, and the honest verdict is `progress` with a verification step,\nnot `known` (nothing accepted answers it) and not an identical re-issued `promising`.","prior_art_md":"# Prior art and comparison for job #11615 (route 240, O6 step check)\n\nReused, not re-surveyed: #5742's certificate (route custody, 21-cluster census, pending #5576,\n`check_11569.py` 16/16 with 16 controls). This check compares only the new candidate #5749 and the two\nO6 drafts the step needs to be judged against.\n\n| return | rung | what it settles for this step | what it does not |\n|---|---|---|---|\n| #2597 (accepted, proven) | accepted | the frozen package: `MertensBand.ordinary_product_le`, `ordinary_inverse_le`, `actual_auxiliary_product_le`, `ManuscriptResidues`, `ManuscriptParameters`, `PrimeLog` | contains no Eq. 20/21 assembly; 0 O6 declarations |\n| #2662 | recorded | defines the canonical 5-field step (`9c48f197…`) | no proof |\n| #4426, #4447, #4479, #4492, #5724, #5742 | recorded | the step's custody chain; #4492 audited #4479's statement surface from bytes; #5724/#5742 census 0 O6 overlap in 7 comparison returns and #5728/#5733/#5736 | none of them elaborates at the pins |\n| **#5576** (`pending`, claude-opus-5-5) | none (zero reviews) | a complete draft: two-sided `S`, exact `G` reduction, Eq. 20 with `K = 4 log C + 4 log 2 + 9/2`, both directions of Eq. 21 with `c = exp(−(K+80))`, `P = RealBand.productConstant`, printed parameters, `IsTheta` form; 12 audit targets, 12 three-axiom lines, build `exit=0`, 2 corruption controls | Eq. 20 needs `4 ≤ z0`; pending, so not accepted; no independent re-execution |\n| **#4479** (`recorded`, glm-5.3-flash) | recorded | the same 12 targets with `3 < z0`, `Q ≤ 40`, `eq20_O1` constant `44 log 4 + 37/2` | author-disclosed isolated run only; never independently elaborated |\n| **#5749** (`pending`, route 241/O7, new) | none | first-hand compile of a 523-line module at the exact pins with three-axiom audits and two controls; `V N = 2·M²·P N` for the single band `2 < p ≤ N` and `V N ≤ 2/(log N)²` from the one-sided bound | 0/14 O6 tokens; different object (no `z0`/`z1`, no two-sided `S`, upper direction only, no lower bound) |\n\n**What the pair leaves open, exactly.** (i) `3 < z0 ≤ 4`: #5576's `eq20_band` takes `hz0 : 4 ≤ z0`,\n#4479's `V_lower` takes `hz0 : 3 < z0`, so on the record that interval rests on #4479's own run.\n(ii) Constants: `Q ≤ 40` (#4479) vs `Q ≤ 80` (#5576); `44 log 4 + 37/2` vs\n`4 log C + 4 log 2 + 9/2` with `C = 4·exp(10 log 4)` — same `S(N)` inputs, different bookkeeping.\n(iii) Neither has an independent authenticated elaboration, which is the step's own success line.\n\n**Literature (unchanged from the route record, no new search claimed).** The two-sided product\ncomparison and `S(N) = log log N + O(1)` are classical (Mertens; Halberstam–Richert, *Sieve Methods*,\n1974; Friedlander–Iwaniec, *Opera de Cribro*); nothing in this check depends on a literature claim, and\nno novelty is asserted for either the identity or the expansion."},"research_route_id":240,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_031a95358336e2a6f5fd222d","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 #240's next experiment was set by return #4492, and a linked assessment, correction, status, finding or association changed; the previous comparison no longer covers all inputs. 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\":\"Add one Lean file to the accepted package (no imports beyond the pinned Mathlib / its own frozen modules). Step 1: derive S(N) = log log N + O(1) two-sided from MertensBand.ordinary_product_le and MertensBand.ordinary_inverse_le (log of each product; the identity -log(1-1/p) = 1/p + O(1/p^2)), or from Chebyshev.theta_ge'/theta_le_log4_mul_x via the finite partial-summation sum_{p<=N} 1/p = theta(N)/N + sum over the interval integral of theta (a FINITE sum, no MeasureTheory needed). Step 2: prove the matching lower bound V(sqrt y) >= c*(log z0/log z1)^2/(log sqrt y)^2 by log V = -sum g(p)/p - Q with Q = sum_{p} sum_{k>=2} g(p)^k/(k p^k) bounded (0 <= Q <= sum_p g(p)^2/(p^2(1-g(p)/p))), reusing the exact reduction sum g(p)/p = 2 S(sqrt y) - 1/2 + 2(S(z1)-S(z0)) and the printed parameter identities z0=L^A, z1=exp(L L3/(A L2)). Step 3: state both directions of Eq.21 (upper already MertensBand.actual_product_le; lower as above) and the exact Eq.20 O(1) form; keep the numeric constant 2M-1/2 OUT of the statement. Do NOT attempt Mertens (O1), do NOT touch O2/O3/O4/O5/O7, and do not claim a numerics-optimised constant.\",\"compute\":{\"ram_gb\":8,\"disk_gb\":5,\"cpu_hours\":2},\"failure\":\"The two-sided S(N) bound or the matching product lower bound cannot be stated/closed at the pins without importing an uncached module or weakening a hypothesis (for example needing a full Mertens asymptotic). Record the exact definitional/import obstruction on route 240, keep O6 OPEN, keep the exact constant conditional on O1, and preserve the one-sided budget untouched.\",\"success\":\"The added file elaborates at the pinned toolchain; #print axioms for the lower-bound lemma and the two-sided product comparison returns exactly [propext, Classical.choice, Quot.sound] with no sorryAx; a snapshot shows Eq.21 both directions and the Eq.20 O(1) statement present with the printed parameters and the small primes 2,3 retained; the O1 dependency is confined to the exact constant 2M-1/2 (recorded as still conditional).\",\"question\":\"Can a matching LOWER bound for the actual auxiliary product V(sqrt y) - equivalently the two-sided weighted prime-harmonic expansion sum_{p<=sqrt y} g(p)/p = 2 log log sqrt y + 2(log log z1 - log log z0) + O(1) - be formalised in the pinned package toolchain from interfaces already present (MertensBand.ordinary_product_le + ordinary_inverse_le, or Chebyshev.theta_ge' + theta_le_log4_mul_x), so that the printed Eqs.20-21 O(1)/asymp form holds WITHOUT Input M, leaving only the exact constant 2M-1/2 conditional on O1?\",\"shortfall\":\"This host cannot run the elaboration half: the served lean-toolchain.txt pins leanprover/lean4:v4.35.0-rc3, the folder-local Lean is 4.34.1 core-only, and the served module fails at line 29 with \\\"unknown module prefix 'MertensBand'\\\"; Mathlib at the pinned revision 331d5244f0d3aad530d9ab00ded135b4c7691502 and the accepted package's built oleans are absent (no compiled artifacts are served). The pins are reachable (the 4.35.0-rc3 aarch64 asset is 596161714 bytes), but a first build also needs Mathlib's compiled artifacts plus the package's 39 modules, above this assignment's 5 GB disk allowance. Needed: a host holding the pinned toolchain plus that Mathlib revision and the package oleans, where the remaining work is the ~0.25 h first-hand re-run of the served module (sha256 8d882bfe2f7b1e2343561fcaceb292fb88650efa860e02f9fa9cc03aa6e1a5c7) capturing its twelve #print axioms lines and the statement snapshot. The statement half is already verified from the served bytes.\",\"budget_hours\":0.5,\"required_tools\":[\"lean\",\"lake\"],\"required_sources\":[]}\n\nEarlier step check #5742: reuse its conclusions. Compare only the new candidates listed below and references needed to assess them; do not survey the whole project again.\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5749 (route 241, result, pending): **What the evidence changes.** O7's product step is closed in the pinned kernel, from the package's own one-sided input. In `ShortConstruction.lean` (523 lines, lean4 v4.35.0-rc3, Mathlib 331d5244) the exact identity `V N = 2 * (MertensBand.ordinaryProduct N)^2 * P N` is proved for the integer band `2 < p <= N`, `V N = (1/2)*prod_{2