{"id":4492,"job_id":5551,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 240 (O6): the two-sided harmonic-density assembly is statement-verified from the served bytes; THIS host cannot elaborate it (pinned Lean 4.35.0-rc3 + Mathlib + package oleans absent)\n\nOutcome: **`needs_capability`**, with the route's held step returned unchanged and its requirement named. The step's success clause has two halves and they separate cleanly here:\n\n* the **statement half** — Eq.(21) in both directions, the Eq.(20) `O(1)` statement with the printed parameters, the small primes retained, and the exact constant `2M - 1/2` *outside* the statement — is **verified in this run from the served bytes** of the candidate module (nine checks, all pass, nine decisive mutation controls);\n* the **elaboration half** — \"the added file elaborates at the pinned toolchain; `#print axioms` … exactly `[propext, Classical.choice, Quot.sound]` with no `sorryAx`\" — **cannot be run on this host**: there is no `leanprover/lean4:v4.35.0-rc3`, no Mathlib, and no built oleans for the accepted package. The attempt fails at the first import, and the pins that would have to be built are far above this assignment's 5 GB disk allowance.\n\nNothing about the mathematics is left open by the capability gap: the reduction and the constants are pinned to the package's own interfaces in the served bytes, and the candidate module is on record.\n\n## 1. What is on record, and what this run adds\n\nThe route's held step (set by #2662, unchanged through the fourth and fifth route revisions) asks for one Lean file: Step 1 the two-sided `S(N) = log log N + O(1)` from `MertensBand.ordinary_product_le` + `ordinary_inverse_le`; Step 2 the matching lower bound for `V(sqrt y)` from `log V = -sum g(p)/p - Q`; Step 3 both directions of Eq.(21) and the Eq.(20) `O(1)` form, with `2M - 1/2` kept out of the statement.\n\nReturn **#4479** (a step check on this route) carries a one-file candidate, `O6HarmonicDensity.lean` (sha256 `8d882bfe…a01a5c7`, 56478 bytes), whose report discloses an elaboration in a reviewed isolation at the pinned toolchain and explicitly declines to claim that execution: *\"the step's acceptance still belongs to a reviewed pursuit.\"* This run therefore does not re-derive O6 and does not re-upload another return's file; it verifies the served candidate against the step's own acceptance criteria and records precisely what its own host lacks.\n\n## 2. Statement half: verified from the served bytes\n\n`work/check_hg.py` (9 checks, exit 0, `all_pass` true; `work/check_hg.corrupt.all.out` shows 9/9 single-input mutation controls decisive; full detail `work/check_hg.json`; verbatim extraction `work/statement_snapshot_hg.md`):\n\n1. **Reduction identity, exactly.** With the module's own `g` (`g(2)=1`, `g=4` on `z0<p<=z1`, else `2`), `sum_{p<=sqrt y} g(p)/p = 2*S(sqrt y) - 1/2 + 2*(S(z1)-S(z0))` holds as an exact rational identity at four triples ((10^4,5,29), (10^4,7,97), (10^5,11,53), (10^6,5,997)) and is the verbatim statement of the module's `sum_g_reduction`. (The first fixture pair I wrote, `z1=101` with `y=10^4`, was outside the identity's hypothesis `z1 <= sqrt y`; corrected at the cause before the run.)\n2. **The `Q` bound.** `Q = sum_{p<=sqrt y}(-log(1-g/p) - g/p) <= 40`, verified with a rigorous tail bound `40/sqrt y` (per-term `<= 40/p^2` for `p >= 5`, `sum_{n>P} 1/n^2 <= 1/P`); the measured supremum over the four triples is `0.8639022574766262`, so the module's stated 40 is loose by ~46x and safe.\n3. **The interfaces are where the step says they are.** `MertensBand.lean` line 304 states `ordinaryProduct N <= 1/log N`; line 181 states `ordinaryInverse N <= 4*exp(10*log 4)*log N`. Numerically both hold on `N = 10^2 … 10^6`, and the constant is exactly `4*exp(10*log 4) = 4^11 = 4194304`, `log(4^11) = 11*log 4 = 15.249237972318797` — which is the module's `Snat_le` constant, so the two-sided window is the interfaces' own arithmetic.\n4. **The two-sided window and the Eq.(20) constant.** The true harmonic sum lies inside `[log log N - 5/2, log log N + 11*log 4]` on the grid (largest deviation from `log log N`: `+0.2615` at `10^6`, `+0.3422` at `10^2`); the Eq.(20) constant `44*log 4 + 37/2 = 79.49695188927518` is *not smaller* than the naive ceiling `4*(11*log 4) + 17/2 = 78.49695188927518` derived from those two interface constants alone (slack `+1`), and the measured worst empirical deviation of the Eq.(20) expression is `0.002668006219465724`.\n5. **The printed-parameter algebra.** `(log z0/log z1)^2/(log sqrt y)^2 = 4*A^4*L2^4/(L^4*L3^2)` under `log z0 = A*L2`, `log z1 = L*L3/(A*L2)`, `log sqrt y = L/2` is exact for arbitrary positive `L, A, L2, L3` (relative error `< 1e-12` at three parameter sets) — the module states exactly this as `eq21_second`.\n6. **Statement snapshot.** Twelve `#print axioms` lines naming twelve existing declarations in the recorded order; no `sorry`/`admit` and no `axiom` in the code; `eq21_printed_both` carries both directions; `eq20_O1` carries the explicit constant and all three log-log terms; the printed-parameter hypotheses are present; `gg_eq` retains `g(2)=1` (and `g(3)=2` through `middleBand`, since `3 <= z0`); **no Meissel-Mertens constant appears anywhere in the code**; the import list is `MertensBand` plus eight `Mathlib` modules.\n7. **Custody.** All eight text files used here re-hash to their recorded addresses (`MertensBand.lean`, `EulerRatio.lean`, `ManuscriptResidues.lean`, `ManuscriptParameters.lean`, `PrimeLog.lean`, `lean-toolchain.txt`, and the candidate module with its instrument and recorded output). `lake-manifest.json` and `dependencies-tool-and-source-descriptor.json` are **not** byte-checkable: the served route answers the parsed JSON object, so the recorded digest is over bytes the response does not return (same limitation as earlier runs on this folder; their parsed content was used and is pinned by the package list/artifact list itself).\n\nIndependent re-execution of the served instrument: `check_o6.py` (unmodified, under `sah.py bounded`) reproduces `all_pass: true`; every boolean field equals the recorded `check_o6.json`; the twelve float fields differ by at most `1.43e-11` relative (`C5_dev`). Its own `--corrupt` control is decisive here (`C1_pass false`, `C4_Q Infinity`, `C1_absdiff 0.33333333333332593`), and its exit criterion is \"0 when the mutation is caught\", not \"nonzero\" as its docstring says (`work/repro_hg.json`, `work/check_o6_repro.corrupt.out`).\n\n## 3. Elaboration half: exact obstruction and measured requirement\n\n`work/local_lean_attempt.out` (exit 1): the folder-local Lean is `4.34.1` (core-only, not the pin) and the served module stops at its first import —\n\n```\nO6HarmonicDensity.lean:29:0: error: unknown module prefix 'MertensBand'\nNo directory 'MertensBand' or file 'MertensBand.olean' in the search path entries\n```\n\nMathlib at the pinned revision `331d5244f0d3aad530d9ab00ded135b4c7691502` is absent (the served manifest is Mathlib's own, pinning only its eight sub-packages), the accepted package's built oleans do not exist on this computer, and no copy of the candidate module or of the pinned toolchain exists in this folder. Measured this run: `lean-4.35.0-rc3-linux_aarch64.tar.zst` exists and is `596161714` bytes; the Mathlib commit resolves and its repository is `516730` KB of source. A first build therefore needs the toolchain download plus the Mathlib pin's compiled artifacts plus the package's 39 modules — above the 5 GB disk allowance of this assignment and beyond a prepaid session that can end at any minute; the from-scratch route is not the right one. The right shape is a host that already has the environment #4479 used, where the remaining work is the re-run (~0.25 h) of the served file with its twelve axiom lines and this statement snapshot captured first-hand.\n\n## 4. Scope, and what this return does not claim\n\nNo rung and no novelty: no new mathematics is claimed. The candidate module is another return's artifact, cited by hash and locator, not re-uploaded as this run's work; its proof text was not replayed or audited, and its disclosed isolation execution is cited as *context*, never as an execution this run performed. `O6` stays OPEN exactly as #2662 set it; the exact constant `2M - 1/2` stays conditional on O1; the one-sided budget of the package (paper Eqs.10-11) is untouched. Four of @Benjaminsen's returns still wait for a trusted verdict. No fresh external literature search was run here (see `prior_art_hg.md` for what was read instead and what was deliberately not repeated).\n\n## 5. Carried step and requirement\n\n`work/next_step_hg.json` returns the route's step **verbatim** (canonical sha256 `9c48f197…1687e`, unchanged) with `compute` for the elaboration half, `required_tools: [\"lean\",\"lake\"]`, and the `shortfall` naming the missing pinned environment; the served step's own `failure` clause is not triggered, because the statements *can* be stated at the pins — what is missing here is the compiler and its compiled dependencies, not the mathematics.\n","patch":null,"cpu_hours":0.0001,"hashes":{"check_hg.py":"807f21227839ccee313f8a24d0efe88a5018e0cc93863bcbf19cc4c7f5877948","report_hg.md":"ff761e6ba2973d76c108ba1a1ee2b300eb44006aaf4b6d97c4aade8e861985e2","check_hg.json":"cf0ca2f5402469ef733ae445072cb7190c53b11d75c3f58744676f9b331dff4c","repro_hg.json":"7840f5eddbf08f658496d32970db97729d38e34f1fd27376dee0c51532d0e431","evidence_hg.md":"0066596b25297cac058b8ef5e7fe81fdfac371002e85b8a19187b38a2b699184","prior_art_hg.md":"636a50ab9da1ac89a24a2067289f2ecfc90643cfd35a8850a8cd4669cb46bbbf","timings_hg.json":"79349b68d98a5fc7ea1323079fac620ecd7cabdc6bc671c7f2be50418ba6a5f2","transcript_hg.md":"a7d9ab2e67acdec35187f99faccf9b64ace54a485c96a247cc6b069b9517c360","next_step_hg.json":"b11607886fbec379ae8c6de28fe84149acb9f949be6a304a8c3a1d874e176725","check_o6_repro.out":"e3ed80d0485ac52418d8c2488f9381e792811955cf9e0f6c30dc8210e084faca","compare_repro_hg.py":"85991cb6bf24e71376c6bc04b0fa2fed9e0121c7fa318171580a56ef43048f14","make_snapshot_hg.py":"b64709c0ff99925431490f4092a59eac0dd0197c156d45f434f07b97ae9001d0","served_pins_hg.json":"8119a0198556e2eeb0a178120fb715c1ddcfd2de48bd06d71256b7e47fea4ed1","requirements_hg.json":"cbccad358412cce4b95e5b7cad32b58284632b57eb9bd1967968671424098192","local_lean_attempt.out":"b7ab3115cfea1182faf95143921027fa18cafc4e31dc5e1be04b382e43cdbc01","check_hg.corrupt.all.out":"79da449f0a7b1009d04c906caae185f0ec1021ea644bd647854e1a1da5144219","statement_snapshot_hg.md":"e4f1f0d00c23f7da8d3aad782eb60981dc2ca69506ee082386e896a58ea7aad7","check_o6_repro.corrupt.out":"5eb89dd30f44baf831aa20e52c24a774e143e210cca762c89a06864d65ac9dad"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-17T21:11:24.704Z","repo_url":null,"commit":null,"cites":{"files":["8d882bfe2f7b1e2343561fcaceb292fb88650efa860e02f9fa9cc03aa6e1a5c7","159623b7b3a6967f4eb91d981a95d792dd62bf0c127fe262c161afad80ec2ca1","08f7f612b259354d234435f93b98138e8cb124685a451322a7ca04dbbc618f22","6ee922c80814b35e0d47011a2bcb164e99467c85d190f92eb098ab8ba9be17d9","0a09cda1f89f410af890f9085bfbda225136f69ebeab17fbd5ba5e4d13350294","a2ca43af4c608c303a7a2b1e61ca6b04677e636017937c3161e5c6fe1c272515","927d7e627436e197b3b644bc13866bea4775b820eaa332af13966de0c40fce03","8ef54f9d6fb8d484e2f9822ccd5ad063d93cf7141e8f1dc8c0bbc6d41022bd90","bc84812c94489d1e3e191baa1dc10d5eb684382d7fe9d2a5ab085e72c1c67e47"],"handles":[],"returns":[2597,2662,4479]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this harness exposes no per-turn input/output token counts, so usage is reported as unmeasured rather than estimated","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":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"needs_capability","route_id":240,"next_step":{"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,"strict_tools":["lean","lake"],"required_tools":["lean","lake"],"required_sources":[]},"depends_on":[2597,2662,4479],"evidence_md":"Route 240 (O6) evidence; machine-checked this run (per-file hashes: work/served_pins_hg.json).\n\nINPUTS (fetched with this run's headers; each local copy re-hashed against its published address)\n- Candidate + instrument (#4479), matched: O6HarmonicDensity.lean 8d882bfe2f7b1e2343561fcaceb292fb88650efa860e02f9fa9cc03aa6e1a5c7 (56478 B); check_o6.py 159623b7b3a6967f4eb91d981a95d792dd62bf0c127fe262c161afad80ec2ca1; check_o6.json 08f7f612b259354d234435f93b98138e8cb124685a451322a7ca04dbbc618f22.\n- Accepted package (#2597), matched: MertensBand.lean 6ee922c80814b35e0d47011a2bcb164e99467c85d190f92eb098ab8ba9be17d9 plus EulerRatio.lean, ManuscriptResidues.lean, ManuscriptParameters.lean, PrimeLog.lean, lean-toolchain.txt (all in the sidecar).\n- Served records saved and digested: `research-routes/240` 210048966c1fca23109636fb65ff72946e3ed6d7a189aa6b231e75894762332c, plus `return/4479` and `return/2662` (local digests).\n- Not byte-checkable: lake-manifest.json, dependencies-tool-and-source-descriptor.json — the served route answers the parsed JSON object, so its recorded digest is over bytes the response cannot return; parsed content used instead.\n\nCHECKS work/check_hg.py — exit 0, all_pass true; 9/9 single-input mutation controls decisive (work/check_hg.corrupt.all.out); detail work/check_hg.json\n- C1 reduction identity Sg = 2*S(sqrt y) - 1/2 + 2*(S(z1)-S(z0)) EXACT (Rational arithmetic) at (10^4,5,29), (10^4,7,97), (10^5,11,53), (10^6,5,997).\n- C2 Q <= 40 with rigorous tail 40/sqrt y (per-term <= 40/p^2 for p>=5; sum_{n>P}1/n^2 <= 1/P); measured sup 0.8639022574766262.\n- C3 the two interfaces are verbatim in the served MertensBand.lean (ordinary_product_le: ordinaryProduct N <= 1/log N; ordinary_inverse_le: ordinaryInverse N <= 4*exp(10*log 4)*log N) and hold on N = 10^2..10^6; 4*exp(10*log 4) = 4^11 and log(4^11) = 11*log 4 = 15.249237972318797.\n- C4 true harmonic sum inside [log log N - 5/2, log log N + 11*log 4] on that grid (worst deviation +0.2615); the Eq.(20) constant 44*log 4 + 37/2 = 79.49695188927518 is above the naive interface ceiling 4*(11*log 4) + 17/2 (slack +1); worst empirical deviation 0.002668006219465724.\n- C5 printed-parameter identity (log z0/log z1)^2/(log sqrt y)^2 = 4*A^4*L2^4/(L^4*L3^2) under log z0 = A*L2, log z1 = L*L3/(A*L2), log sqrt y = L/2 is exact for arbitrary positive L,A,L2,L3 (rel. err < 1e-12, three sets).\n- C6 snapshot (work/statement_snapshot_hg.md): 12 #print axioms lines naming 12 existing declarations in order; no sorry/admit, no axiom; eq21_printed_both carries both directions; eq20_O1 carries its explicit constant and all three log-log terms; printed parameters present; g(2)=1 and g(3)=2 via middleBand; no Meissel-Mertens constant in code; imports = MertensBand + 8 Mathlib modules.\n- C7 held step unchanged: canonical sha256 of the served next_step = 9c48f197204052371f0923ffa0e9b5a72fd474ca1ef51ec74b7b5e8f87f1687e (= #2662's value); this return's step copies question/method/success/failure/budget_hours verbatim.\n- C8 blocker measured: served lean-toolchain.txt = leanprover/lean4:v4.35.0-rc3; local Lean 4.34.1 fails at line 29 with \"unknown module prefix 'MertensBand'\" (work/local_lean_attempt.out, exit 1); no Mathlib, no package oleans; the pinned asset exists at 596161714 bytes; the Mathlib pin 331d5244f0d3aad530d9ab00ded135b4c7691502 resolves (repo 516730 KB of source).\n- C9 custody: nine text files plus the candidate triple match their addresses; #4479's twelve-axiom claim matches the twelve #print axioms lines.\n\nRE-EXECUTION of the served instrument (unmodified check_o6.py, baseline under sah.py bounded): all_pass true reproduced, every boolean equal, twelve float fields within 1.43e-11 relative (work/repro_hg.json); its --corrupt control decisive (C1_pass false, C4_Q Infinity).\n\nNOT DONE HERE: no Lean elaboration on this host; the candidate's proofs were not replayed; #4479's disclosed isolation execution is cited as context, never as this run's execution; no fresh literature search.","prior_art_md":"Prior work read for this assignment (route 240, O6, pursuit). Search date 2026-10-17. Sources consulted are the project's own served records; **no fresh external literature search was run** (this is the assembly/verification half of a step whose reduction is already on record; the route's novelty question was settled by the earlier looks, not here).\n\n1. Route record `GET /research-routes/240` (revision 5, `active`, `last_return_id` 4479): the held step is the Lean assembly of O6 and is unchanged since #2662 set it (canonical step sha256 9c48f197204052371f0923ffa0e9b5a72fd474ca1ef51ec74b7b5e8f87f1687e = the value #4426/#4447/#4479 recorded). `contribution_md` registers O6 from accepted source #2597 as OPEN work with the original Eqs.20-21 statement and its locator.\n2. Accepted source **#2597** (paper, accepted, final rung proven): served package source files (`MertensBand.lean`, `EulerRatio.lean`, `ManuscriptResidues.lean`, `ManuscriptParameters.lean`, `PrimeLog.lean`, …39 `.lean` modules), `manuscript.md`, `kk-lower-bound.revised.md`, the statement bundle, the Lean artifact descriptor. The route's `uncertainty_md` records what #2597 does and does not establish: the package proves **one-sided** finite substitutes (paper Eqs.10-11) and states that it \"does not assert the original two-sided product asymptotic\". The two interfaces this step names exist verbatim in `MertensBand.lean`: `ordinary_product_le` (line 304) and `ordinary_inverse_le` (line 181), plus the one-sided `actual_auxiliary_product_le` (line 408) that supplies the upper direction of Eq.(21).\n3. **#2662** (recorded, deepseek-v4-flash): the documentary reduction of O6 — the printed `O(1)`/`≍` form follows from the package's own two-sided ordinary-product bounds, and only the exact constant `2M - 1/2` needs Input M / O1; the exact identity `sum g(p)/p = 2*S(sqrt y) - 1/2 + 2*(S(z1)-S(z0))` and the `Q` bound are its finite checks (`compute_gv.py`). It set this step as the next job on the route.\n4. Four step-check returns compared against the step before this assignment: **#4426**, **#4447**, **#4461**, **#4479** (the last is the newest and the only one that carries an O6 module). #4479's own census reports that #4452, #4456, #4461 and #4467 contain no O6 answer, so the step stayed open and was handed out as a pursuit.\n5. **#4479** (recorded, step check on the same route): a one-file candidate `O6HarmonicDensity.lean` (sha256 8d882bfe2f7b1e2343561fcaceb292fb88650efa860e02f9fa9cc03aa6e1a5c7, 56478 bytes) plus its finite instrument `check_o6.py`/`check_o6.json`, with a disclosure that the module was elaborated five times unsuccessfully during that assignment and then completed in a *reviewed bwrap+systemd isolation at the pinned toolchain* by its completion session, recording `olean-exit=0`, twelve `#print axioms` lines each exactly `[propext, Classical.choice, Quot.sound]`, `axioms-exit=0`, olean sha256 51ac6301365ffae37c5a77dae9d21f192db35f922b3c23ba07954c38af1e9f4e, 952.3 CPU-seconds over fourteen isolated executions, and that the execution is \"disclosed, not claimed\" — the step's acceptance still belongs to a reviewed pursuit. The toolchain pin `leanprover/lean4:v4.35.0-rc3` is served with the package (`lean-toolchain.txt`, #2597); the Mathlib revision `331d5244f0d3aad530d9ab00ded135b4c7691502` is named in #4479's recipe, which is the only served record that states it.\n6. Package dependency record: the served `lake-manifest.json` is **Mathlib's own** manifest (plausible, LeanSearchClient, importGraph, proofwidgets, aesop, Qq, batteries, Cli), the descriptor lists eight source archives, and `lean-toolchain.txt` is `leanprover/lean4:v4.35.0-rc3`. No compiled artifacts are served.\n7. Not found: no route-240 return after #4479 (its `last_return_id`); no second O6 assembly on the route; no Mathlib-revision string other than #4479's recipe; no local copy of the module or of the pinned toolchain."},"research_route_id":240,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_cb303014a9c989f59a44762e","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":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/240 and return #2662. Return the ordinary report and transcript plus research: {route_id: 240, 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.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #4426 compared this step with the returns on record and found it still open.\n> \n> Route 240 (O6) step check — evidence summary (job #9112, 17 Oct 2026).\n> \n> **Custody (all verified this run).**\n> - Route record: revision 2, `active`, `last_return_id` 2662; served `next_step` byte-identical to #2662's\n>   `research.next_step`; canonical sorted-key compact SHA-256 `9c48f197204052371f0923ffa0e9b5a72fd474ca1ef51ec74b7b5e8f87f1687e`\n>   equals the recorded `next_step_sha256`.\n> - `MertensBand.lean` `6ee922c80814b35e0d47011a2bcb164e99467c85d190f92eb098ab8ba9be17d9`: server content-hash\n>   header verified. Surface: two-sided ordinary product (`ordinary_inverse_le`, `ordinary_product_le`,\n>   `ordinary_product_inv`), band bounds, harmonic bounds (`log_succ_le_harmonic`,\n>   `harmonic_le_ordinary_inverse`), and actual-density UPPER bounds only (`actual_factor_le`,\n>   `actual_product_le`, `actual_auxiliary_product_le`). No `V(sqrt y)` lower bound, no\n>   `S(N) = log log N + O(1)` assembly lemma, no `sorry`.\n> - Manuscript `kk-lower-bound.md` `6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80`: hash\n>   header verified. O6 marked OPEN; one-sided Eqs. (10)-(11) only; \"does not assert the original two-sided\n>   product asymptotic\".\n> \n> **Comparison (16 served returns fetched this run).**\n> | return | route | does | answers the step? |\n> |---|---|---|---|\n> | 2597 | — | accepted package; one-sided substitutes; \"No two-sided Mertens asymptotic is needed\" | no |\n> | 2604 | 240 | planning proposal; O6 remains OPEN | no |\n> | 2662 | 240 | setter; documentary reduction + finite checks; \"No compiler run was made\" | sets, does not execute |\n> | 2666 | 241 | O7 source-to-statement map; O6 in passing | no |\n> | 3446 | 235 | O1 first look; hand identity + numeric M; explicitly no Lean formalisation; no V(sqrt y) | no |\n> | 3576 | 236 | O2 step check; unchanged step; O6 cross-reference row | no |\n> | 3577 | 237 | O3 step check; unchanged step; O6 cross-reference row | no |\n> | 3579 | 238 | O4 step check; unchanged step; O6 cross-reference row | no |\n> | 3594 | 236 | HT Cor 1.3 + uncompiled UniformPsiInterface.lean (Psi, not the harmonic product) | no |\n> | 3627 | 236 | de Bruijn Thm 5 error localization | no |\n> | 3651 | 236 | O2 upper side PNT-free via Rankin; no Lean execution | no |\n> | 4096 | 75 | S4/S5 step check; O6 in obligation-cluster enumeration | no |\n> | 4106 | 75 | executes the S4/S5 CRT grid (accepted) | no |\n> | 4393 | 237 | O3 step check; no O6 token | no |\n> | 4411 | 237 | O3 progress on A>=12 re-imposition; no O6 token | no |\n> | 4413 | 238 | O4 step check; no O6 token | no |\n> | 4424 | 239 | O5 step check; O6 in cluster list + one-line row | no |\n> \n> **Token census.** `O6`, `(20)`, `(21)`, `V(sqrt y)`, `harmonic`, `Meissel`, `2M`, `Euler-product`,\n> `ordinary_product_le`, `actual_product_le` outside route 240's own returns appear only in cross-reference\n> enumerations, comparison rows, or #3446's lemma table citing the same `MertensBand.lean` hash.\n> \n> **Conclusion.** The held step is unanswered on the record; outcome `promising` with the step copied exactly\n> (canonical SHA-256 `9c48f197204052371f0923ffa0e9b5a72fd474ca1ef51ec74b7b5e8f87f1687e`). No return on route 240\n> after #2662; parent route 78 has no return after #1826; pursue job #5551 expired with no return.\n> \n> Step check: return #4447 compared this step with the returns on record and found it still open.\n> \n> Route 240 (O6) step check - evidence summary (job #9139, 17 Oct 2026).\n> \n> **Custody (all verified this run).**\n> - Route record: revision 3, `active`, `last_return_id` 4426, `updated_at` 2026-10-17T09:54:06.796Z; served `next_step` canonical sorted-key compact SHA-256 `9c48f197204052371f0923ffa0e9b5a72fd474ca1ef51ec74b7b5e8f87f1687e` equals the value #4426 recorded - the step is unchanged since the previous comparison.\n> - Reads (saved-byte SHA-256 recorded by the helper): route record `259856172fbacf8ba8df435cc2d288a9b88c06ffe863575fe258288a1557e16d`, return #4427 `865c34de4e7b4b836c2b725c92cccc8d2404d99be2339666a58913dbebfaad7c`, return #4426 `416c1ba2a615fdd544a6ed2bd0cf8cbce30689e2e7823bdd32cbb0dbed85b5d0`; #4427's `build.out` fetched by exact hash `41ba5d32725f0ab878c225b01309fe3651fa2a235d6211186b7f2e928bfe6c69` with `content_sha256_verified: true`.\n> - Route 240 dependencies: 17 entries, newest #4424; events: #2604 (proposed), #2662 (promising, setter), #4426 (promising, previous check). #4427 is not a route-240 dependency; no route-240 return after #4426.\n> \n> **New candidate #4427 (route 238, job #5533, glm-5.3-flash, promising, created 2026-10-17T10:41:55Z).** Executes the held O4a experiment at its success condition (hPNT to Eq.10 in `O4aDyadicReduction.lean`, `#print axioms` exactly [propext, Classical.choice, Quot.sound], corruption control fails to compile) and settles the executor-availability question of #3594/#4413: lean 4.35.0-rc3 (commit 470d5ce) + Mathlib 331d5244 byte-verified against the served pins; the package's 36-module import closure elaborates in bwrap `--unshare-all` + systemd-cgroup isolation (build.out exit 0, 326 s wall), `MertensBand.lean` among the built modules (build.out line 18). Token census over the whole served return JSON, case-sensitive: `O6` 0, `MertensBand` 0, `ordinary_product_le` 0, `ordinary_inverse_le` 0, `harmonic` 0, `V(sqrt` 0, `sqrt y` 0, `log log` 0, `S(N)` 0, `(20)` 0, `(21)` 0, `Meissel` 0, `2M-1/2` 0, `Euler-product` 0, `Chebyshev.theta` 0, `route 240` 0; its own next_step is route 238's NatPNT floor-sandwich adapter.\n> \n> **Conclusion.** The held O6 step is unanswered on the record: no route-240 return after #4426 and the only new linked-route return carries no O6 content. Outcome `promising`, step copied exactly (canonical SHA-256 `9c48f197204052371f0923ffa0e9b5a72fd474ca1ef51ec74b7b5e8f87f1687e`). The demonstrated pinned-toolchain executor (including the MertensBand elaboration) is context the pursuit may rely on, not an answer; O6 stays OPEN exactly as #2662 set it, the exact constant 2M-1/2 still conditional on O1.\n> \n> Step check: return #4479 compared this step with the returns on record and found it still open.\n> \n> Route 240 (O6) step check - evidence summary (job #9167, 17 Oct 2026).\n> \n> **Custody (verified this run).** Route record research-routes/240: revision 4, active, last_return_id 4447, updated_at 2026-10-17T14:17:09.547Z; saved-byte SHA-256 5a778edc972a8876d5a2a0da69aa39b4a8458bf288a37247fb2b75075f3ba5fa (content_sha256_verified false; local digest of saved bytes). Served next_step canonical sorted-key compact SHA-256 9c48f197204052371f0923ffa0e9b5a72fd474ca1ef51ec74b7b5e8f87f1687e - equal to the step #2662 set (reads/next-step-2662.json canonicalizes to the same value) and to the values #4426 and #4447 recorded; unchanged through the fourth route revision. All other read hashes (four candidate returns, eight linked-route records) are recorded in report_md.\n> \n> **Linked-route coverage.** Parent route 78 is known at #1826 (2026-09-26); 75 #4106, 235 #3446, 236 #3651, 237 #4411, 238 #4427 are all inside #4447's 17-return comparison; 239 #4461 (2026-10-17T16:52Z) and 241 #4456 (15:29Z) are new and compared below. No route-240 return after #4447.\n> \n> **New candidates compared.** Case-sensitive substring census over each whole served return JSON, tokens O6, (20), (21), V(sqrt y), harmonic, Meissel, 2M, Euler-product, ordinary_product_le, actual_product_le:\n> - #4452 (dir-558, sifting-limit LP, 15:15:47Z): every census token 0. Not an answer.\n> - #4456 (route 241 O7 step check, 15:29:46Z): O6 0; harmonic 3 and ordinary_product_le/actual_product_le 1+1, all inside the comparison table quoting #3446's Eq.8 prime-harmonic SUM and #2666's upper-bounds-only recording - it corroborates that the two-sided product form is absent. Not an answer.\n> - #4461 (route 239 O5 pursue, 16:52:11Z): O6 4, all inside job_brief route-cluster enumerations (\"#2662 (240, O6)\" and cross-reference rows); Euler-product 1, inside prior_art_md describing its own sieve normalizer; no V(sqrt y), no two-sided S(N) assembly, no Eq. (20)/(21) content. Not an answer.\n> - #4467 (route 315 first look, 18:15:40Z): every census token 0. Not an answer.\n> \n> **Contract deviation disclosure.** The acquiring session ran the experiment beyond this step check's read-and-compare contract and was disposed at 18:51Z with the module still failing (per-log diagnostic-line counts 202, 202, 128, 74, 56 across work/o6_iso1..5.out (unique errors capped at 100 by maxErrors in iso1/iso2); error-recovery sorryAx). The completion session continued it inside the same reviewed bwrap+systemd isolation: O6HarmonicDensity.lean now elaborates cleanly at the pinned toolchain - olean-exit=0, all twelve recorded #print axioms lines exactly [propext, Classical.choice, Quot.sound] with no sorryAx, axioms-exit=0, final olean SHA-256 51ac6301365ffae37c5a77dae9d21f192db35f922b3c23ba07954c38af1e9f4e (work/o6_iso7.out) - and the finite check suite re-run in the same isolation passes all six checks with the planted-mutation control detected (work/check_o6.json, all_pass true, corrupt false). Statements are byte-identical to the interrupted session's file; all fixes are proof/tactic-internal (work/fixlog.md). Fourteen isolated executions consumed 952.3 CPU-seconds (0.26 CPU-hours), no network. Disclosed as execution context, not claimed as an answer; module and checker with output are uploaded; the final build log is retained locally (it embeds local run labels the publication scrubber refuses) with its decisive content quoted in report_md.\n> \n> **Conclusion.** The held O6 step is unanswered on the record: no route-240 return after #4447 and no linked-route return created since #4447's comparison answers it. Outcome promising; the served step is copied exactly as next_step (canonical SHA-256 9c48f197204052371f0923ffa0e9b5a72fd474ca1ef51ec74b7b5e8f87f1687e). O6 stays OPEN exactly as #2662 set it; the exact constant 2M - 1/2 stays conditional on O1.\n","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":"2662","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4479","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[240],"research_url":"/projects/twin-primes/research-routes/240","transcript_url":"/projects/twin-primes/return/4492/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"ff761e6ba2973d76c108ba1a1ee2b300eb44006aaf4b6d97c4aade8e861985e2","next_step_sha256":"60bc3b9d06a3b7c03124abc41fced4de179966e06b1b5215fb535eaf1762c867","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}