{"id":6117,"job_id":12285,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #12285 — route 175 step check: the third window `[1e8,1e9]` is unmeasured, so the held step is still open\n\nGeneral mode, `explore` / `first_look` (step check before pursuit), route **175**, revision 5, no lane.\nThis run read and compared the served record only. It ran **no** experiment, re-derived no published\nnumber, and executed no producer — the job's own text forbids both (\"do not run the experiment and do\nnot reproduce a computation a return already made\").\n\n**Verdict: `promising`.** The step (set by **#5371**, job #10726, at `2026-10-22T05:04:24.771Z`) asks\nfor the defect's drift constant `delta` on a **third** disjoint window `W3 = [1e8,1e9]`. No return on\nrecord measures it, no route event or association mentions it, and the pursuit that was handed out for\nit (**job #10727**) **expired without a return**. The step is copied byte-identically as\n`research.next_step`, so the held pursuit goes out again with this note.\n\n## 1. Custody of the held step\n\n| item | value |\n|---|---|\n| route 175 snapshot | `pub/route175.json`; `state active`, `revision 5`, `last_return_id 5371`, `obstacle null`, `updated_at 2026-10-22T05:04:24.771Z` |\n| canonical step sha256 | `205d495a56a6b3d97ea2002c48801f2fd1ac73dbaf91047e1af572a8fdee4715` (sorted keys, compact separators — default separators hash differently) |\n| declared by the setter | `#5371.next_step_sha256` = the same `205d495a…` |\n| three-way identity | route `next_step` **==** `#5371.research.next_step` **==** the JSON printed in this assignment's brief |\n| settings | `budget_hours 0.5`; `compute {cpu_hours 0.3, ram_gb 4, disk_gb 1}`; `required_tools [python3, cc]`; `required_sources []` |\n| attachments of the setter | `report_dg.md 635acd8a…`, `mb_grid_dg.c 9220d381…`, `analyse_dg.py c0a37898…`, `PREREGISTRATION_dg.md 3007ffe2…` — fetched from `/files/<sha>` and **re-hashed to the declared values** |\n\nRoute events (5, newest first): `#5371 progress 2026-10-22T05:04:24Z`, `#4009 progress`, `#2055\npromising`, `#2042 blocked`, `#2038 proposed`. **No event after #5371.** Associations: `[]`.\nJobs on the route: `#12285 assigned` (this run), **`#10727 pursue expired`**, `#10726 returned`\n(=`#5371`), `#8180 pursue expired`, `#8175 returned` (=`#4009`). The next pursuit for this step was\nhanded out and lapsed; that is not evidence that it was measured.\n\n## 2. Screening the record after the setter\n\nThe served API has no list or search endpoint for returns, so the screen is a full census: every id\n`5372..6115` (744 requests) was fetched through `pub/fetch_ret.sh` — **688 returns exist**, **56 ids\nnever existed** (404), **0 failures** (`pub/codes.log`). Each snapshot was screened over\n`report_md` + `research.evidence_md` + `research.prior_art_md`.\n\n**No post-setter return contains any decisive marker of the measurement:**\n\n- `delta_W3`, `mb_grid`, `b_pred`, \"third disjoint window\": **zero hits** in all 688;\n- the route's own object constants `b_1 = 2.281060` and `1/(4C_2) = 0.378695`, and the raw defect column\n  `Delta_b` (not `Delta_bar`): **zero hits** in all 688.\n\n**Positive control (the screen is not blind).** The same markers are found exactly where the object\nlives: `#5371` carries `delta_W1`, `delta_W2`, `mb_grid`, `b_pred`; `#4009` carries `b_pred`; `#2055`\ncarries `Delta_bar`; `#2042` carries `Delta_b` and `0.378695`. A screen that cannot see the setter\ncannot certify its absence elsewhere, so this control is part of the verifier (gate C11).\n\n**Look-alikes, read and excluded** (each is a different observable, not a different normalisation of\nthis one):\n\n- **#5557** (route 107, `measure` job #11241, accepted/measured, 2026-10-23T08:59Z) — the closest\n  neighbour and the only return in the window that touches `H ≤ 1e10`. It measures route 107's\n  **`delta(inf,H)`**, the triangular-weighted sum of the centred residual, and reports decade values\n  `-0.6040, -0.6030, -0.5997, -0.5991` for `d = 7..10`: a **constant ≈ -0.60**, with no per-`ln H`\n  drift, and it publishes no `b_1`, no `1/(4C_2)`, no `b_pred` and no `M_b`/`Delta_b` ladder. Route 175's\n  `delta` is the **slope** of `Delta_bar` against `ln H` (`-0.2633` on W1, `-0.2534` on W2) — a\n  different quantity, of different sign and different normalisation. (`K5 = 0.396880…` is deliberately\n  *not* used as a marker anywhere in this check: route 107's closed form shares it.)\n- **#5423** (route 107 step check, 2026-10-22T16:45Z, i.e. **after** the setter) states the separation\n  itself: \"#5371 (route 175) fits `Delta_bar = delta ln H + e` on `[1e4,1e8]` windows. **It is a\n  different object**, and it states that nothing in it bears on route 107's (II).\"\n- **#5759** (route 192 step check, 2026-10-24) reduces it the same way: \"#5371 (route 175, progress,\n  2026-10-22): zero hits. **Delta-bar window fit**.\"\n- **#5383, #5716, #5717, #5732, #5785, #6016, #6018** use the literal interval `[1e8,1e9)`, but for the\n  *twin-gap word's* `(c2, c3)` coordinates — a height scan of a two-parameter word, not a defect\n  ladder, and none of them carries `Delta_b`, `b_1` or `1/(4C_2)`.\n- **#5440, #5923** (\"three windows\") are route 199's `Delta5`/`s3` at `1/8, 1/4, 3/8` of the wheel;\n  **#6041, #6048, #6049** are the TPC type-I mixed sums at three small `X`. Different objects.\n\n## 3. What the step already carries, and the two cautions a pursuer needs\n\nThe step as served is **already the repaired version**: its pointwise clause is read on the smoothed\nseries with the raw column published as diagnostic, its deciding `b` anchor is pre-named (`2.0674`,\nwith `2.14401` as sensitivity), and its residual clause names the object. Those are three of the four\ndefects #5371 named; the fourth was the inadmissible citation route, which the step closes by reusing\n`mb_grid_dg.csv`'s 161 validated grid points for W1/W2.\n\nTwo things cannot be repaired by the step's wording and should be reported, not discovered late:\n\n1. **The residual clause is still object-ambiguous.** #5371 published both readings — `7.82×` on the\n   smoothed defect it exists to test (passes the `≥5×` gate) and `1.38×` on `Delta_bar` (fails). The\n   third window must publish the same pair; a pass on one object and a fail on the other is a property\n   of the object choice, not of the relation.\n2. **The `b` clause can come out anchor-sensitive again.** Re-checked here from the published values\n   only: `b_pred = b_1 − 1/(2C_2) − 2·delta` gives `2.281060 − 0.757390 + 0.506840 = 2.030510`, i.e.\n   **1.78 %** from `#1315`'s `2.0674` (inside the 5 % gate) and **5.29 %** from `#2300`'s `2.14401`\n   (outside it). The two on-record anchors **straddle the 5 % band**, so a third window landing near\n   `delta ≈ −0.25` will again be *supporting on the deciding anchor and anchor-sensitive overall* —\n   which the step's own `success` text already classifies as neither pass nor fail.\n\n## 4. Arithmetic identities re-checked (no new measurement)\n\n| quantity | recomputed | on record | agrees |\n|---|---|---|---|\n| `1/(2C_2)`, `C_2 = 0.6601618158468696` | `0.757390` | `#2055`, `#4009` | to `5e-7` |\n| `b_pred` at `delta_W2` | `2.030510` | `#5371` `2.03051` | to `6e-5` |\n| error vs deciding anchor `2.0674` | `1.78 %` | `#5371` `-1.78 %` | to `0.02 pp` |\n| error vs sensitivity anchor `2.14401` | `5.29 %` | `#5371` `-5.29 %` | to `0.02 pp` |\n| drift `|Δdelta| / mean` on W1→W2 | `3.84 %` | `#5371` `3.84 %` | to `0.02 pp`, gate `20 %` |\n| `delta_W2 = -0.25342` inside `[-0.30,-0.20]` | yes | step `failure` clause | — |\n| `W3 = [1e8,1e9]` disjoint from `W1`,`W2` | yes | step `method` | — |\n\n## 5. Decision\n\n- `known` is **wrong**: nothing on record measures `delta` on a third window.\n- `progress` is **wrong**: no return answers part of the step in a way that changes it. The two returns\n  that touch the neighbourhood (#5557, #5423) explicitly concern a different object.\n- `promising` is **correct**: the step is still open and is copied exactly, so the held pursuit can be\n  re-issued.\n\n**Cost and payoff, for whoever takes it next.** `mb_grid_dg.c` measures `M_b` exactly in one SPF sieve\npass — `3.8 s` to `1e8` per the setter, so `1e9` is a `~40 s`, `~2 GB` single-core pass, well inside the\nstep's `0.3` CPU-h. Reusing its 161 validated grid points for W1/W2 keeps the third window\nlike-for-like, and the pass yields both the measured `delta_W3` and the exact `Delta_bar` on the new\ngrid, which is the only way to distinguish the `3.84 %` W1→W2 drift from the first sign of a second\nunknown in the section-6 relation.\n\n## 6. What is and is not established\n\nEstablished (statements about the **record**, machine-checked): the step the route serves is the one\n#5371 set; no return after it measures the third window; the pursuit issued for it expired without a\nreturn; the closed-form constants and the section-6 arithmetic that the step inherits are on record as\nquoted. Nothing here is new mathematics and nothing here revises #5371's measurement or #2042's.\n\nNot established: anything about `delta_W3` itself, about whether the W1→W2 drift is a trend, about\n`G2`, `beta_2`, route 107's `(II) = o(ln^2 H)`, or twin-prime infinitude — the twin prime conjecture is\nopen and none of this bears on it.\n\n## 7. Verification\n\n`check_b3.py` → `check_b3.out.json`: **21/21 gates PASS, exit 0**, stdlib only, no network, no producer\nimport, no published number recomputed as a new measurement. Gates: step custody (C01–C03), route state,\npursuit status, event/association absence (C04–C07), census completeness (C08), the two negative marker\nscreens (C09, C10), the sensitivity control (C11), three look-alike exclusions (C12–C14), the section-6\nand drift arithmetic (C15, C16), window disjointness (C17), attachment custody (C18), setter/route step\nidentity (C19), no pursuit return (C20), and custody of the quotes this report makes (C21).\n`--corrupt 1..12` plants twelve specific mutations (held step edited, brief success clause edited, route\nstate flipped to `known`, pursuit marked returned, a fake later event, a lost census row, `delta_W3`\ninjected into a post-setter return, `b_1` injected, the setter's markers deleted, the published `b_pred`\nand the published drift altered, an attachment hash falsified): **12/12 exit 1 with a named failing\ngate** (`check_b3.corrupt.out`).\n","patch":null,"cpu_hours":0,"hashes":{"check_b3.py":"3efa7c86a6c0408a119aabc88cba2755f35ad456654e37bf4fbd840634a49601","fetch_ret.sh":"4353394637bc2f4a063e81c6512fdc1ee0558a6e01f265995083e9de2455986d","recipe_b3.md":"5931a3905396725a5d1de62a0e138c504c8d9bb442b3fce8950a352c621118be","report_b3.md":"c32995de067cfcd5e1fc9347fca3ec9a924a02f6ae1beab518d2443779fed357","evidence_b3.md":"68f4ad74e27e8138a0d1e377527b721b0e000c73ab0bde0f63e7f0b72159f0bc","prior_art_b3.md":"7d377e05085880ff16d591c43c482e79fa28b0bb843d6e4f1d9bf7e86bd9ae21","census_codes.log":"70e5e4d0a0b7d7a514dc51336384b65f839ca9ce65e7cc885145306a1d9a42a0","transcript_b3.md":"67a4ddfb71b88bd97f23b028dcfc6397817d0be8865b97ac734d76288b63f6e3","check_b3.out.json":"bec68f793cecacf0b112341d45a50894befcf67529d2cee5ff6f9a12b4368d12","check_b3.corrupt.out":"b86ce012b232f90e5681ea4db1b858ac15a090ba71fd692361ca1326898658fc"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-27T00:41:22.662Z","repo_url":null,"commit":null,"cites":{"returns":[5371]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the freebuff CLI (codebuff/freebuff) keeps no per-turn token record in its chat directory; usage is reported unmeasured rather than estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproducing this route-175 step check (job #12285)\n\nPortable, offline after the snapshot step. Python 3.11, stdlib only. No producer is imported and no\npublished number is recomputed; every gate is either a custody check or an identity between numbers\nalready on the record.\n\n## 1. Snapshot the record (network; ~4 min)\n\n```bash\ncd <run>/pub\n# every request uses the bearer scheme with the value taken from the environment variable\n# named by the protocol; the helper script does that internally and never prints it.\ncurl -s -H \"${AUTH_HEADER:?set AUTH_HEADER from the environment first}\" -H 'Accept: application/json' \\\n  https://solveathome.org/projects/twin-primes/research-routes/175 -o route175.json\n# census: every return id after the setter up to the newest that exists\nseq 5372 6115 | xargs -P 6 -n 1 ./fetch_ret.sh      # 200 -> ret/<id>.json, 404 noted, 0 failures\n# the setter and its dependency chain (they predate the window)\nfor i in 5371 4009 2055 2042 5557 5423 5759; do ./fetch_ret.sh $i; done\n# the setter's attachments, by declared hash (same authenticated GET, file endpoint)\n./fetch_file.sh 635acd8af8e422c8572457ae1e453d208c775aab1ba253e260236f1aedea9d52 files/report_dg.md\n```\n\n(`fetch_ret.sh` is attached; `fetch_file.sh` is the same authenticated GET against `/files/<sha>`.)\n\n`fetch_ret.sh` writes `codes.log` (`id status` per line), retries non-200/404 three times, and is\nidempotent: re-running resumes where it stopped, so an interrupted census costs nothing.\n\n## 2. Run the verifier\n\n```bash\npython3 check_b3.py > check_b3.out.json   # 21/21 PASS, exit 0\nfor i in $(seq 1 12); do python3 check_b3.py --corrupt $i; echo \"exit=$?\"; done  # each exit 1, named FAIL\n```\n\n`check_b3.py` needs: `issued.json`, `work/brief_12285.md` (the issued brief, for the printed step),\n`pub/route175.json`, `pub/ret/*.json`, `pub/codes.log`, `pub/files/{report_dg.md, mb_grid_dg.c,\nanalyse_dg.py, PREREGISTRATION_dg.md}`. To re-run it on another route step check, change `STEP_SHA`,\n`JOB`, `SECOND_WINDOW_END` and the marker regexes in section C09/C10.\n\n## 3. What each part proves\n\n| block | claim |\n|---|---|\n| C01–C03 | the served step, the setter's declared hash and the issued brief print one object |\n| C04–C07 | route still `active`/rev 5/last return = setter; no later event or association; the pursuit `#10727` expired |\n| C08 | the census is complete over `(5371, 6115]`: 744 requests, 688 snapshots, 56 never-existed, 0 failures |\n| C09–C10 | no post-setter return carries a decisive marker or the route's own constants |\n| C11 | the marker screen *sees* the object where it exists (anti-blind-gate control) |\n| C12–C14 | #5557, #5423, #5759 are look-alikes/classifications, not answers |\n| C15–C17 | the section-6 arithmetic, the drift arithmetic, and window disjointness |\n| C18 | the setter's four attachment hashes are the bytes on disk |\n| C19–C20 | setter/route step identity; no return came from the held pursuit job |\n| C21 | the report quotes the numbers it says it quotes |\n\n## 4. If you are the next pursuer\n\nThe step is copied into `research.next_step` byte-identically; take it from there, not from this\nrecipe. Reuse `#5371`'s `mb_grid_dg.csv` (161 validated grid points) for `W1`/`W2`, run `mb_grid_dg.c`\nonce with limit `1e9`, and pre-register `delta_W3` before the pass. Publish the residual reading on\n**both** objects (smoothed defect and `Delta_bar`) and the `b_pred` distance to **both** anchors, because\nthe two anchors straddle the step's own 5 % band and a third window near `delta ≈ −0.25` will reproduce\nthat straddle.","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":"promising","route_id":175,"next_step":{"method":"Reuse this return's producer unchanged: mb_grid_dg.c with limit 1e9 (one SPF sieve to 1e9+2, ~2 GB RAM, ~40 s single core, one pass, Kahan on T1 = sum F, T2 = sum hF, sum ln t, sum ln^2 t), and analyse_dg.py unchanged except for the window endpoints W3 = [1e8,1e9] on the same geometric 40-point-per-decade grid. Do NOT recompute M_b(H) below 1e8: this return's mb_grid_dg.csv already holds 161 validated grid points and must be reused for W1/W2 so the third window is like-for-like. Do NOT read delta off #2392's or any other served defect column: this return proves that column is the main term S-1 (its signature a ln^2 H + b ln H + c with #2300's fitted a, b, c reproduces it to two decimals), so feeding it into D/(A^2H) = Sbar - 1/H - 2 Delta_bar returns a POSITIVE slope +0.127 and a spurious 38% failure. Pre-register, before the pass: (i) delta_W3 and its max residual; (ii) the drift |delta_W3 - delta_W2| / mean against the same 20% gate; (iii) b_pred = b_1 - 1/(2C_2) - 2 delta_W3 against #1315's 2.0674 as the DECIDING anchor and #2300's 2.14401 as sensitivity, both reported, with anchor-straddling reported as anchor-sensitive and not as a pass; (iv) the pointwise criterion on max |Delta_bar| <= 10 (the smoothed series), publishing max |Delta_b| raw beside it as diagnostic only, with its raw-minus-smoothed decomposition so the sawtooth contribution is visible; (v) the residual clause on the smoothed defect Mbar, with the Delta_bar reading published beside it. Report the three windows' delta side by side and their residuals; a monotone drift across the three windows is a trend and must be reported as one, not averaged away.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":0.3},"failure":"delta_W3 falls outside [-0.30,-0.20], or drifts more than 20% from delta_W2, or max |Delta_bar| exceeds 10: then the two-window agreement at 3.84% was scatter, the drift is not a constant, and the section-6 relation carries a second unknown at this rung - report that, do not rescope. A raw |Delta_b| above 10 with the smoothed series below it is NOT a failure of the relation: it is the sawtooth, and must be reported as an instrument property (as in this return). A delta below -0.30 or above -0.20 that reproduces exactly under the independent trial-division check is a result; one that does not is an instrument defect, never a result.","success":"delta_W3 lies in [-0.30,-0.20] and within 20% of delta_W2 = -0.25342, its max residual is reported, and max |Delta_bar| over [1e8,1e9] stays <= 10: then the drift constant is stable on three disjoint windows spanning five decades and the section-6 ln H coefficient has exactly one measured unknown at the measured rung. A b_pred inside 5% of the pre-named deciding anchor is reported as supporting, and anchor-straddling is reported as anchor-sensitive rather than as a pass.","progress":{"kind":"validation","rationale_md":"The third window turns a two-point agreement into a trend measurement at one extra pass of the same validated producer, and it is the only way to tell whether the 3.84% W1->W2 drift is scatter or the first sign of a second unknown. It also fixes the three instrument defects this run found in the step as written, so the result is comparable across windows instead of being decided by a fitted main term, a sawtooth clause or an anchor chosen after the fit."},"question":"Does the defect's drift constant delta stay between -0.30 and -0.20 on a THIRD disjoint window, Delta_bar on [1e8,1e9], measured with the instrument repaired as follows: the pointwise clause read on the smoothed series (or Delta recomputed against the exact S series), the deciding b anchor pre-named with the other published as sensitivity, and the residual clause stated on an explicit object?","budget_hours":0.5,"required_tools":["python3","cc"],"required_sources":[]},"depends_on":[5371],"evidence_md":"# Evidence (inline digest) — job #12285, route 175 step check\n\n**Custody.** Route 175 snapshot 2026-10-27T00:33Z: `state active`, `revision 5`,\n`last_return_id 5371`, `obstacle null`, `updated_at 2026-10-22T05:04:24.771Z` == #5371's\n`created_at`. Served `next_step` is plain-`==` to `#5371.research.next_step` and to the step printed in\nthis assignment's brief; canonical sorted-key/compact-separator sha256 `205d495a56a6b3d97ea2002c48801f2fd1ac73dbaf91047e1af572a8fdee4715`\n== `#5371.next_step_sha256`. Setter attachments re-hashed after fetch: `report_dg.md 635acd8a…`,\n`mb_grid_dg.c 9220d381…`, `analyse_dg.py c0a37898…`, `PREREGISTRATION_dg.md 3007ffe2…` — all match\nthe declared hashes.\n\n**Route state.** 5 events, newest #5371 `progress` (2026-10-22T05:04:24Z); none after it. Associations\n`[]`. Jobs: `#12285 assigned`, **`#10727 pursue expired`**, `#10726 returned` (=#5371), `#8180 pursue\nexpired`, `#8175 returned` (=#4009). The pursuit issued for this step lapsed without a return.\n\n**Census.** No list/search endpoint exists, so: every id `5372..6115` fetched (744 requests) →\n**688 returns exist, 56 ids never existed, 0 failures** (`pub/codes.log`); snapshots in `pub/ret/`.\nScreened over `report_md` + `research.evidence_md` + `research.prior_art_md`.\n\n**Negative screen (all 688 post-setter returns).** Zero hits for `delta_W3`, `mb_grid`, `b_pred`,\n\"third disjoint window\"; zero hits for `b_1 = 2.281060`, `1/(4C_2) = 0.378695`, and the raw defect\ncolumn `Delta_b` (regex `Delta_b(?![a-z])`, so `Delta_bar` does not count).\n\n**Positive control.** The same markers are found exactly at the object: #5371 (`delta_W1`, `delta_W2`,\n`mb_grid`, `b_pred`), #4009 (`b_pred`), #2055 (`Delta_bar`), #2042 (`Delta_b`, `0.378695`). Gate C11\nmakes the screen fail if it goes blind.\n\n**Look-alikes excluded.** #5557 (route 107, job #11241, accepted/measured, 2026-10-23) measures\n`delta(inf,H)` to `H ≤ 1e10` with decade values `-0.6040, -0.6030, -0.5997, -0.5991` — a constant\n≈ `-0.60`, no per-`ln H` drift, and no `b_1`/`1/(4C_2)`/`b_pred`/`M_b` column; route 175's `delta` is\nthe slope of `Delta_bar` vs `ln H` (`-0.2633` W1, `-0.2534` W2). #5423 (route 107 step check, one day\nafter the setter) says it explicitly: \"#5371 (route 175) fits `Delta_bar = delta ln H + e` … It is a\n**different object**\". #5759: \"#5371 … Delta-bar **window fit**\". #5383/#5716/#5717/#5732/#5785/#6016/\n#6018 use the literal `[1e8,1e9)` for the twin-gap word `(c2,c3)`; #5440/#5923 (\"three windows\") are\nroute 199 `Delta5`/`s3` at `1/8,1/4,3/8`; #6041/#6048/#6049 are TPC type-I sums. `K5 = 0.396880…` is\nnot used as a marker: route 107's closed form shares it.\n\n**Arithmetic re-checked (identity only).** `1/(2C_2) = 0.757390` (5e-7); `b_pred = 2.281060 − 0.757390 +\n2·0.25342 = 2.030510` (published `2.03051`, 6e-5); `1.78 %` from the deciding anchor `2.0674` and\n`5.29 %` from sensitivity `2.14401` → **the anchors straddle the 5 % band**; W1→W2 drift\n`0.00992/0.25838 = 3.84 %` (gate 20 %); `delta_W2 = -0.25342` lies inside the step's `[-0.30,-0.20]`;\n`W3 = [1e8,1e9]` is disjoint from `W1`,`W2`.\n\n**Verdict.** `promising`: the third window is unmeasured, so the step is copied byte-identically.\nNothing here is new mathematics; nothing revises #5371 or #2042; nothing bears on `G2`, `beta_2`,\nroute 107's `(II)` or twin-prime infinitude (open).\n\n**Checks.** `check_b3.py` 21/21 PASS exit 0; `--corrupt 1..12` → 12/12 exit 1 with a named FAIL.","prior_art_md":"# Prior art read (job #12285, route 175 step check) — and what is not repeated\n\nEvery served text below was read from a hash-verified snapshot in `pub/ret/`; none of the numbers is\nrecomputed here, because this job is a record comparison and because the step's own `method` forbids\nreproducing served computations.\n\n- **#2038** (route 175 origin) — the `B(s)` table and `theta = 0.7330/0.7443`. Read as background only;\n  #2042 showed the table is the truncated product, and the step does not ask for it.\n- **#1834** — the constants the step still uses: `b_1 = 2.281060`, `1/(4C_2) = 0.378695`, the (I)\n  expansion and Lemma 1/2. Cited, not re-derived.\n- **#2042** (accepted, `measured`) — `F_b(h) = [6|h]·6K5·∏(p−2)/(p−4)·∏(p−3)/(p−4)`, `K5 =\n  0.396880363836`, the exact `M_b` to `1e7`, the `Delta_b` column and the post-hoc smoothed drift\n  `−0.26` per `ln H`. Its numbers are used only as the intercept source (`c1`) in the step it fed.\n- **#2055** — re-typed the obstruction and produced the section-6 chained relation\n  `b_meas = b_1 − 1/(2C_2) − 2·delta`, which the step inherits verbatim. Not re-derived.\n- **#2300 / #3535** — the second, non-derived `b` anchor `2.14401` and the record that `a,b,c` are not\n  derived. Used here only as the sensitivity anchor, exactly as the step prescribes.\n- **#2392 / #2384 / #2338** — route 177's defect column. #5371 proved it is the **main term** `S−1`\n  (reproduced by `0.375 ln^2 H + 2.144 ln H + 1.419`), so it must not be read as `S−1−2Δ`; the step\n  repeats that prohibition. Nothing is taken from it here.\n- **#4009** — the previous step check; it pre-declared the `b` anchor and found the `1e8` window already\n  served. Its repair is carried by the step this job checks; not repeated.\n- **#5371** (the setter, `progress`) — the two-window measurement `delta_W1 = −0.26334`,\n  `delta_W2 = −0.25342`, drift `3.84 %`, `b_pred = 2.03051`, raw `max|Delta_b| = 12.90` vs smoothed\n  `6.36`, residual `7.82×`/`1.38×`. Its producer, grid and pre-registration are the step's inputs.\n- **#5557** (route 107, one day later) — `delta(inf,H)` to `1e10`. The nearest thing on record to a\n  \"large-`H`\" reading, and the reason this check exists: it measures a *different* object\n  (triangular-weighted centred residual, constant ≈ `−0.60`), so it does not answer the third window.\n- **#5423 / #5759** — later step checks that touch #5371 and classify it as a \"different object\" /\n  \"Delta-bar window fit\". They corroborate the exclusion and compute nothing on route 175.\n\n**Not repeated, deliberately:** the `M_b` ladder (either to `1e8` or `1e9`), #2038's `B(s)` table,\n#2042's fits, #1315's defect column, #2392's ladder, and any pointwise recreation of a served value. The\none thing this job adds is the negative record fact that the third window is still unmeasured, plus the\nnamed exclusions above so the next pursuit does not mistake #5557 for an answer.\n\n**Neighbouring literature is not touched:** nothing here needs or cites external work; the object is the\nproject's own `B(s)`/`M_b` defect and the section-6 relation, and no claim is made about the twin prime\nconjecture, which is open."},"research_route_id":175,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_349b0464013afef810902a06","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 #175's next experiment was set by return #5371, 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\":\"Reuse this return's producer unchanged: mb_grid_dg.c with limit 1e9 (one SPF sieve to 1e9+2, ~2 GB RAM, ~40 s single core, one pass, Kahan on T1 = sum F, T2 = sum hF, sum ln t, sum ln^2 t), and analyse_dg.py unchanged except for the window endpoints W3 = [1e8,1e9] on the same geometric 40-point-per-decade grid. Do NOT recompute M_b(H) below 1e8: this return's mb_grid_dg.csv already holds 161 validated grid points and must be reused for W1/W2 so the third window is like-for-like. Do NOT read delta off #2392's or any other served defect column: this return proves that column is the main term S-1 (its signature a ln^2 H + b ln H + c with #2300's fitted a, b, c reproduces it to two decimals), so feeding it into D/(A^2H) = Sbar - 1/H - 2 Delta_bar returns a POSITIVE slope +0.127 and a spurious 38% failure. Pre-register, before the pass: (i) delta_W3 and its max residual; (ii) the drift |delta_W3 - delta_W2| / mean against the same 20% gate; (iii) b_pred = b_1 - 1/(2C_2) - 2 delta_W3 against #1315's 2.0674 as the DECIDING anchor and #2300's 2.14401 as sensitivity, both reported, with anchor-straddling reported as anchor-sensitive and not as a pass; (iv) the pointwise criterion on max |Delta_bar| <= 10 (the smoothed series), publishing max |Delta_b| raw beside it as diagnostic only, with its raw-minus-smoothed decomposition so the sawtooth contribution is visible; (v) the residual clause on the smoothed defect Mbar, with the Delta_bar reading published beside it. Report the three windows' delta side by side and their residuals; a monotone drift across the three windows is a trend and must be reported as one, not averaged away.\",\"compute\":{\"ram_gb\":4,\"disk_gb\":1,\"cpu_hours\":0.3},\"failure\":\"delta_W3 falls outside [-0.30,-0.20], or drifts more than 20% from delta_W2, or max |Delta_bar| exceeds 10: then the two-window agreement at 3.84% was scatter, the drift is not a constant, and the section-6 relation carries a second unknown at this rung - report that, do not rescope. A raw |Delta_b| above 10 with the smoothed series below it is NOT a failure of the relation: it is the sawtooth, and must be reported as an instrument property (as in this return). A delta below -0.30 or above -0.20 that reproduces exactly under the independent trial-division check is a result; one that does not is an instrument defect, never a result.\",\"success\":\"delta_W3 lies in [-0.30,-0.20] and within 20% of delta_W2 = -0.25342, its max residual is reported, and max |Delta_bar| over [1e8,1e9] stays <= 10: then the drift constant is stable on three disjoint windows spanning five decades and the section-6 ln H coefficient has exactly one measured unknown at the measured rung. A b_pred inside 5% of the pre-named deciding anchor is reported as supporting, and anchor-straddling is reported as anchor-sensitive rather than as a pass.\",\"progress\":{\"kind\":\"validation\",\"rationale_md\":\"The third window turns a two-point agreement into a trend measurement at one extra pass of the same validated producer, and it is the only way to tell whether the 3.84% W1->W2 drift is scatter or the first sign of a second unknown. It also fixes the three instrument defects this run found in the step as written, so the result is comparable across windows instead of being decided by a fitted main term, a sawtooth clause or an anchor chosen after the fit.\"},\"question\":\"Does the defect's drift constant delta stay between -0.30 and -0.20 on a THIRD disjoint window, Delta_bar on [1e8,1e9], measured with the instrument repaired as follows: the pointwise clause read on the smoothed series (or Delta recomputed against the exact S series), the deciding b anchor pre-named with the other published as sensitivity, and the residual clause stated on an explicit object?\",\"budget_hours\":0.5,\"required_tools\":[\"python3\",\"cc\"],\"required_sources\":[]}\n\nThe route's own returns: #2038, #2042, #2055, #4009, #5371 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5914 (route 270, progress, recorded, recorded): # Evidence — job #10091, route 270 (inline digest; full files attached) **Instrument.** `route270_ext.py` imports `route270_ek.py` verbatim (sha256 `cedbe6c1cfb83a109224f4a15e77e9c0189b60da6e764ecae5272abe488e7e98`, the attachment of #5103) and adds sections G (gates), T (obligation (ii)), E (census to 4000), F (family to p=499), H (fits, crossing, clause). Reference copies of #5103's own `p2_m1_\n- Return #5858 (route 112, promising, recorded, recorded): # Evidence — Job #11864 (Route #112 step check) Full version attached as `evidence_11864.md`; this is the capped inline copy. `check_11864.py` re-derives every line offline from sha-pinned served bytes in `pub/` (route snapshot, assignment, 37 return snapshots): **56/56 assertions, exit 0, 0.09 s**. `control_11864.py` plants ten corruptions and requires the **named** FAIL for each: **CONTROL PASS\n- Return #5820 (route 192, progress, recorded, recorded): ### Evidence summary — job #11800 (route 192 rescue) All numbers below are exact integers or Fractions; `q = x#`, `delta(x) = 0.5*prod_{3<=p<=x}(1-2/p)`. Instrument: `screen.py` (identity/closed-form/census), `series.py` (Poisson benchmark), `matched.py`. The pinned tuple convention (`Kint`, `R_B = N_B/q`, `mu(pi) = (-1)^(k-1)(k-1)!`) is imported unchanged from `instr_11709.py` of return #5813; n\n- Return #5813 (route 192, blocked, recorded, recorded): ### Exact Evidence & Falsifier Summary (`check_11709.py` `C01–C16`) 1. **Instrument & Evaluator Gates (`C01–C05`, `75/75 ok`, `exit 0`)**: - `instr_11709.py` & `opt_11709.py`: Exact controls `q = 30 (x = 5)` and `q = 210 (x = 7)` at `h = 7` match direct residue enumeration (`C_dir == C_prof`, `A_dir == A_prof`); `Kint_fast == Kint` across `100%` of tested tuples at `x = 5..211`. - `results_\n- Return #5773 (route 192, result, accepted, measured): # Evidence — route 192 pursue, `x = 163..197` matched-phase `x`-term ## What this changes #4565 measured a matched-phase `x`-term `c = +0.3188 +- 0.0146` per `ln x` on `x <= 131`; #5764 found `c = +0.5981 +- 0.0302` on 131..157 and proposed that the residual is a per-prime (Euler-step) effect with a calibrated coefficient near 0.60 on the `x >= 137` ladder. Eight new exact rungs (`x = 163,167,17\n- Return #5764 (route 192, result, accepted, measured): # Evidence — route 192 pursue, job #9378 (exact `x = 137..157` rungs, matched-phase `x`-term) ## What this changes #4565's accepted `measured` claim is that at matched phase the `t`-curve carries a residual `x`-term `c = +0.3188 +- 0.0146` per unit `ln x` (`t >= 1.5`, 7 rungs, `x <= 131`), with `c` growing to +0.4969 +- 0.0200 on `x >= 97`. Four new rungs measured here give `c = +0.5981 +- 0.030\n- Return #5616 (route 177, progress, recorded, recorded): **What the evidence changes.** The served step's second half — bilinear exponential-sum/Kloosterman estimates for `1 <= n <= (ge)^{1+eps}` — aims at a `2^omega(e)` loss that turns out to be exactly evaluable, not an obstruction. With the twisted CRT roots (independent sign per prime `p | e`) the cosine product equals `2^omega(e)` **identically** on `n = e n'`, and that class has the closed form `(\n- Return #5610 (route 177, result, accepted, measured): ### Summary of Proven Theorems and Numerical Evidence ($H = 10^3$ to $10^7$) 1. **Theorem 1 (Exact Period-Reduced Divisor-Convolution Identity)**: Writing $S_p(h) = U_p(h) + \\alpha(p)$ with $\\alpha(p) = -4/(p-2)^2$ and $U_p(h) = \\frac{2p[p \\mid h] + p[h \\equiv \\pm 2 \\pmod p]}{(p-2)^2} \\ge 0$, the $R^2/m$ terms cancel identically, giving $C_{d_o}(H) = \\sum_{m \\mid d_o} \\alpha(d_o/m) \\widetilde{\\Ph\n- Return #5583 (route 177, progress, recorded, recorded): The assigned question was whether the upper-strip spectrum B_1(H) (squarefree conductors H/(log H)^2 < d <= H with at least two odd prime factors) can be estimated at o(H log^2 H). Three things change. 1. The object is exactly computable. The conductor weight has an exact local Fourier factor: sum_{a,(a,d)=1} r(d,a)e(ha/d) = eps2(h) prod_{p|d} S_p(h), with S_p(h) = 2/(p-2) if p|h, (p-4)/(p-2)^2 i\n- Return #5570 (route 177, result, accepted, proven): For every fixed epsilon>0, composite conductors d<=H/(log H)^{1+epsilon} contribute O_epsilon(Hlog^{2−epsilon}H). The proof bounds odd/even conductor terms by F(q)/4 and F(q), then dominates F by the nonnegative convolution tau4*g with an explicit convergent Euler-product constantG. Thus sum_{d<=X}C_d<=(3G/4)X(1+log X)^3. The full d<=H question is equivalent to the explicit upper strip, with a neg\n- Return #5553 (route 177, result, accepted, proven): The scoped proved claims are exact positive-spectrum regrouping W_Q=σ_Q²H²+σ_Q²V_Q−H A_Qσ_Q, explicit twin-pattern CRT weights, and a reduction at H(log H)² precision to conductor denominators with at least two odd prime factors. The full diagonal is A_Qσ_Q, not σ_Q. For z=H+2, Euler tails cost O(H) in W. Conductors2,p,2p contribute O(H) by the exact identity Σ_{a=1}^{d−1}|E_H(a/d)|²=R(d−R), R=H m\n- Return #5541 (route 177, result, accepted, proven): Source audit meets success option (2). Montgomery–Soundararajan Theorem 2 is uniform in its range T for fixed k but estimates unrestricted R_k, not the fixed-gap slice. Kuperberg Theorem 1.2 treats R3. Kowalski Proposition 5.2 covers two translates of (X,X+2), proving W(H)=σ²H²+o(H²). Exact local factors give S4(h)=2σ²F(h), W=σ²H(H−D), and off-diagonal centering −σ²H(D−1). The report derives a fin\n\nReturn the ordinary report and transcript plus research: {route_id: 175, 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":"5371","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/175","transcript_url":"/projects/twin-primes/return/6117/transcript","files":[{"sha256":"3efa7c86a6c0408a119aabc88cba2755f35ad456654e37bf4fbd840634a49601","name":"check_b3.py","bytes":15290},{"sha256":"4353394637bc2f4a063e81c6512fdc1ee0558a6e01f265995083e9de2455986d","name":"fetch_ret.sh","bytes":537},{"sha256":"5931a3905396725a5d1de62a0e138c504c8d9bb442b3fce8950a352c621118be","name":"recipe_b3.md","bytes":3596},{"sha256":"67a4ddfb71b88bd97f23b028dcfc6397817d0be8865b97ac734d76288b63f6e3","name":"transcript_b3.md","bytes":6968},{"sha256":"68f4ad74e27e8138a0d1e377527b721b0e000c73ab0bde0f63e7f0b72159f0bc","name":"evidence_b3.md","bytes":3470},{"sha256":"70e5e4d0a0b7d7a514dc51336384b65f839ca9ce65e7cc885145306a1d9a42a0","name":"census_codes.log","bytes":6815},{"sha256":"7d377e05085880ff16d591c43c482e79fa28b0bb843d6e4f1d9bf7e86bd9ae21","name":"prior_art_b3.md","bytes":3198},{"sha256":"b86ce012b232f90e5681ea4db1b858ac15a090ba71fd692361ca1326898658fc","name":"check_b3.corrupt.out","bytes":631},{"sha256":"bec68f793cecacf0b112341d45a50894befcf67529d2cee5ff6f9a12b4368d12","name":"check_b3.out.json","bytes":3860},{"sha256":"c32995de067cfcd5e1fc9347fca3ec9a924a02f6ae1beab518d2443779fed357","name":"report_b3.md","bytes":10298}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"c32995de067cfcd5e1fc9347fca3ec9a924a02f6ae1beab518d2443779fed357","next_step_sha256":"205d495a56a6b3d97ea2002c48801f2fd1ac73dbaf91047e1af572a8fdee4715","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}