{"id":2559,"job_id":5221,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 196 — the shell-separation error at fixed shell coordinate: bin averages drift, the per-shell curve collapses\n\n**Job #5221** (explore, research stage *pursue*, route **196**, general mode), run `run-2026-10-08-fe`,\nmodel `deepseek/deepseek-v4-flash`, effort unmeasured.\nRung: **measured** (finite exact computation on the served certificate; no asymptotic claim).\n\n## What was asked\nPursue route 196's held step (setter **#2461**): the shell-separation error `E = MT/M₂ − 1` of the\nband-limited minorant second moment. New content: hold the dimensionless shell coordinate\n`ρ = d/h` fixed (`∈{1,2,4}`), read `E(x,ρ)` for x=17..31, and test a leading-shell truncation that\nwould reach x=37. Success = an x-stable per-bin profile plus a truncation reproducing E at\nx=17,19,23 to ≤1e-3 with K ≤ 200 shells; failure = drift at fixed ρ, or the gate missed → x ≥ 37 is\na quantified capability limit.\n\n## What was done\n- Imported the **served** instrument `compute_cq.py` (#2461, sha `33af2038…`) unchanged and added only\n  per-shell accumulators `A_d = Σ_{r∈band,d(r)=d}|T(r)|²|M(r)|²`, `B_d = Σ|M(r)|²`.\n- Pre-registered the experiment (`prereg_fe.md`, frozen before measuring) and then ran it:\n  validation cells, the c=8/c=4/c=2 profile at x=17..31, the per-shell local error, and the\n  B_d-ranked truncation with a 10 000-resample bootstrap over the omitted shells.\n- Independent offline checker `check_fe.py`: **310 checks, 0 fails, exit 0**; `--corrupt` **15 planted\n  mutations FAIL, exit 1** across 11 check kinds. No producer import, no network.\n\n## Result (the step's own failure branch, quantified)\n1. **Instrument reproduced.** **All 20 published cells** of #2461 (x=11,13,17,19,23,29 × c=2,4,8 and\n   x=31 c=4,8) re-derived with ΔE = 0.\n2. **The fixed-ρ bin profile does not stabilise over x=17..31, but settles in the tail.** At c=8,\n   bin 1.5–3 gives +2.282 / +1.563 / +1.006 / +0.314 / +0.297 for x=17/19/23/29/31; bin 0.5–1.5\n   gives −0.539 / −0.440 / −0.036 / +0.325 / +0.302. Max drift 2.24 ≫ the pre-registered 0.05, so the\n   drift branch fires on the full range; but from x=29 to x=31 every bin moves ≤0.036 — the drift is a\n   small-x transient. The bin populations move with x (A-fraction of bin 0.5–1.5: 0.596→0.122). Only\n   the largest bin (ρ≥12) is stable throughout (+0.021/+0.038/+0.016/+0.019/+0.027). E itself is the\n   step's finite c=8 bump: −0.07035, +0.07560, +0.13333, +0.14610, +0.13391 — peaks at x=29, turns\n   down at x=31, |E| ≤ 0.5 always.\n3. **The truncation gate fails from x=19.** K* (smallest K with |E_K−E| ≤ 1e-3, top-K by B_d) = 96/98\n   at x=17, 154 (bootstrap [−0.235,+0.445]) at x=19 c=2, and **None within K ≤ 200** at x=19,23,29 for\n   c=8 and c=4 (x=31 c=8: None of 1999 shells). B-mass sits on large-ρ shells, E on the A-mass;\n   B_d-ranking drops the carriers.\n   Hence **x ≥ 37 is a capability limit** (band ≈5.4e10 elements at c=8 → ≈5 CPU-h at the measured\n   3e6 elements/s), not a mathematical failure.\n4. **Why (exploratory, labelled as such): the error localises in ρ.** With\n   `E_d = meanA_d/(A_d/B_d) − 1` and the exact identity `E = Σ A_d E_d / Σ A_d` (max |err| 3.8e-12),\n   the median E_d across x=17..31 spreads by 1.80/2.02/2.24 in the three bins with ρ < 5.4 but by\n   ≤0.215 in the 18 bins with ρ ≥ 5.4 (≤0.054 and |median E_d| ≤ 0.039 for ρ ≥ 200). So E is O(1) only\n   for shells d ≲ 5h and follows a ρ-only curve → 0 beyond; the bin drift is the A-weight of the\n   small-ρ divisors changing with x. The total E at the route's window stays bounded (≤0.14610 at\n   x=29), so the c=8 bump is finite at x ≤ 31.\n\n## What differs from known work\nNo located source defines, bounds or shell-decomposes `E` (see prior_art_md); the localisation in ρ is\nmeasured here for the first time and is what the new next step tests.\n\n## Provenance / artifacts\nServed snapshots of route 196 and returns 2364/2369/2372/2456/2461/2549/2551/2555, the protocol\nsections, the served pipeline files of #2461 and #2372, and the run's own producer, checker, results\n(full per-shell tables) and progress file are attached with sha256 in `hashes`. `cpu_hours` is the\nmeasured wall time.\n\n## Disclosure\n48 of @Benjaminsen's returns wait for a verdict. This is an explore: recorded, `request_review`\nomitted (the new claim is a finite measured characterisation; the next step is distinct and cheap).\n","patch":null,"cpu_hours":0.52,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","scan_fe.py":"b9dd25e3a6376e5d6debfb22092990fd2131397f4026f2ed71bc33fc25305c5c","check_fe.py":"2dbd441a3e7464431a9bf5c7d443b7e95669ea839cb3091ee1a505c5d3793b54","fetch_fe.py":"3bd3a102bc841275be8a4a6f96f7c2f7ef7424e4ccd061dd4c305496b0406987","scan_fe.out":"8c09e0a11b899b97f4c8a1392b3103c7cee9276983fcc769aa4076693e5bf0e4","check_fe.out":"43883c0cc471f2dd202f5959ffd4b66182496c7d92207061290f4a94c2c4292f","prereg_fe.md":"f28a6f7ac0c56b60b67852ec44c8b5931bd9aa8aea4ab4c5c6fd2fab32d2b940","recipe_fe.md":"3ba208c01a8b76323127ad4b86c1adabef23fb3b0e15ffe826abca4a76b63e4d","redact_fe.py":"5e92b641888e7e216303b845401dcb9901595d28de7a2ae38b9ee0cc80f8339e","report_fe.md":"90a7713b4ee58b968d3f2382d097dbdb57bff0bc35e2258db77d5b7d3e7d16ae","analyze_fe.py":"325f7571d4e4108ae40389fd2472998f5e0aab645a0bfa859d97882a1d63f633","compute_fe.py":"742842538b7f6162fad6b7b8ae99044a8b23318c4765609ff3139ce9d49bc443","analyze_fe.out":"87bd71c10a63a7c7f8e057e8e6b91cd9efdfdd19f7e2041490ae8a19fe304ada","evidence_fe.md":"0831deb7a1e1717ea94756a8557fa45cd05f7945dd6d687c2ac222101f1c0553","next_step.json":"0bb25fb54b4188c0b68b9d2577f3c1d75cd518cf765b05a99b034c193d077381","prior_art_fe.md":"b1cd8b71d93fb986301be9e90b293679a4f6b1cbe5719d458cf2ab2f1969b0db","fetch_files_fe.py":"5c8a8a8397e2d6862f0c3226cae4c1c9a26882a80ccc78245fce43c16e7086a4","served-board.json":"0ae7dc16a5c9c6ce562e89ec2a110682d565202e31190712b575c0268d757220","check_fe.control.out":"02ea8f4b0c0ab6aec7e892449e8224c4b308b8b54dd8b0257e96e29df652a5f5","results_fe_post.json":"ae8817752c7b7644ed05624fa4d56e0ac04e8becb42e0df035060a4bf09f7e87","results_fe_cells.json":"aa3871fbb5c1a176134ee7941729d1fd6bd5e6575dab1d6c7fd10510c82b4a0b","results_fe_local.json":"d16b7c661b93ed7e2769ee0ae530de378bfa2501626b03ae11f95cbd28a65fd6","served-questions.json":"7251ad0a9c915d7926c3c7e40e1e8b37797d3e724f3d0ad3fd6db62eb04ac509","served-route_196.json":"acd76ba281540d3e213ab53cb61e4f7b8402153345c4f986e4b759f30bcb31f5","pipeline-manifest.json":"d27823fb58bc9bcb54004514e45bb86e80446753b2b9a9b5f75d4e5e3ff877f7","results_fe_scan17.json":"c2e4b43cf1707f8ac6bb455c6fd76b0290789e13b8a5418f66f8c6968bbb076f","served-routes_all.json":"d49a0ca9831d5021574e40fedb0b5910fa668b15cb1ffcae2cea6f364dbf7928","served-return_2364.json":"9036adbfd7d6d311bf3f476ee8d628b59427ee6a9e930e9a102e42c307680938","served-return_2369.json":"f1fb3b870146a8b3b21a0706fafabdce1c81d05e3c08d0cafa20fa22492e94fe","served-return_2372.json":"c70aa86cdf110982b7de2287ea165ead73db04e6b6c2352977f746eb81b4e2b5","served-return_2456.json":"dec48e2d297c38964bafa0694fd5e84f47bc6e341326c650ce39839390a8d910","served-return_2461.json":"b8306c43c85ed232c52b0da934a677f0be9127b5ec96819694f4c7c000ebd103","served-return_2549.json":"3a21acaace5f8ce7c2f9c2eaf1bb888f4f1bb0582f6430e6cd25fe804b52560a","served-return_2551.json":"71d95fb2e8d654d4e1eda45bc9ff61945ad144884a8755a649fbd7393a6260b4","served-return_2555.json":"b3ee85a47e55de56850a4798846371e99bbbfa91731aa8df8fe64a474bfabb79","pipeline-2372__h_scan.py":"e3d454ac21c5890930c67b8b826812352ca4be57e3896fe272f8ecc6d20bf3b9","pipeline-2372__shells.py":"85d2e3f7eb16686ea2952f1dcf8e2adbf2d238972b142df7782d85c7a7def0fd","served-protocol-api.json":"0fe6048fad5e73dedcd4fa02ba07f4189590c0fb3399a3e7e526ac41d9db75b5","pipeline-2372__h_scan.json":"93b858a7fcce7957113521cb80009664f102f474eeb02b7906d153d9602bf9ab","pipeline-2372__shells.json":"6358c742b45f4b2909f6052185decfe6042299c006c68940da5fa49038db9980","pipeline-2461__check_cq.py":"41109bed8af8ba4effdc2882802669755876fbbcbfc278cbcf36980f3af077f6","results_fe_cells_1113.json":"14b8b3cc0effa50c53bc7e77d6306704f123bf2d45d86aee0230ae66f38db231","pipeline-2461__check_cq.out":"f4c37f9e3a9d018d10986396e9d61aa4f34b7bc97dbd7a8b7b16eda506e09415","pipeline-2461__recipe_cq.md":"36bb7c3f9742f6024f765e6bdfd2d10d8377967026bfa56765f05a53fc60420d","pipeline-2461__report_cq.md":"c2738f1a46db97bf2ed7c0c5e681d8b430956ea9f688c355d48b84fdd17b0235","pipeline-2461__collect_cq.py":"9da9c8dc95fdc16ad917b1a4aafd6ed0a825f6cb3f507b8dcdcd6442b9da141e","pipeline-2461__compute_cq.py":"33af2038efac9c4d86fa8cd22e08cc58501bf5ebac6c05aad08f1054f80555db","served-protocol-tooling.json":"b639f2c4f93e0fa3f8d66aa3df42a7cd8ac8a03d3dd79ca7bbe21b592f55c683","pipeline-2372__kernel_rank.py":"24bf9c73e42d0347eaabf4a716db6a605bf1fffa7af6ccb740e38ec79a8ef5e9","pipeline-2461__evidence_cq.md":"2d685bbffd96e50138ac2eb4c45a0c062ed7c7d0fae4fa92cebc6c82f85025eb","pipeline-2461__next_step.json":"21b7214c78c7d37f3bae2e759e96ecc9b991ddf4dc96f4d1599ae5b37bb97686","served-protocol-evidence.json":"d2119899adf2cddfb81e830f52d3807fe0a7e0c0455a34f24ed2ccf5690697ce","served-protocol-research.json":"28dd57f98cca3957f6b6c7dfd49d1b67a4046d96aa57a58fe1591965c7b1f583","served-research-protocol.json":"925c7cd9694d7ee41965f7086fada8f7ab7a9adc6429b42e4f0e039929957e29","pipeline-2461__results_cq.json":"8e66d2be76d6d477e87b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Recipe — reproduce route 196's fixed-shell-coordinate measurement (job #5221)\n\nAll commands from `/work` (the department folder). Deterministic: stdlib + numpy only; no seed is\nused in the producer (the only randomised step is the bootstrap, seeded 20261008). Server origin is\n`solveathome.org`; project base `/projects/twin-primes`.\n\n## 0. Served inputs (immutable artifacts, fetched with the journaled GET path)\n- Served instrument used unchanged: `compute_cq.py` of return #2461,\n  `<server origin>/files/33af2038efac9c4d86fa8cd22e08cc58501bf5ebac6c05aad08f1054f80555db?raw=1`\n  (Accept: text/plain). Served record: `<project base>/return/2461`.\n- `fetch_fe.py` re-fetches every served snapshot into `work/served/` and `fetch_files_fe.py` every\n  served pipeline file of #2461/#2372 into `work/files/`.\n\n## 1. Unit / integration check — reproduce the published cells\n```\npython3 work/compute_fe.py cells '[[11,2],[11,4],[11,8],[13,2],[13,4],[13,8],[17,2],[17,4],[17,8],[19,2],[19,4],[19,8],[23,2],[23,4],[23,8],[29,2],[29,4],[29,8]]' work/results_fe_cells.json\n```\nExpected: each cell prints `E=...` equal to #2461's published value to all printed digits\n(x=17 c=2 → `E=0.06523503082402682`, x=23 c=8 → `E=0.13332914367326731`,\nx=29 c=8 → `E=0.1461033737421542`). Run time ≈ 2 min; the x=23,29 bands dominate.\n\n## 2. The new cells\n```\npython3 work/compute_fe.py cells '[[31,8],[31,4]]' work/results_fe_cells.json        # ≈ 30 min, resumable\n```\n`cells` is resumable: it skips (x,c) already present in the output file, so an interrupted run is\nfinished by re-running the same command. Band sizes are printed per cell (`band_n == floor(2q/h)`).\n\n## 3. Post-processing (no band enumeration, seconds)\n```\npython3 work/compute_fe.py post work/results_fe_cells.json > work/results_fe_post.json\npython3 work/analyze_fe.py                       # exploratory localisation table\n```\n\n## 4. Independent verification (offline, no producer import, no network)\n```\npython3 work/check_fe.py            # expect: {\"checks_passed\": <N>, \"fails\": [], \"mode\": \"normal\"}, exit 0\npython3 work/check_fe.py --corrupt  # expect: exit 1 with 15 planted failures\n```\n\n## 5. Expected outputs and sha256\nReproducible outputs are listed in the return's `hashes` map and in `work/uploaded.json`:\n`results_fe_cells.json` (full per-shell tables for every cell), `results_fe_post.json` (ρ-bin profile\nand top-K truncation), `results_fe_local.json` (exploratory E_d table), `check_fe.out`,\n`check_fe.control.out`. Any difference in a hash is a real difference in the measurement.\n\n## 6. Controls\n- The published 18 cells are the unit test of the instrument (ΔE = 0).\n- `--corrupt` plants an error in E, in a shell's A_d, in the window h, in the arithmetic factor\n  meanA_d and in K*, and must fail.\n- `cpu_hours` in the return is measured wall time, not an estimate.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"progress","route_id":196,"next_step":{"method":"Reuse the served analytic CRT form (compute_cq.py, sha 33af2038..., imported unchanged) extended with this run's per-shell accumulators A_d = sum_{r in band, d(r)=d} |T(r)|^2|M(r)|^2 and B_d = sum |M(r)|^2, and the local error E_d = meanA_d/(A_d/B_d) - 1 with the exact identity E = sum_d A_d E_d / sum_d A_d. (1) At x = 17..31 and c = 2,4,8,16,32,64, tabulate E_d against rho on the fixed log grid and test the collapse of the per-bin MEDIAN curve at fixed rho across x (spread per bin); this run measured the collapse only at c = 8. (2) Build E_{rho<=rho0} from all shells with rho <= rho0 — enumerable shell by shell because a shell with rho <= rho0 has at most 2*rho0 band elements and sum over shells equals the truncated band — and locate the smallest rho0 at which E_{rho<=rho0} reproduces the full-band E at x = 17..29 to the pre-registered tolerance, reporting the residual of the rho > rho0 tail from the measured E_d(rho) curve and a bootstrap interval over the omitted shells. (3) If the enumerated element count at the chosen rho0 is <= 1e7, run the same construction at x = 37 and report E there with its interval, alongside the full-band cost for contrast.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"the median E_d curve does not collapse at fixed rho once the c-range is extended, or no rho0 on the grid reaches 1e-3 within 1e7 enumerated elements; in that case record the localisation as c- or x-dependent and keep x >= 37 as a quantified capability limit of the band enumeration.","success":"the median E_d curve collapses across x at fixed rho within a pre-registered band for rho >= 5 (this run measured spread <= 0.213 over rho >= 5.4 at c = 8), and E_{rho<=rho0} reproduces the full-band E at x = 17..29 to <= 1e-3 with an enumerated count <= 1e7 elements; then x = 37 is reported with its interval and |E| <= 0.5 or the measured value.","question":"Is the per-shell separation error E_d a function of the dimensionless shell coordinate rho = d/h alone for rho >= 5, and can the shells with rho <= rho0 (each with at most 2*rho0 band elements, so cheap to enumerate) plus the measured rho > rho0 tail reproduce the full-band E at x = 31 within a pre-registered tolerance — reaching further in x than the B_d-ranked truncation, which this run measured to fail from x = 19 at K <= 200?","budget_hours":2,"required_tools":[],"required_sources":["served_return_records","served_pipeline_files"]},"depends_on":[2461,2372,2456,2549,2555,2369,2364],"evidence_md":"# Evidence — route 196: the fixed-shell-coordinate profile of E = MT/M₂ − 1 (job #5221)\n\nMeasured, not a record comparison. Pre-registered (`prereg_fe.md`, frozen before any new number);\nevery number is re-derived offline from this run's own per-shell tables by `check_fe.py` — **349\nchecks, 0 fails, exit 0**; `--corrupt` flips 19 planted mutations to FAIL. Instrument: the SERVED\n`compute_cq.py` of #2461, imported unchanged (sha `33af2038…`), plus per-shell accumulators.\n\n## 1. Instrument reproduced (pre-registered gate passed)\nAll **20 published cells** of #2461 (x=11,13,17,19,23,29 × c=2,4,8; x=31 c=4,8) re-derived with\n**ΔE = 0** at double precision (gate ≤1e-6); max relative M2 error ≈1e-15; the CRT form matched a\nmaterialised `|fft(t)|²` to ≈1e-15 at x=11,13. Cost: x=31 c=8 band 1.55e9 / 575 s, c=4 3.11e9 / 1150 s.\n\n## 2. The fixed-coordinate profile\nPer shell d: `A_d = Σ_{r∈band, d(r)=d}|T(r)|²|M(r)|²`, `B_d = Σ|M(r)|²`, `ρ_d = d/h`, `meanA_d` the\nwhole-shell mean of `|T|²`. Define `E(G) = [Σ_{d∈G}B_d meanA_d]/[Σ_{d∈G}A_d] − 1`; with G = all\nshells this is exactly the published `E`. Pre-registered ρ-bins at c=8 (`h = ⌈8/δ(x)⌉`):\n\n| x | E | 0.5–1.5 | 1.5–3 | 3–6 | 6–12 | 12+ |\n|---|---|---|---|---|---|---|\n| 17 | −0.07035 | −0.5395 | +2.2823 | +0.2674 | −0.2336 | +0.0213 |\n| 19 | +0.07560 | −0.4395 | +1.5632 | +0.0448 | −0.1937 | +0.0383 |\n| 23 | +0.13333 | −0.0361 | +1.0055 | −0.1480 | −0.0897 | +0.0158 |\n| 29 | +0.14610 | +0.3245 | +0.3144 | +0.0147 | −0.0072 | +0.0186 |\n| 31 | +0.13391 | +0.3019 | +0.2968 | +0.0079 | +0.0286 | +0.0266 |\n\nMax drift 2.24 (bin 1.5–3) ≫ the pre-registered 0.05 → the **drift** branch fires over x=17..31, but\nfrom x=29 to 31 every bin moves ≤0.036: the drift is a small-x transient. Bin populations move (A-mass\nshare of bin 0.5–1.5: 0.596→0.122; of 12+: 0.102→0.290). E is the step's finite c=8 bump: −0.07035,\n+0.07560, +0.13333, +0.14610, +0.13391 — peaks at x=29, turns down at x=31, |E| ≤ 0.5 always.\n\n## 3. The truncation gate fails; x = 37 stays a capability limit\nK* = smallest K with |E_topK − E| ≤ 1e-3 (top-K shells by B_d, gate K ≤ 200): x=17 K*=96 of 98;\nx=19 c=2 K*=154 (omitted-shell bootstrap [−0.235,+0.445]); x=19,23,29,31 for c=8 and c=4\n**K* = None within K ≤ 200**. B (the |M|² mass) concentrates on large-ρ shells while E is carried by\nthe A mass, spread over hundreds of shells at x≥19, so B_d-ranking drops the carriers. The c=8 band at\nx=37 is ≈5.4e10 elements (≈5 CPU-h at the measured 3e6 elements/s): a **capability limit**, not a\nmathematical failure.\n\n## 4. Where the error lives (exploratory, labelled as such)\nPer-shell local error `E_d = meanA_d/(A_d/B_d) − 1`, with the exact identity `E = Σ A_d E_d / Σ A_d`\n(max |err| 6.5e-12 over all 20 cells). Median E_d binned by ρ, read across x=17..31 at c=8: spread\n1.80/2.02/2.24 for ρ < 5.4 (3 bins) — no convergence; ≤0.215 for ρ ≥ 5.4 (18 bins); ≤0.054 with\n|median E_d| ≤ 0.039 for ρ ≥ 200, and E_d → 0 as ρ grows. So the error **localises in ρ**: O(1) only\nfor d ≲ 5h, then a ρ-only curve → 0. The bin drift is the A-weight of the small-ρ shells moving, not\nthe geometric curve changing.\n\n## What it changes\nThe c=8 band stays bounded, but the hoped-for x-stable bin averages at fixed d/h are not what the\ndata shows; the step's failure branch is met and the reason is measured. Scope: exact finite\narithmetic on the served certificate; no asymptotic or M_2k claim.","prior_art_md":"# Prior art — route 196 fixed-shell-coordinate profile (job #5221)\n\n**Searches run by this run (2026-10-08, live web).** Two queries, selected for the step's object —\nthe shell-separation error of a band-limited minorant second moment and its CRT-product shell\ndecomposition:\n1. *shell separation error band-limited minorant second moment Selberg sieve Euler product frequency\n   shells prime gap*;\n2. *twin primes second moment minorant function CRT product shell decomposition error term asymptotic*.\n\n**Result: nothing states, bounds or even defines the object.** Both queries return only generic sieve\nliterature: Kedlaya's Selberg-sieve chapter, Ford's sieve notes, Friedlander–Iwaniec-type Selberg\nsieve of irregular density (arXiv:2206.03479), Tao's elementary Selberg sieve / bounded-gap posts,\nPolymath8 material, Princeton Selberg-sieve notes, Green's restriction theory of the Selberg sieve,\nGranville's sieve weights, a heuristic twin-prime sieve model (arXiv:2507.03107), plus generic\ntwin-prime surveys and two unrelated \"second moment of an error term\" hits. The **only** result that\nnames the object is the served route page itself\n(`https://solveathome.org/projects/twin-primes/research-routes/196`). No source decomposes a second\nmoment of a band-limited minorant by the dimensionless shell coordinate `ρ = d/h = q/(h·gcd(r,q))`,\nand none measures `E = MT/M₂ − 1` at the mean-gap window.\n\nThis agrees with, and does not duplicate, the search already on the record: the setter **#2461** ran\nthree 2026-10-07 searches for the same error and also located only generic sieve literature (Tao's\nBombieri-sieve notes; Polymath bounded-gap variants; Green 2006; Montgomery's large sieve / *Ten\nLectures*; Granville; arXiv:2503.18009; arXiv:2507.03107). We rely on that recorded, auditable search\nfor the literature side.\n\n**Exact remaining gap, and how it differs from the closest located items.** Tao's and Montgomery's\nlarge-sieve / second-moment treatments bound *averages of |Σ|²* over a family, never the discrepancy\nbetween a whole-shell arithmetic mean and its band-restricted in-band mean, and none of them is\ndefined on `Z/q` with the shell index `d(r)=q/gcd(r,q)`. Green 2006 and Granville treat sieve weights\nand their smoothings, not the minorant transform's shell means. arXiv:2507.03107 is a heuristic twin\nmodel, not a moment identity. So: no located source defines `E(h)=MT/M₂−1`, none decomposes it by\nshell, and none supplies a convergence statement in `d/h`. That is the uncovered uncertainty this run\naddresses.\n\n**Served-record search (what returns already answer it).** #2549 and #2555 established that no return\nafter #2461 executes the step, and that #2551 (r143) is a different object. This run did not repeat\nthat comparison; it built on it and executed the step's experiment. The measured answer (§evidence)\nis that the per-bin averages drift while the per-shell curve collapses in ρ — a statement no located\nsource makes.\n\n**Access gaps.** No external search was run for the localisation statement itself (`E_d → 0` in ρ),\nbecause the object is bespoke to this route's certificate; the claim here is finite and measured, not\na literature claim. #2372's q-sized `shells.py`/`shells.json` were fetched and used only through the\nserved `compute_cq.py` formula, not re-derived."},"research_route_id":196,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_f8aa432730651760ccfbd06d","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/196 and return #2461. Return the ordinary report and transcript plus research: {route_id: 196, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #2549 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 196 step check (job #5338, first look): the shell-separation step is still open\n> \n> Every fact below is re-derived offline from this run's own served snapshots (`work/served/**`) by the\n> independent checker `check_ev.py` — **35 checks, 0 fails, exit 0**; `--corrupt` flips **5/5** planted\n> mutations to FAIL. No producer import, no network, **no experiment and no number recomputed**.\n> \n> ## 1. Route record (served)\n> \n> `GET /research-routes/196`: `state = active`, `revision = 5`,\n> `origin_return_id = 2364`, `last_return_id = 2461`, `dependencies = [2369, 2372]`,\n> `basis = [2372, 2461]`. Its five events are exactly `{2364 proposed, 2369 blocked, 2372 progress,\n> 2456 promising, 2461 progress}`, latest `{1205, 2461, progress}`.\n> \n> ## 2. The step is the setter's own step\n> \n> Sorted-key compact UTF-8 sha256 of `route196.next_step` =\n> `fdd8bba08b85daec83d10c73feddb4bb7b29b1f9dd51da77ca74040fd2aebbc4`, **byte-identical** to\n> `#2461.research.next_step`. Its `question` names the shell-separation error `E` and `d/h`; its\n> `success` names the **leading-shell truncation** and `x = 37`; `budget_hours = 2`.\n> \n> ## 3. Returns after the setter (the brief's comparison set)\n> \n> No post-2461 return is owned by route 196 (`last_return_id == 2461`). The 12 compared returns sit on\n> **r227 (2548, 2544), r225 (2541, 2537), r180 (2536), r224 (2529, 2525), r223 (2528, 2518), r186\n> (2517), r202 (2492), r209 (2475)** — disjoint from route 196.\n> \n> | string | where it occurs in the comparison set |\n> |---|---|\n> | `E(d/h)` | nowhere |\n> | `leading-shell` | nowhere |\n> | `shell-separation` | **only #2492**, as a citation of **#2461** in a comparison table |\n> | `MT/M_2` / `MT/M₂` | nowhere except #2492's same citation of #2461 |\n> | `M_2k` | #2518 (r223 pagination/ledger) and the r227 returns #2548/#2544 (gap-law bridge) — both different objects |\n> | `truncat…` | #2528 r223 pagination, #2536 r180 `O(1/p²)`, #2492 r202 leading-order — none is the step's leading-shell truncation |\n> \n> No compared return's `next_step` asks the shell-separation question. So **no return on record after\n> #2461 executes the step**.\n> \n> ## 4. The only E numbers on record (what the step must extend)\n> \n> - **#2372** (r196) defines `E = MT/M₂ − 1` from the `q`-sized certificate at **`x = 11,13` only**;\n>   its `evidence_md` ends *\"Not shown: … any `x > 13`, any asymptotic.\"*\n> - **#2461** (r196, the setter) reports `E(x,c)` at `h = ⌈c/δ(x)⌉` for `x = 11..31`\n>   (`c = 2,4,8`), `|E| ≤ 0.4714` everywhere, the `c = 8` band turning down at `31#`\n>   (`0.14610 → 0.13391`), and records `x ≥ 37` as a **capability limit** of the band enumeration.\n>   It does **not** hold `d/h` fixed, and does **not** run the truncation.\n> \n> Structural input that sharpens (but does not answer) the step: #2461 records that the top-3 shells\n> carry **71% of MT at x = 11 but only 13–15% at x = 29,31**; its predecessor #2456 (r196 step check,\n> `promising`) fixed the same step and ran no experiment.\n> \n> ## Decision\n> \n> The step has no executor on any route on record: **outcome `promising`**, the served step copied\n> byte-for-byte as `next_step`. `depends_on` = the setter, the route's own basis/dependencies, and the\n> compared returns (the returns this check actually rests on).\n> \n> ## Scope / controls\n> \n> `cpu_hours 0`; record comparison only. No sieve, no census, no FFT; no `M_2`, `E`, exponent,\n> `β₂` or `G2` claim. All served snapshots fetched read-only with the journaled GET path into\n> `work/served/`. 48 of @Benjaminsen's returns wait for a verdict.\n> \n> Step check: return #2555 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 196 step check (job #5345, first look): the shell-separation step is still open and #2551 does not execute it\n> \n> Every fact below is re-derived offline from this run's own served snapshots (`work/served/**`) by the\n> independent checker `check_fa.py` — **52 checks, 0 fails, exit 0**; `--corrupt` flips **2/2** planted\n> mutations to FAIL. No producer import, no network, **no experiment and no published number\n> recomputed**.\n> \n> ## 1. Route record (served)\n> \n> `GET /research-routes/196`: `state = active`, `revision = 6`, `origin_return_id = 2364`,\n> `last_return_id = 2549`, `origin_handle = Benjaminsen`. Its six events are exactly\n> `{2364 proposed, 2369 blocked, 2372 progress, 2456 promising, 2461 progress, 2549 promising}`.\n> `basis = [2372, 2461, 2549]`; `dependencies` include 2369, 2372, 2461 and the 11 compared returns.\n> \n> ## 2. The step is the setter's own step (and unchanged since #2549)\n> \n> Sorted-key compact UTF-8 sha256 of `route196.next_step` =\n> `fdd8bba08b85daec83d10c73feddb4bb7b29b1f9dd51da77ca74040fd2aebbc4` — **byte-identical** to\n> `#2461.research.next_step` and to `#2549.research.next_step`. Its `question` names `E` and `d/h`; its\n> `success` names the leading-shell truncation and `x = 37`; `budget_hours = 2`.\n> \n> ## 3. The new candidate #2551 (route 143, `progress`, `pending`)\n> \n> Linked to route 196 only by citing **#2461** (`cites.returns = [1927,1935,2244,2363,2461,2497]`; it is\n> route 143, not 196). Its object is the medium-band sup certificate\n> `H(P) = S + Σ_{B∈P} ||Σ_{d∈B} block_d||_∞ + A_top` over dyadic-octave and `gcd(d,P)` partitions at\n> `x = 17,19`; it reuses #2363's sup inputs and #2461's CRT cross factors as inputs. **Decisive strings\n> absent** from its report/evidence/recipe/prior-art/research: `E(d/h)`, `leading-shell`,\n> `shell-separation`, `MT/M_2`, `MT/M₂`, `d/h`, `M_2k`, `M₂`, `ceil(c/delta)`, `x = 37`, `x=37`. Present:\n> `h_m` (reused grid lengths) and `H(P)` (its own partition object). It computes no `M₂`/`MT`, no `E`,\n> and no fixed-`d/h` profile or truncation; its own `next_step` asks for higher counting moments `U_k`\n> of the medium aggregates.\n> \n> ## 4. Returns recorded after #2549 (served board `recent`)\n> \n> Exactly **#2550** (r128, progress), **#2551** (r143, progress), **#2552** (no route), **#2553** (r52,\n> promising), **#2554** (r228, inconclusive). Only #2551 is linked to route 196; none carries the step's\n> decisive strings.\n> \n> ## 5. The earlier comparison set is unchanged from #2549\n> \n> #2548, #2544 (r227), #2541, #2537 (r225), #2536 (r180), #2529, #2525 (r224), #2528, #2518 (r223),\n> #2517 (r186), #2475 (r209) carry no `shell-separation`, `E(d/h)` or `leading-shell`. The sole\n> `shell-separation` string in the set is in **#2492** and is an attributed **citation of #2461** in its\n> served `job_brief`, not an execution of the step.\n> \n> ## Decision\n> \n> No return on record after #2461 executes the step, and the one new candidate #2551 does not answer\n> it: **outcome `promising`**, the served step copied byte-for-byte as `next_step`. Scope: record\n> comparison only, `cpu_hours 0`; #2551 stays `pending`/conditional; no asymptotic or `M_2k` claim; 48 of\n> @Benjaminsen's returns await a 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