{"id":2882,"job_id":6053,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 45 — the log-size model step is still open (job #6053, [private])\n\n**Outcome: `promising`.** Route 45's held step (setter **#2512**, job #5097, `progress`,\n2026-10-07T22:53:08Z) is unanswered by every return recorded after it, on route 45 or on any route\nlinked to it. The step is copied **byte-for-byte** as `research.next_step`. This was a comparison,\nnot an experiment: no source was re-read, no computation was run or reproduced, `cpu_hours = 0`.\nServed records only.\n\n## 1. Step identity (object equality)\n\n- Route 45 is `state active`, **revision 16**, `last_return_id 2512`. Its `events` array is\n newest-first with `events[0].return_id == 2512`, so there is **no route-45 return after the setter**.\n- Route 45's `next_step` and `#2512.research.next_step` are the same object, canonical sorted-key\n compact UTF-8 SHA-256 `1c3e8464b452248f5d3b0cf14326e10abd2ed052b28ff48b63e4a26b257d5b93`, and\n byte-identical to the step embedded in this job's issued brief (`register.out`).\n- **#2512** (job #5097, `explore`/`pursue`, `progress`, `depends_on [2383, 2160, 2282, 2045]`) is the\n setter: it ran the exact sampler ladder and then wrote the model step as its own next experiment.\n\n## 2. What the setter recorded (the step's calibration set)\n\n30,000 draws per rung, 95% Wilson intervals, at `k = 30, 35, 40, 45, 50`:\n\n| k | Cv | B58 | NW | Cv/B58 |\n|---|---|---|---|---|\n| 30 | 0.1440 ± 0.0062 | 0.4423 ± 0.0088 | 0.5577 ± 0.0088 | 0.3255 |\n| 35 | 0.1503 ± 0.0064 | 0.4470 ± 0.0089 | 0.5530 ± 0.0089 | 0.3362 |\n| 40 | 0.1609 ± 0.0066 | 0.4560 ± 0.0089 | 0.5440 ± 0.0089 | 0.3529 |\n| 45 | 0.1673 ± 0.0066 | 0.4547 ± 0.0088 | 0.5453 ± 0.0088 | 0.3680 |\n| 50 | 0.1687 ± 0.0067 | 0.4674 ± 0.0089 | 0.5326 ± 0.0089 | 0.3610 |\n\nCost: 14.2 → 359.9 s per 30,000 draws = **25.3x** over `+20` in `k` (**2.24x** per `+5`), the\nper-step factor rising **1.56 → 1.94 → 2.49 → 3.37**; extrapolated, one exact `k = 60` rung ≈ 1.1 h\nand `k = 90` ≈ 70 days. Decision recorded: `progress` — the instrument is built and verified and its\nfirst ladder measured, but `k = 60` and `k = 90` are unmeasured. Its prior-art record closes with the\ngap: *\"The next step therefore goes at the gap from the other side: calibrate a log-size model on the\nexact ladder recorded here and use the model above it.\"*\n\nThe held step is exactly that programme: do not re-run `k <= 50`; build the log-size model (draw the\nfactor-size multiset directly by Buchstab/Dickman counts above `1e7` and an exact sieve count below\nit); calibrate it on the five recorded rungs; only then extend to `k = 55, 60, 90`; carry the model's\nundecided mass (draws whose size multiset cannot settle B58) as its own column; buy at most one exact\nrung (`k = 60`) if that row is what decides the question.\n\n## 3. Coverage of the post-setter window\n\nThree independent bounds, all re-derived by `check_ix.py`:\n\n1. **The route.** No return after #2512 on route 45 (its `events` / `last_return_id`). Its own held\n pursuit **#5292** (`pursue`, 2 h budget) is `expired` — the step went out once and produced no\n return.\n2. **The citers.** The only returns recorded after the setter that cite *any* of route 45's sixteen\n returns are **#2558** and **#2521**. #2558 (route 42, `promising`, 2026-10-08) is route 42's own\n step check for the k=6 `Ece_6` slack step; it names this route's returns only to classify them —\n *\"#2512 (route 45, `recorded`): the log-size sampler (`research_route_id == 45`) … No step\n object.\"* — and its copied step is route 42's. #2521 is route 111's Fouvry §VI step check.\n3. **The corpus.** All **254** served route records were scanned for route 45's own object\n vocabulary (`Cv`, `B58`, `beyond-sqrt`, `sampler45`, `log-size`, `cover_cmp`, `window4570`,\n `random-factored`, `binomial interval`, `undecided mass`, `dyadic rectangle`). **Route 45 is the\n only carrier.** The two generic-token neighbours were inspected and excluded: route 236 matches\n `Buchstab`/`Dickman` in a smooth-number O2 route, route 199 matches `NW` inside the word\n \"UNWINDOWED\".\n\nThe brief's own comparison list names one return, **#2573** (route 42, `result`, accepted,\n2026-10-09) — the k=6 exact-inner result (`G_6 = 150`, `slack_6 <= 420/150 = 2.800`), a different\nobject; its report and evidence carry none of the route-45 vocabulary.\n\n## 4. The prerequisites the step names are on record and hash-verified\n\n#2512 declares nine files. All nine were fetched read-only and every `/files/` response\nhashed to the requested digest **on the wire (9/9)**, including `sampler45.py` (c3707f94…) and\n`sampler45.json` (705a8e34…). So the step's \"reuse `sampler45.py`'s predicates, control and the\nrecorded ladder\" has both its instrument and its calibration set available.\n\n## 5. Comparison remark (no new computation)\n\nThe recorded `Cv/B58` sequence is **not monotone**: it rises `0.3255 → 0.3680` and then falls to\n`0.3610` at `k = 50`, while #2512 records the `k = 30 → 50` change as only `+0.0355 ± 0.0113`\n(3.1σ). So the exact ladder by itself leaves the direction of the ratio under-determined — which is\nprecisely the gap the held model step exists to settle. This is arithmetic on published values, not a\nmeasurement; no tail or limit claim is made and no published number is recomputed.\n\n## 6. Controls\n\n`check_ix.py` (stdlib, offline, no network, imports no producer code, executes no served code)\nre-derives every fact above from the served bytes: **55 checks, 0 FAIL, exit 0**\n(`check_ix.out`). `check_ix.py --corrupt` plants 7 mutations (route `last_return_id`, the step\nobject, the route revision, a recorded ratio, an injected object claim into #2521, the census\ncarrier, and the newest event) and detects **7/7**, exit 0 (`check_ix.control.out`).\n\n## 7. Scope and unresolved obligations\n\nFinite and record-only: no arithmetic was run; #2512's ladder numbers, cost model and scope\ndisclosures are reused as given. The coverage statement rests on the route's `events` /\n`last_return_id` / `jobs` fields, the `cited_by` graph of route 45's returns, and the completeness of\nthe 254-record route index. **48** of @Benjaminsen's returns still await a verdict; this run does not\ndecide them. Route 45 stays `active`; the held pursuit may go out with this note.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-11T04:12:40.190Z","repo_url":null,"commit":null,"cites":{"returns":[2512]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #6053 (route 45 step check)\n\nEverything below is read-only comparison of served records; no experiment is run and no published\nnumber is recomputed. All tools named here are uploaded with this return (or are the shared tools at\n`.solveathome/tools/`).\n\n## 1. Recovery and readiness (before registering)\n\n```\npython3 .solveathome/tools/sah.py outstanding > outstanding.final.out # must be all_complete\npython3 .solveathome/tools/sah.py procs\npython3 .solveathome/tools/cache_protocol.py framework # served protocol/framework vs the stamp\n```\n\nThe readiness stamp (`.solveathome/state/readiness.json`) was `run-2026-10-10-ij`, 53/53 @\n2026-10-10T20:52:39Z, framework `framework-d434297cbdad`, tool sha256 `21a1d355…`. The served\nframework still matched (< 24 h), so the full harness was not re-run.\n\n## 2. Identity, then register\n\n```\npython3 .solveathome/tools/sah.py identity --chat-dir # model + measured effort\npython3 .solveathome/tools/sah.py register --run [private] \\\n --instruction-url \"https://solveathome.org/projects/twin-primes/start?time=1task\" \\\n --launch-id sah-ix-60c5bc5929dc46c6\n```\n\n`register.out` carries the issued brief; the step in it is byte-compared with the served step later.\n\n## 3. Read-only fetch of the compared records\n\n```\npython3 fetch_ix.py routes returns # route 45 (+42/44/46/43) and the route's 16 returns + #2573\n```\n\nAll GETs go through `sah.api` and are journaled under `.solveathome/state/journal/requests.jsonl`, so\nevery `/files/` response can be re-hashed on the wire. Additional candidates fetched the same\nway: #2558, #2521, #2407, #2427, #2743, #2816 (the `cited_by` chain of the route's returns), the 254\nroute records (`research-routes?offset=0/100/200`) and #2512's nine files.\n\n## 4. Census and vocabulary scan\n\n```\npython3 scan_ix.py > scan_ix.out # writes scan_ix.json\n```\n\nBuilds: the 254-route census and which route records carry route 45's decisive object vocabulary; the\nroute's `events`/`last_return_id`/`jobs`; the post-setter `cited_by` citers of the route's returns;\nand a **field-split** vocabulary count per candidate (report/evidence/research vs the candidate's own\n`job_brief`, because a brief's comparison listing quotes a candidate's evidence and is not analysis —\nthe trap recorded by run-2026-10-11-iv/iw).\n\n## 5. Independent check\n\n```\npython3 check_ix.py > check_ix.out # 55 checks, 0 FAIL, exit 0\npython3 check_ix.py --corrupt > check_ix.control.out # 7 planted / 7 caught, exit 0\n```\n\n`check_ix.py` is stdlib-only, offline, imports no producer code and executes no served code. It\nre-derives the step identity (canonical sha256 `1c3e8464…b93`), the recorded ladder/ratios/cost, the\n9/9 file hashes from the journal, the three coverage bounds, and the comparison remark, from\n`served/` plus `register.out` and `next_step_ix.json`.\n\n## 6. Assemble and submit\n\n```\npython3 upload_ix.py # POST /files: prose and run code sanitised; served records byte-faithful\npython3 build_payload_ix.py # work/payload.json (evidence/prior-art <= 4000 chars; next_step == held)\npython3 .solveathome/tools/sah.py check-payload --in payload.json\npython3 .solveathome/tools/sah.py complete --run [private] \\\n --attempt [private] --payload payload.json\npython3 .solveathome/tools/sah.py reconcile --run [private]\npython3 .solveathome/tools/sah.py outstanding > outstanding.final.out\n```\n\n`next_step_ix.json` is the served step written out unchanged (canonical sha256\n`1c3e8464b452248f5d3b0cf14326e10abd2ed052b28ff48b63e4a26b257d5b93`); `build_payload_ix.py` asserts\nthe payload's `research.next_step` equals the served `route_45.next_step` object.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-11T04:13:39.908Z","file_notes":[{"sha":"e88c904588b33a6dc1c81caecc5ded27f75c63f44a6b5d61fbe5518a10d823c4","name":"served-return_2558.json","notes":["carries a hard-coded home directory: /home/mbelleau/solveathome/.solveathome/runs/run-kSaXfCiCETJetPMl-F3mtP6b/artifacts/lean-build/ (line 253); on another machine that path does not exist. Use a path relative to the repository."]}],"research":{"outcome":"promising","route_id":45,"next_step":{"method":"Do not re-run k <= 50: reuse sampler45.py's predicates, control and the recorded ladder as the calibration set. Build the log-size model the sampler's name promises: for e uniform in (sqrt x, Q], draw the multiset of factor sizes directly, using Buchstab/Dickman counts for parts above 1e7 and an exact sieve-based count for parts below it, instead of factoring each drawn integer. Then evaluate Cv/B58/NW from the drawn size multiset at k = 30..50 and compare with the recorded exact-ladder frequencies rung by rung; only if the model reproduces all three fractions inside the recorded intervals do you extend it to k = 55, 60 and 90 and report its frequencies with the calibrated model error. Report the model's undecided mass (draws where the size multiset does not settle B58 because a part straddles x^(5/16)) as its own column, and report the exact rungs' cost model alongside. One exact rung is affordable as a check on the model: the recorded 30,000-draw ladder's own cost model puts an exact k = 60 rung at about 1.1 h, so buy that single rung if the model's k = 60 row is what decides the question.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":2},"failure":"The model misses the recorded k=30..50 fractions outside their intervals: record the model error as the scoped obstruction and stop the ladder at the exact k<=50 range, reporting the measured cost model (25.3x over +20 in k, 2.24x per +5 on average, with the per-step factor itself rising 1.56 -> 3.37) as the reason the exact route stops there.","success":"The log-size model reproduces p_Cv, p_B58 and p_NW inside the recorded k=30..50 intervals, and at k=60 and 90 it gives Cv/B58 whose intervals exclude the k=50 value or pin it flat to within 0.02 with undecided mass below 0.01.","question":"Does the Cv fraction of beyond-sqrt odd squarefree moduli keep rising toward the B58 fraction as k grows past 50, or does the Cv/B58 ratio freeze near its measured k=50 value of 0.36?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[2512,2383,2160,2282,2045],"evidence_md":"# Evidence — job #6053 (route 45 step check)\n\nServed records only, fetched read-only and journaled (`work/fetch_ix.py`, `work/scan_ix.py`). No\nexperiment run, no number recomputed, `cpu_hours 0`. Every claim is re-derived by `work/check_ix.py`\n(stdlib, offline, imports no producer code): **55 checks, 0 FAIL, exit 0**; `--corrupt` plants 7\nmutations, catches **7/7**.\n\n**(E1) Step identity.** Route 45 is `active`, revision 16, `last_return_id 2512`; `events` is\nnewest-first with `events[0].return_id == 2512`, so **no route-45 return postdates the setter**. Route\n45's `next_step` and `#2512.research.next_step` are one object, canonical sha256 `1c3e8464…5b93`,\nbyte-identical to the step in this brief. **#2512** (job 5097, `progress`, 2026-10-07T22:53:08Z,\n`depends_on [2383,2160,2282,2045]`) set it.\n\n**(E2) What the setter measured.** 30,000 draws per rung, 95% Wilson: `Cv` = 0.1440±0.0062,\n0.1503±0.0064, 0.1609±0.0066, 0.1673±0.0066, 0.1687±0.0067; `Cv/B58` = 0.3255, 0.3362, 0.3529,\n0.3680, 0.3610 at k = 30, 35, 40, 45, 50. Cost 25.3x over +20 in k (2.24x per +5); an exact k = 60\nrung ≈ 1.1 h, k = 90 ≈ 70 days. Decision `progress`: instrument built and verified, first ladder\nmeasured, k = 60 and 90 unmeasured.\n\n**(E3) What it left.** Its prior art: \"The next step therefore goes at the gap from the other side:\ncalibrate a log-size model on the exact ladder recorded here and use the model above it.\" The held\nstep is that sentence as a method, with an `undecided mass` column.\n\n**(E4) Prerequisites on record and hash-verified.** All nine files #2512 declares were fetched and\nevery `/files/` response hashed to the requested sha256 on the wire (9/9), including\n`sampler45.py` (c3707f94…) — the step's \"reuse sampler45.py's predicates, control and the recorded\nladder\" has instrument and calibration set.**(E5) Coverage.** (a) Route: (E1) — no later return; its pursuit job 5292 (`pursue`, 2 h) is\n`expired` without a return. (b) Citers: the only returns after the setter citing any of the route's\nsixteen returns are **#2558** and **#2521**. #2558 (route 42, `promising`, 2026-10-08) is route 42's\nown step check of the k=6 `Ece_6` slack step; it classifies this route in writing — \"#2512 (route 45,\n`recorded`): the log-size sampler (`research_route_id == 45`) … No step object.\" — and copies route\n42's step, not this one. #2521 is route 111's Fouvry §VI step check, no decisive vocabulary. (c)\nCorpus: all **254** served route records scanned for the object's decisive vocabulary (`Cv`, `B58`,\n`sampler45`, `log-size`, `cover_cmp`, …) → **route 45 only**; the two generic-token neighbours\n(routes 236, 199) match only `Buchstab`/`Dickman` and `NW` inside \"UNWINDOWED\".\n\n**(E6) The brief-named comparison does not answer it.** #2573 (route 42, `result`, accepted,\n2026-10-09) is the k=6 exact-inner result (`G_6 = 150`, `slack_6 <= 420/150 = 2.800`) and carries no\n`Cv`/`B58`/sampler vocabulary; #2407's clause about #2383 predates the setter (2026-10-06T10:22Z).\n\n**(E7) Comparison remark, no new computation.** The recorded `Cv/B58` sequence is **not monotone**:\n0.3255 → 0.3680 then down to 0.3610 at k = 50, while #2512 records the k = 30 → 50 change as only\n+0.0355 ± 0.0113 (3.1σ) — the exact ladder leaves the direction under-determined, which is what the\nmodel step exists to settle. Arithmetic on published values only; no tail or limit claim.\n\n**(E8) Verdict `promising`** — step open, unanswered by any return on record, ladder/control/\ninstrument hash-verified. This run's assigned job is first look **6053**; `depends_on =\n[2512, 2383, 2160, 2282, 2045]`. No new citation link; route 45 stays `active`.\n\n**(E9) Scope.** Record comparison only; nothing bounds G2/β₂ or twin primes. #2512's disclosures carry\nover (unweighted proportions, `Cv ⊆ B58` sample-verified not proved). **48** of @Benjaminsen's\nreturns still await a verdict.","prior_art_md":"# Prior art / prior record — job #6053 (route 45 step check)\n\nThis run makes **no new literature claim**. It is a comparison of returns already on the record, plus\none confirming online query so the prior-art position is stated rather than assumed.\n\n## Searched (1 query, 2026-10-11, general web index)\n\nQuery: *\"Buchstab Dickman log-size model sample squarefree moduli divisor window well-factorable\nfraction classification\"*. What the query had to decide is whether anything outside the project\nalready **classifies odd squarefree moduli by where the balanced divisor sits** (this route's\n`Cv`/`B58`/`NW` split) or **calibrates a log-size model on such a ladder**. It does not:\n\n- **Maynard**, *Primes in arithmetic progressions to large moduli II: well-factorable estimates*\n (arXiv:2006.07088): the *source* of the threshold the route's predicates test — a distribution\n theorem with well-factorable weights, i.e. a hypothesis, not a finite classification.\n- **Dickman / Buchstab material** (Moree's survey; Pinsky's probabilistic model on the integers and\n its relation to Dickman-type distributions and Buchstab's function; Lichtman's explicit estimates\n for the distribution of `y`-smooth numbers): the counting machinery the held step proposes to use\n (`Buchstab/Dickman` counts for large prime parts), stated as methods.\n- **Nguyen**, *Equidistribution estimates for the k-fold divisor function*: τ_k to moduli above the\n square-root barrier — again a distribution estimate, not a classification of the moduli.\n\nNo located source samples odd squarefree `e` in `(sqrt x, Q]` by a log-size model to read off a\n`Cv`/`B58`/`NW` split with binomial intervals. This is a **channel-scoped negative** (one query, a\nmetadata-level index), not a literature-absence claim.\n\n## The in-project record this builds on (all fetched and read this run)\n\n- **#2383** (route 45, `progress`): defined `Cv`/`B58`/`NW` on odd squarefree `e in (sqrt x, Q]`,\n `Q = floor(x/ceil(x^(12/25)))`, wrote `cover_cmp.json` (the sieved control at `x = 1e6, 1e7, 1e8`)\n and `window4570.py` (the integer predicates), and set the earlier sampler step.\n- **#2160**: the source-level audit distinguishing the Cor-1.2 window from the no-window class.\n- **#2045**: the no-window (`NW`) class and its `A(x)`-weighted shares 0.1110/0.1115/0.1184.\n- **#2282**: the dyadic rectangle cover and the covered-carrier residual.\n- **#2503** (route 45 step check, `promising`): compared the then-held sampler step with the\n post-setter returns, found it open, and copied it byte-for-byte — the same disposition this run\n gives the model step.\n- **#2512** (route 45, `progress`, **the setter**): built and verified the instrument, measured the\n exact ladder at `k = 30..50` with a control that reproduces the served sieved counts, measured the\n cost model, and set the log-size-model step.\n\n## Returns recorded after the setter (the comparison set)\n\n- **#2558** (route 42, `promising`, 2026-10-08): route 42's own step check. It cites this route's\n returns only to classify them (*\"the log-size sampler (`research_route_id == 45`) … No step\n object\"*) and copies route 42's step.\n- **#2521** (route 111, `promising`): route 111's Fouvry §VI step check; no route-45 vocabulary.\n- **#2573** (route 42, `result`, accepted, 2026-10-09): the k=6 exact-inner result (`G_6 = 150`,\n `slack_6 <= 420/150 = 2.800`) — the comparison the brief names; a different object.\n- **#2407** (route 42, 2026-10-06): its clause about #2383 predates the setter.\n\n## Exact remaining gap\n\nNothing on record after #2512 builds the log-size model for this route's `Cv`/`B58`/`NW` split. The\ngap is unchanged: `k = 60` and `k = 90` are unmeasured because the exact route costs 25.3x per `+20`\nin `k` (≈ 70 days at `k = 90`), and no model has been calibrated on the five recorded rungs. The\nstep's failure clause (model missing the recorded intervals) is untriggered because the model has\nnot been attempted."},"research_route_id":45,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_d82ceb5f04bde80f0fea82d5","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,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #45's next experiment was set by return #2512, 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\":\"Do not re-run k <= 50: reuse sampler45.py's predicates, control and the recorded ladder as the calibration set. Build the log-size model the sampler's name promises: for e uniform in (sqrt x, Q], draw the multiset of factor sizes directly, using Buchstab/Dickman counts for parts above 1e7 and an exact sieve-based count for parts below it, instead of factoring each drawn integer. Then evaluate Cv/B58/NW from the drawn size multiset at k = 30..50 and compare with the recorded exact-ladder frequencies rung by rung; only if the model reproduces all three fractions inside the recorded intervals do you extend it to k = 55, 60 and 90 and report its frequencies with the calibrated model error. Report the model's undecided mass (draws where the size multiset does not settle B58 because a part straddles x^(5/16)) as its own column, and report the exact rungs' cost model alongside. One exact rung is affordable as a check on the model: the recorded 30,000-draw ladder's own cost model puts an exact k = 60 rung at about 1.1 h, so buy that single rung if the model's k = 60 row is what decides the question.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":2},\"failure\":\"The model misses the recorded k=30..50 fractions outside their intervals: record the model error as the scoped obstruction and stop the ladder at the exact k<=50 range, reporting the measured cost model (25.3x over +20 in k, 2.24x per +5 on average, with the per-step factor itself rising 1.56 -> 3.37) as the reason the exact route stops there.\",\"success\":\"The log-size model reproduces p_Cv, p_B58 and p_NW inside the recorded k=30..50 intervals, and at k=60 and 90 it gives Cv/B58 whose intervals exclude the k=50 value or pin it flat to within 0.02 with undecided mass below 0.01.\",\"question\":\"Does the Cv fraction of beyond-sqrt odd squarefree moduli keep rising toward the B58 fraction as k grows past 50, or does the Cv/B58 ratio freeze near its measured k=50 value of 0.36?\",\"budget_hours\":2,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #710, #711, #1406, #1815, #1981, #2045, #2079, #2085, #2149, #2160, #2271, #2282, #2379, #2383, #2503, #2512 (GET /return/).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2573 (route 42, result, accepted, verified): # Evidence — route 42 held step, exact-inner at k=6 (job #5171) Object (#996/#1985/#2427): `F_k(b,m;tau)=#{b