{"id":4426,"job_id":9112,"problem_id":1,"lane_id":null,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Job #9112 — route 240 step check (O6): no recorded return answers the two-sided harmonic-product assembly step\n\n**Outcome: `promising`.** Route #240's held step — set by return #2662 and unchanged since — asks for one Lean\nfile in the accepted package that (1) derives the two-sided `S(N) = log log N + O(1)` from the package's own\nproduct interfaces, (2) proves the matching lower bound `V(sqrt y) >= c*(log z0/log z1)^2/(log sqrt y)^2` for\nthe actual auxiliary product, and (3) states both directions of Eq.21 and the Eq.20 `O(1)` form with the exact\nconstant `2M - 1/2` kept out of the statement. After comparing every return the served brief lists for this\nroute and its linked routes, every return citing the step's setter (#2662) or the route's planning return\n(#2604), and each linked route's latest recorded return, the step is still open. The step is copied exactly as\n`next_step`; the held pursuit goes out with this note. Documentary step check: `cpu_hours = 0`, no Lean build,\nno compiler run, no experiment, no published computation reproduced, no new theorem, no fresh literature query.\n\n## 1. The step under check, and its custody\n\nRoute 240 served record (fetched 17 Oct 2026): revision 2, state `active`, `last_return_id` 2662, parent route\n78 (state `known`, revision 4, `last_return_id` 1826, `next_step` null — route 78 records the ledger/bridge\nconstant and stays closed as the route's own note says). The served `next_step` is byte-identical to #2662's\n`research.next_step`, and its canonical sorted-key compact SHA-256 equals #2662's recorded\n`next_step_sha256` (`9c48f197204052371f0923ffa0e9b5a72fd474ca1ef51ec74b7b5e8f87f1687e`), verified this run.\nThe step (abridged): add one Lean file to the accepted package (no imports beyond the pinned Mathlib / its own\nfrozen modules); Step 1 two-sided `S(N) = log log N + O(1)` from `MertensBand.ordinary_product_le` +\n`MertensBand.ordinary_inverse_le` (or from `Chebyshev.theta_ge'`/`theta_le_log4_mul_x` via finite\npartial summation, no MeasureTheory); Step 2 the matching lower bound\n`V(sqrt y) >= c*(log z0/log z1)^2/(log sqrt y)^2` via `log V = -sum g(p)*(1/p) - Q` with\n`0 <= Q <= sum_p g(p)^2/(p^2(1-g(p)*(1/p)))`, reusing the exact reduction\n`sum g(p)*(1/p) = 2 S(sqrt y) - 1/2 + 2(S(z1) - S(z0))` and the printed parameter identities `z0 = L^A`,\n`z1 = exp(L*L3/(A*L2))`; Step 3 both directions of Eq.21 and the Eq.20 `O(1)` form, `2M - 1/2` out of the\nstatement; do NOT attempt Mertens (O1), do NOT touch O2/O3/O4/O5/O7, no numerics-optimised constant. The\ncopied `next_step` JSON carries the full method, success and failure rules.\n\n## 2. Route 240's own record\n\n- #2604 (recorded): the route's planning proposal. Source/dependency/formalization-planning only; it states\n  \"O6 remains OPEN\"; no Lean file, no mathematics executed. Does not answer the step.\n- #2662 (recorded): the setter. Documentary reduction of Eqs.20-21 to the two-sided ordinary product plus a\n  bounded `Q`, with finite checks C1/C2/C2b/C3 (`compute_gv.py`), and it says explicitly \"No compiler run was\n  made; the missing lemma is a bounded formalization proposal, not a proof.\" It sets the step; it does not\n  execute it.\n- After #2662: no return on route 240 (`last_return_id` 2662; the route's jobs are #5416 returned, #5551\n  expired with no return, and #9112 = this assignment). Parent route 78 has no return after #1826.\n\n## 3. Linked-route sweep (16 returns, every one fetched from the server this run)\n\nLinked routes per the served route graph and citation index: 75, 236, 237, 238, 239 (via #2662's\n`route_dependents` and `cited_by`), plus the shared-premise routes 235 (O1) and 241 (O7) named by the brief.\nPer-return characterization from the served records:\n\n- #3576 (236, recorded): read-only step check on #2645's uniform-Psi step; unchanged step. Names O6 only as a\n  cross-reference row (\"two-sided auxiliary-product/O(1) assembly proposal; no compiler run\").\n- #3577 (237, recorded): read-only step check on #2651's general-A step; unchanged step; O6 cross-reference\n  only.\n- #3579 (238, recorded): read-only step check on #2653's dyadic-PNT step; unchanged step; O6 cross-reference\n  only.\n- #3594 (236, recorded): HT p417 Cor. 1.3 verification, hash-pinned source inventory, and an explicitly\n  **uncompiled** conditional `UniformPsiInterface.lean` draft — a smooth-count Psi interface, not the harmonic\n  product; no compile, no axiom audit. O2 scope only.\n- #3627 (236, recorded): de Bruijn (1966) Theorem-5 error localization; the primary source is image-only on\n  that host. O2 only.\n- #3651 (236, recorded): the O2 upper side is dischargeable PNT-free via an explicit Rankin bound; \"no Lean\n  execution\" by its own terms. O2 only.\n- #4096 (75, recorded): S4/S5 step check; that step unanswered; O6 appears only in a cross-reference\n  enumeration of #2604/#2662.\n- #4106 (75, accepted, verified): executes route 75's fourth/fifth-order inclusion-exclusion CRT grid (exact\n  integer enumeration). No harmonic-product or Eq.20/21 content.\n- #4393 (237, recorded): O3 step check; held step unchanged and copied; no O6 token at all.\n- #4411 (237, recorded): O3 progress on the two places `SmoothBudget.lean` re-imposes A>=12; no O6 token.\n- #4413 (238, recorded): O4 step check; three linked-route returns do not answer the dyadic-count reduction;\n  no O6 token.\n- #4424 (239, recorded): O5 step check on the general-kappa `SieveUpperBound` interface; its served-file\n  census covers the four sieve files (no harmonic-density token there); O6 appears only in the\n  obligation-cluster list and a one-line comparison row restating #2662's finding.\n- #3446 (235, recorded): the O1 first look. Hand-derives the Abel identity for `sum 1/p`, proves `E(t) = O(1)`\n  uniformly from package lemmas, and validates the Meissel-Mertens constant numerically to 1e8 — with the\n  explicit scope statement \"no Lean formalisation and no kernel claim is made here (this container has no\n  Mathlib)\". It never touches the weighted density product `V(sqrt y)`, the `g(p)` reduction, or Eq.21's\n  comparisons. Relevant but separate: it strengthens the O1 side, not this step.\n- #2666 (241, recorded): the O7 source-to-statement map plus one bounded missing lemma for the Eq.28 short\n  construction; serves copies of the package files for reading; mentions O6 only in passing (\"exactly as O6's\n  first look found for the harmonic ledger\").\n- #2597 (accepted, proven): the accepted package itself. Its report proves the one-sided product bound and\n  states \"No two-sided Mertens asymptotic is needed\"; the served manuscript's obligation list marks\n  \"O6. OPEN — Original harmonic-density expansion and two-sided Euler-product comparisons (Eqs. 20-21)\".\n\nNo return in the sweep adds a Lean file, records a compile, states a `V(sqrt y)` lower bound, or assembles the\ntwo-sided weighted expansion.\n\n## 4. Served-byte census (the package surface the step consumes)\n\n- `MertensBand.lean` SHA-256 `6ee922c80814b35e0d47011a2bcb164e99467c85d190f92eb098ab8ba9be17d9` (fetched via\n  the content-addressed file endpoint; the server content-hash header verified this run): contains the\n  two-sided ordinary product (`ordinaryInverse`, `ordinary_inverse_le`, `ordinaryProduct`,\n  `ordinary_product_pos`, `ordinary_product_inv`, `ordinary_product_le`), the band versions\n  (`band_product_eq_ratio`, `band_product_le`), the harmonic bounds (`log_succ_le_harmonic`,\n  `harmonic_le_ordinary_inverse`), and the actual-density **upper** bounds (`actual_factor_le`,\n  `actual_product_le`, `actual_auxiliary_product_le`). It contains **no lower bound for the actual density\n  product**, **no `S(N) = log log N + O(1)` assembly lemma**, and no `sorry`. The inputs the step names\n  exist; the step's target does not exist in the module.\n- Served manuscript `kk-lower-bound.md` SHA-256\n  `6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80` (content-hash header verified): its\n  product-calculation section states Eqs. (10)-(11) and closes with \"The proof establishes a sufficient\n  one-sided estimate. It does not assert the original two-sided product asymptotic.\"; its obligation list\n  marks O6 OPEN; the verbatim (20)/(21)/(22) block sits at the cited locator.\n\n## 5. Token census\n\nKeyword census over the 16 fetched served return records (report text plus research evidence of each):\n`O6`, the `(20)`/`(21)` tags, `V(sqrt y)`, `harmonic`, `Meissel`, `2M`, `Euler-product`,\n`ordinary_product_le` and `actual_product_le` occur outside route 240's own returns only in cross-reference\nenumerations, one-line comparison rows, or #3446's package-lemma table (which cites the same\n`MertensBand.lean` hash). No return records a computation, compile, Lean artifact or statement touching the\ntwo-sided `V(sqrt y)` lower bound.\n\n## 6. What would change this answer\n\n- Any return that adds the step's Lean file (elaborating at the pinned toolchain, `#print axioms` clean) with\n  the two-sided assembly and the `V(sqrt y)` lower bound, or that records the step's success/failure outcome\n  on route 240.\n- Executing #3446's O1 next step (the Stieltjes/limit formalization for Eq.8's exact constant) would put\n  Step 1's mathematics into Lean — but it is a distinct obligation (Input M/O1), adds neither the `V(sqrt y)`\n  lower bound nor the Eq.21 comparisons, and would not by itself answer this step.\n- The expired pursue job #5551 left no return; its budget lapsed without output and is not evidence either\n  way.\n\n## Rung and limits\n\nRung: **measured** — every claim above is a served-byte observation: return records, the route record, the\nmanuscript and `MertensBand.lean` were fetched and hash-recorded this run, the two hash-identity checks\n(next-step canonical SHA-256 equality; server content-hash headers) were executed this run, and the token\ncensus ran over the fetched bytes. No computation, no compile, no reproduction. `cpu_hours = 0`. No files\nuploaded. Known limit: the sweep is bounded by the served brief's shortlist plus the citation and route-graph\nclosure over the linked routes; a return on an unlinked route that silently formalized the step would be\nmissed by design, and the route's own `next_step` would still hold the obligation until a route-240 return\nreplaces it.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-17T09:54:06.796Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2597,2604,2662,2666,3446,3576,3577,3579,3594,3627,3651,4096,4106,4393,4411,4413,4424],"messages":[]},"tokens":{"log":"summary","input":141477,"models":{"glm-5.3-flash":57944},"output":57944,"source":"reported","entries":0,"cache_read":8197316,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Reproduction (read-only; no computation):\n1. Read the route record: GET the project endpoint research-routes/240 (revision 2, active, last_return_id 2662).\n2. Read the returns: GET the project's return endpoint for 2597, 2604, 2662, 2666, 3446, 3576, 3577, 3579, 3594, 3627, 3651,\n   4096, 4106, 4393, 4411, 4413, 4424.\n3. Fetch the package file by its content address 6ee922c80814b35e0d47011a2bcb164e99467c85d190f92eb098ab8ba9be17d9\n   (MertensBand.lean) and the manuscript document kk-lower-bound.md from the project's docs; check both\n   X-Content-SHA256 headers against the hashes quoted in the report.\n4. Verify step custody: serialize the route's next_step as canonical sorted-key compact JSON and hash it;\n   the SHA-256 must be 9c48f197204052371f0923ffa0e9b5a72fd474ca1ef51ec74b7b5e8f87f1687e, equal to #2662's\n   recorded next_step_sha256, and the object must equal #2662's research.next_step.\n5. Census: scan each fetched return's report text and research evidence for O6, (20), (21), V(sqrt y),\n   harmonic, Meissel, 2M, Euler-product, ordinary_product_le, actual_product_le; inspect every hit's\n   surrounding text and confirm each is a cross-reference enumeration or comparison row.\n6. Compare each return against the step's three parts (two-sided S(N); V(sqrt y) lower bound; Eq.21 both\n   directions + Eq.20 O(1) form) and classify. Nothing in the record executes any of the three parts.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"promising","route_id":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":2,"disk_gb":1,"cpu_hours":0},"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?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[2597,2604,2662,2666,3446,3576,3577,3579,3594,3627,3651,4096,4106,4393,4411,4413,4424],"evidence_md":"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\nenumerations, 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\nafter #2662; parent route 78 has no return after #1826; pursue job #5551 expired with no return."},"research_route_id":240,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_b3c0a54fd6019875476b972b","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":"malaiwah","job_brief":"Step check before pursuit. Route #240's next experiment was set by return #2662, 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\":\"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\":2,\"disk_gb\":1,\"cpu_hours\":0},\"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?\",\"budget_hours\":0.5,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #2604, #2662 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\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 #4106 (route 75, result, accepted, verified): Exact integer CRT grid extended from #2306 to the fourth- and fifth-order inclusion-exclusion terms answers route 75's S4/S5 step in the POSITIVE and identifies the order. Object: reflected pair {C-d, C+d}, forbidden d = +-C mod p, window length W at offset s; S_j = sliding sum of the j-subset event indicators, N = W - S1 + S2 - S3 + S4 - S5 exactly, and the odd Bonferroni partial sums are non-dec\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- Return #3446 (route 235, promising, recorded, recorded): # evidence — route 235 (O1) first look: elementary Abel closure of Eq.8 from package lemmas **Obligation.** Route 235 (rev 2): can `sum_{p<=t} 1/p = log log t + M + o(1)` (t -> infinity, primes, inclusive real cutoff, `M` a constant) be discharged inside the accepted package's pinned toolchain from finite Mertens-band bounds, without a foreign toolchain? Bounded errors and differences suffice; a \n- Return #2666 (route 241, promising, recorded, recorded): Route 241 / O7 first look. The published manuscript (`manuscript.md` `6c287c18…` = docs paper `kk-lower-bound.md`) explicitly retains O7 OPEN; accepted source #2597 (`manuscript_main_goal`, `integer_maximum_binding`, `manuscript_integer_main_goal`; receipt31/reviews695-696) establishes only the three main claims. This run adds a scoped source-to-formal-statement map and one bounded missing lemma; \n\nReturn the ordinary report and transcript plus research: {route_id: 240, 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":"2604","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2662","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2666","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3446","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3576","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3577","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3579","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3594","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3627","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3651","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4096","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4106","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"4393","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4411","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4413","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4424","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4439,"handle":"malaiwah","status":"accepted"},{"id":4447,"handle":"malaiwah","status":"recorded"},{"id":4479,"handle":"malaiwah","status":"recorded"}],"route_dependents":[239,240],"research_url":"/projects/twin-primes/research-routes/240","transcript_url":"/projects/twin-primes/return/4426/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"1808c938e71c4e27ea28c14ee88cbc525922a7ba07918d3105ab8e64cdce3167","next_step_sha256":"9c48f197204052371f0923ffa0e9b5a72fd474ca1ef51ec74b7b5e8f87f1687e","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}