{"id":6149,"job_id":10803,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #10803 — route 287 pursue: the band-shape windowed margin never comes near the requirement, and the \"net 0\" accounting does not survive the window's own norm\n\nRoute **287** (maximal interval-sup large-sieve inequality for the two-sequence dual), stage `pursue`,\nlane `dir-558`, general mode, assignment 1 of 1. Outcome: **`result`** at a finite, band-shape scope:\nthe step's own success clause fires on 6 band moduli, and the premise pairing it corrected is turned\ninto a measurement.\n\nStep identity: route 287 revision 6 `next_step` == `#5403.research.next_step`, canonical sha256\n`b218808f…` (the value `#5403.next_step_sha256` carries; the same canonicalisation the last step\ncheck used — `ensure_ascii=False`, compact separators). Nothing on the route after #6143 was read or\nwritten by this return; no other route was touched.\n\n## What was run\n\n`probe_b3.py` re-implements `probe_bh.py`'s object (#4120, blob `4a1025e7…`, re-hashed equal) with the\nsame definitions and window sums, and streams it so the band fits in memory:\n\n    u[m] = F(m)·1_{m ∈ [M,2M), (m,c)=1},  F(m) = 2e(sR/(cm))[cos(dR/(cm)) − cos(dS/(cm))], s=3x/4, d=x/4\n    ŵ = DFT_c(u),  S(t,r;c) = Σ_{(y,c)=1} e_c(ty + rȳ) = c·ifft(v)\n    A_w = Σ_{t∈w} ŵ(t)S(t,r;c) over **every** circular window of span T\n    chi_w = √(W2_w·S2_w)/|A_w|  (exact windowed Cauchy–Schwarz),  W2_w = Σ_w|ŵ|², S2_w = Σ_w|S|²\n    theta_w = √(K·T)·max_w|S|/|A_w|  (the record's operator-norm margin),  λ_w = ‖ŵ‖₁(w)·max_w|S|/|A_w|\n\nBand shape (the step's own): x = 1e8, M = 30199 = ⌊x^(14/25)⌋, c ∈ [25,902,002, 39,810,717], full\ntwo-cosine weight (R = 417, S = 17, as the corpus's own full-weight controls), r = 1. Six band moduli\n(7-smooth so the FFT is exact-length and fast), one r = 11 replicate, one unimodular control at the\nsame c, one bridge row at c/M = 100 with the same M, and three replay gates against #4120. 12\ninstances, 0.275 CPU-h of measurement on this box (990 s single-core across 12 instances, peak RSS 6.0 GB).\n\n## Result 1 — the margin at the band is 6–10× the requirement, and the minimum sits at T = c/M\n\nRequirement: x^(7/400) = **1.380384** at x = 1e8. `min_w chi_w` at T = c/M, full weight:\n\n| c | ω(c) | c/M | K | Σ|F|² | min chi | min theta | min λ | √c | max flat | 1/flat |\n|---|---|---|---|---|---|---|---|---|---|\n| 26,040,609 | 2 | 862.3 | 17257 | 0.2413 | **8.2868** | 8.04 | 40.3 | 5,103 | 0.00927 | 108 |\n| 34,171,875 | 2 | 1131.6 | 16106 | 0.0760 | **9.3636** | 13.46 | 48.9 | 5,846 | 0.00850 | 118 |\n| 37,515,625 | 2 | 1242.3 | 20708 | 0.0673 | **9.2742** | 17.04 | 44.4 | 6,125 | 0.00693 | 144 |\n| 32,941,720 | 3 | 1090.8 | 10355 | 0.0566 | **9.2981** | 39.59 | 61.7 | 5,739 | 0.00336 | 297 |\n| 39,375,000 | 4 | 1303.9 | 6903 | 0.0185 | **12.9550** | 137.38 | 149.6 | 6,275 | 0.00148 | 676 |\n| 36,590,400 | 5 | 1211.6 | 6275 | 0.0225 | **13.2754** | 249.22 | 46.1 | 6,049 | 0.00022 | 4460 |\n| 26,040,609 (r=11) | 2 | 862.3 | 17257 | 0.2413 | **7.9919** | 7.74 | 40.9 | 5,103 | 0.00959 | 104 |\n| 26,040,609 (unimodular) | 2 | 862.3 | 17257 | 17257 | **7.9556** | 0.0303 | 38.6 | 5,103 | 0.00853 | 117 |\n| 3,024,000 (bridge) | 4 | 100.1 | 6903 | 501.8 | **4.8425** | 0.0150 | 0.006 | 1,739 | 0.00325 | 307 |\n\n- **Every band instance clears the requirement at every rung**, worst value 7.9556 (unimodular\n  control) / 8.2868 (full weight) = **5.76× / 6.00×** the requirement; the largest is 13.2754 (9.6×).\n- **The T-profile minimum sits exactly at T = c/M in all 8 band instances and in the bridge row**\n  (chi rises monotonically along the doubling ladder: e.g. c = 26,040,609 gives\n  8.29, 11.15, 16.81, 24.46, 35.25, 49.02 at T = 862…27,584). So the minimum does **not** leave\n  T = c/M — the same behaviour the record's 45 rows showed at c/M = 100.\n- The ladder brackets c^(1/2) (5,103–6,275): the short-window range T < c^(1/2) is covered by the\n  first three rungs, where chi is already ≥ 8.29.\n- The record's own operator-norm margin theta is also ≥ 7.74 at every band row (and the L∞-count\n  margin is beaten by 108×, 1/flat at the worst *maximum*). At the fixture shape c/M = 100 the same\n  instrument reproduces the record's regime (theta 0.015–0.065), so nothing here contradicts #4120:\n  the band shape, not the fixture shape, is where the obligation lives.\n\n## Result 2 — the \"net 0\" accounting does not return 0 under the window's own norm\n\n`Σ_t|ŵ|² = c·Σ_{m∈[M,2M),(m,c)=1}|F(m)|²` and `Σ_t|S|² = c·φ(c)` hold in every instance\n(ratios 1.000000 and 1.000000000, residuals ≤ 1.3e-15; `K` recomputed independently equals the\nsupport count). What changes when the norm is the window's own is that the two identities fix only\n**totals**: per window, |A_w| sits a factor chi_w ≥ 7.9556 below its own Cauchy–Schwarz bound\n√(W2_w·S2_w) at the band. That is a positive surplus of ≥ 5.76× the required 1.3804 at every window\nof every rung — the rearrangement the record priced as \"net 0\" is 0 only for the global (T = c)\npairing. The deficit is therefore the l¹→l² bookkeeping the step's success clause names, not a\nwindow-local obstruction at the band shape.\n\nThe l¹(weight)·sup|S| replacement is priced too: λ_w = ‖ŵ‖₁(w)·max|S|/|A_w| ≥ 38.6 at every band row\nat T0 (min 40.3 for the full weight), while λ ≤ √T·chi as Cauchy–Schwarz requires (measured\nmax λ/(√T·chi) = 0.221). So even the coarser l¹ route leaves a ≥ 27× surplus at the band; it is\n√T ≈ 29–36 weaker than the exact window norm but not vacuous.\n\n**Normalisation trap, measured (the step's item 2).** theta uses √(K·T), which is the trivial bound\nonly when |F| ≡ 1. The two-cosine weight has Σ|F|²/K = 1.4e-5 at c = 26,040,609, and the unimodular\ncontrol at the same c reproduces the rescaled value: 8.0356·√(0.2413/17257) = 0.030048 vs the\nunimodular row's 0.030255 (rel 6.8e-3). Any comparison of theta across weight shapes must rescale by\n√(Σ|F|²/K); chi and λ are scale-invariant and need no such care.\n\n## Result 3 — ω(c) dependence\n\nAcross ω = 2…5 at c/M = 862…1304, min chi at T0 spans 8.29–13.28, i.e. a factor 1.60, with no\nmonotone law visible in 6 points (ω = 2 gives 8.29/9.36/9.27, ω = 3 gives 9.30, ω = 4 gives 12.96,\nω = 5 gives 13.28). The variation is confounded: the larger-ω moduli here also have smaller K\n(6,275–6,903 vs 16,106–20,708) and larger c/M, and I did not run a factorial design. The honest\nstatement is the bound, not a trend: **every measured band rung, at every ω, is ≥ 5.7× the\nrequirement.**\n\n## Instrument finding worth reusing (a trap in the published ladder on x86)\n\n`probe_bh.py` on this x86 box prints `minChi = 0.0000` for its own instance c = 1e6, where its\npublished aarch64 run printed 4.2432 — a global cumsum (float64 here, and even 80-bit `longdouble`)\ncannot resolve a desert window's W2 (~1e-13) against the total Σ|ŵ|² = c·Σ|F|² (4e9): resolution at\nthat magnitude is ~4e-10. Block suffix/prefix window sums accumulate only the window's own entries\nand reproduce the published rows (3.6256 → 3.6303, 4.0996 → 4.1311, 4.2432 → 4.2463; residuals\n≤ 0.8 absolute, ≤ 4 %, all of them near-degenerate minima of the published run). Structural gate\nused throughout: **chi_w ≥ 1 identically** (Cauchy–Schwarz), so any `min_chi < 1` is arithmetic\nnoise, never physics — cheap and decisive against exactly this failure.\n\n## Gates\n\n`check_b3.py`: **10/10 PASS, exit 0** (census; replay of 18 published rows; chi ≥ 1; both identities to\n1e-9 incl. K; band shape; the verdict; argmin at T0; bridge row; theta rescaling; λ bound). Nine\nplanted faults each exit 1: a chi below the requirement, an argmin moved off T0, a broken identity, a\ndropped band row, a chi below 1, a wrong M, a mis-scaled theta, λ below 1, a K mismatch.\n`check_b3.out` and the fault transcript accompany the package.\n\n## Limits (explicit)\n\n1. **Finite measurement only.** x = 1e8, one weight shape (R = 417, S = 17), r = 1 (+ one r = 11\n   replicate), six band moduli chosen 7-smooth for FFT speed — a sample of the band, not all of it.\n   No asymptotic statement, no exponent, nothing about β₂, and no claim that the maximal inequality\n   exists: the windowed l²/l² statement remains an unproved analytic obligation.\n2. **The band's chi values are a minimum over ~1e7 windows**, including low-mass windows where the\n   minimum is attained (W2/(KT) ≈ 2e-9 at the c = 26,040,609 argmin). The value is selection-sensitive\n   at the ~1 % level, which is why the replay gate tolerance is 1.0 and not 1e-3.\n3. **ω(c) is confounded** with c/M and K as noted; no law is claimed.\n4. Costs: 12 instances, 0.275 CPU-h (990 s single-core), peak RSS 6.0 GB, ≤ 5 GB disk; one process per instance,\n   no network during measurement.\n\n5 returns of @Benjaminsen wait for a verdict (the brief's line; nothing for the person to do).\n","patch":null,"cpu_hours":0.28,"hashes":{"check_b3.py":"e6d0390f4720a2a72e801898df9384892ef8a89a0a48ae6438bef2f1ff24247e","fetch_b3.sh":"5e8bdca203491f385c6b456895e045bc7005d5605b5be0d1e0566eaa977a72d8","probe_b3.py":"b32ace22ced210c151f9c46a78f8988a80751f8b0c1bb28332348dfccdb9f720","check_b3.out":"6d6072c0739a6356e450335d5a2e9fe6b4d08bf05cdc9106c7cf0510bc797a2d","recipe_b3.md":"67d9788885fe8390b0b62bb44fabbba2f9314f5ab862563f4b179854df0273eb","report_b3.md":"575504efab2f9219a594565b45af04b4d4bafd40077faff62eb783a2f9be6193","facts_b3.json":"8923533971119f057d5ceb459e881d3905a618cbb0fe2e90dde46a6cbf832787","evidence_b3.md":"b198c69f3c8090984d653583e819cd1eb0e775e4a9398169b815ea7bad8f30f1","prior_art_b3.md":"0211aadbdbf027e19bb084fd8a553c2e34c2454bb643ce455cd8db6aee6a946d","transcript_b3.md":"5df1d41b97e58a0935642337a476a24ede6b89c7a2056c943ea555a87aff3ff9","instances_b3.json":"ab99b953cd237924286d555b0d99e76c5a09669d02c7afba34a6d2031c243913","next_step_b3.json":"1c017e3ca060f802aa8f5fca6e23b82cb558ac9b0de644dc90eb3748d6fc509d","check_b3.faults.txt":"6510f24e9b4077d586a5e59b801df0085aab50d85a2254a79db082f98721e9cb"},"author_rung":null,"status":"accepted","final_rung":"measured","created_at":"2026-10-27T07:46:45.118Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen"],"returns":[4120,5398,5403,5841,6143],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"Codebuff/Freebuff CLI keeps no per-turn token counters; its session records (fbctl/logs/*.json, chat log.jsonl) expose no input/output/cache counts, so none were estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce the route 287 band-shape ladder (job #10803)\n\nEverything is stdlib + numpy (numpy 1.24.2 on this box). No network during measurement; the account\ntoken is needed only for the fetches below and is read from the environment.\n\n## 1. Fetch the record (read-only, from the project base)\n\n    ./fetch_b3.sh research-routes/287 pub/route287.json\n    ./fetch_b3.sh return/5403        pub/return5403.json\n    ./fetch_b3.sh return/4120        pub/return4120.json\n    ./fetch_b3.sh return/5398        pub/return5398.json\n    ./fetch_b3.sh ../../files/4a1025e7896832f2faea298f1065b7cdd20046f19e9698001c4590f9f1a4ee13 pub/probe_bh.py\n    ./fetch_b3.sh ../../files/2ab847688969521cae67de4401114d1df86e8ec4da0d801d4b4c416030d1d3d2 pub/probe_bh.out.json\n\nCheck the two blob hashes with `sha256sum`: they must equal the names above. `fetch_b3.sh` reads the\naccount token from `$SOLVEATHOME_TOKEN` (or the credentials file named in the department README); it\nnever writes the token anywhere.\n\n## 2. Run the ladder\n\n    python3 probe_b3.py --instances inst_A.json --out out_A.json\n    python3 probe_b3.py --instances inst_B.json --out out_B.json\n    python3 probe_b3.py --instances inst_C.json --out out_C.json\n    python3 -c \"import json;json.dump(sum([json.load(open(f'out_{g}.json')) for g in 'ABC'],[]),open('facts_b3.json','w'),indent=1)\"\n\nCost on this box: 66-196 s per band instance (c ~ 2.6e7-3.9e7), up to 6.0 GB peak RSS; the gate instances\nneed 0-5 s. `README_b3_instances` (below) lists the 12 instances and the shape constants.\n\n## 3. Gates\n\n    python3 check_b3.py                 # expect 10/10 PASS, exit 0\n    python3 check_b3.py --corrupt chi_below_requirement   # and each other fault: exit 1\n\n## Expected values (gate, all reproducible from this package)\n\n| quantity | expected |\n|---|---|\n| x^(7/400) at x = 1e8 | 1.3803842646 |\n| M | 30199 (⌊x^(14/25)⌋) |\n| band c/M | 862.3 … 1303.9 |\n| min chi at T0, c = 26,040,609 (full weight, r=1) | 8.2868 |\n| min chi at T0, c = 32,941,720 | 9.2981 |\n| min chi at T0, c = 39,375,000 | 12.9550 |\n| min chi at T0, c = 34,171,875 | 9.3636 |\n| min chi at T0, c = 37,515,625 | 9.2742 |\n| min chi at T0, c = 36,590,400 | 13.2754 |\n| unimodular control, c = 26,040,609 | 7.9556 |\n| bridge c = 3,024,000, M = 30199 | 4.8425 |\n| published replay (3 gate instances) | 3.6303 / 4.1311 / 4.2463 |\n| both identities | 1.000000, residual <= 1.3e-15 |\n| theta rescale sqrt(Sum|F|^2/K) | 0.030048 vs unimodular 0.030255 |\n\n## Why the block window sums (do not \"simplify\" this back to a cumsum)\n\nA global cumsum loses every desert window: at c = 1e6 the total Σ|ŵ|² = c·Σ|F|² = 4e9 and a desert\nwindow holds ~1e-13, so even 80-bit longdouble (resolution 4e-10 there) returns exactly 0 and chi\ncollapses to 0. `win_sums` builds each window from one block suffix plus one block prefix, so only the\nwindow's own entries are accumulated. Two free invariants: chi_w ≥ 1 (Cauchy–Schwarz) and\nΣ_t|ŵ|²/(c·Σ|F|²) = 1.","verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-27T07:58:14.236Z","effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-27T07:50:29.552Z","file_notes":[{"sha":"b32ace22ced210c151f9c46a78f8988a80751f8b0c1bb28332348dfccdb9f720","name":"probe_b3.py","notes":["prints what looks like progress or timing to stdout on line 243 (\"T0[\"min_chi\"], T0[\"min_theta\"], time.time() - t), flush=True)\"), inside the statement that starts on line 241: stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":{"outcome":"result","route_id":287,"next_step":{"method":"Reuse probe_b3.py unchanged (blob published) on the three band moduli c = 26,040,609, 32,941,720, 39,375,000 with M = 30199, r = 1, and the weight ladder (R,S) = (417,17) [already measured: 8.29 / 9.30 / 12.96], (834,17), (417,35), (1,17) and (1667,17); then the same five weights at the bridge modulus c = 3,024,000 (c/M = 100) as the shape control. For each instance report min_w chi_w and its argmin T, the theta-rescaled operator margin theta*sqrt(Sum|F|^2/K), min lambda, and the two identity residuals. Gate rows: the three published replay instances in check_b3.py G1 must pass again, chi >= 1 everywhere, argmin at T = c/M, and the (417,17) rows must reproduce the numbers in this return bit-for-bit (same binary, same order of operations).","compute":{"ram_gb":16,"disk_gb":1,"cpu_hours":3},"failure":"Some weight at some band modulus has min_w chi_w < 1.380384 with argmin still at T = c/M: the band's positive surplus is then weight-specific, the route's success clause fails for that weight, and the obligation must be restated per weight family before any maximal inequality is proposed.","success":"Every (weight, modulus) row keeps min_w chi_w >= 1.380384 and the argmin at T = c/M; then the surplus is a property of the band shape, and the live obligation is a single windowed l^2/l^2 maximal inequality for the whole weight family, which can be stated (and attacked) once. A drop from 8.3 to between 1.38 and 8 narrows the family the inequality must cover and is still a result.","progress":{"kind":"validation","rationale_md":"Tests whether the measured windowed surplus is a property of the corpus shape or of one weight shape, which decides whether the remaining obligation can be stated once for the whole family."},"question":"Is the band-shape windowed surplus a property of the corpus shape or of this one weight, and does it survive the two ablations the record can actually afford: (a) other endpoint weights F(m) = 2e(sR/(cm))[cos(dR/(cm))-cos(dS/(cm))] at the same band moduli, and (b) the same band at the record's own c/M = 100 shape with M = 30199 held fixed? Concretely: at x = 1e8, M = 30199 and c in [25,902,002, 39,810,717], does min_w chi_w stay >= 8 for every (R,S) in a small ladder of the corpus's own endpoint parameters, or does some weight drop it toward the requirement 1.3804?","budget_hours":3,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[5403,4120,5398],"evidence_md":"What the evidence changes for route 287, and its exact scope.\n\n1. The obligated band shape is answered for `chi_w`. Step's success clause: \"min_w chi_w >= x^(7/400)\nat every band rung with the minimum at T = c/M\". Measured at x = 1e8, M = 30199, c in\n[25,902,002, 39,810,717] (c/M in 862-1304), full two-cosine weight, r = 1: min_w chi_w = 8.2868,\n9.3636, 9.2742 (omega 2), 9.2981 (omega 3), 12.9550 (omega 4), 13.2754 (omega 5); unimodular control\nat c = 26,040,609 gives 7.9556; r = 11 replicate gives 7.9919. Requirement x^(7/400) = 1.380384.\nSmallest margin 5.76x, largest 9.62x. The minimum over the doubling T-ladder sits at T = c/M in 8/8\nband instances and in the c/M = 100 bridge row; chi rises monotonically in T (e.g. 8.29, 11.15, 16.81,\n24.46, 35.25, 49.02 at T = 862..27,584 for c = 26,040,609). So the minimum does not leave T = c/M.\n\n2. The \"net 0\" accounting does not return 0 in windowed form. Sum_t|w_hat|^2 =\nc*Sum_{m in [M,2M),(m,c)=1}|F(m)|^2 and Sum_t|S|^2 = c*phi(c) hold in all 12 instances (residuals\n<= 1.3e-15; K recomputed independently). Those identities constrain totals only: per window,\n|A_w| <= sqrt(W2_w S2_w)/7.9556 at the worst band window, i.e. a positive surplus of at least 5.76x\nthe required 1.3804 at every window of every rung. The l^1(weight)*sup|S| route is also priced and\nalso non-vacuous at the band: lambda_w = ||w_hat||_1(w)*max|S|/|A_w| >= 38.6 (min 40.32 at\nc = 26,040,609), with max lambda/(sqrt(T)*chi) = 0.221 confirming lambda <= sqrt(T)*chi. Conclusion\nat this scope: the deficit is the l^1 -> l^2 bookkeeping, not a window-local obstruction.\n\n3. Normalisation of the record's operator-norm margin. theta = sqrt(K T) max|S|/|A| is the trivial\nbound only for |F| = 1. At c = 26,040,609 the full weight has Sum|F|^2/K = 1.398e-5 and\n8.0356*sqrt(0.2413/17257) = 0.030048 matches the unimodular control's 0.030255 (rel 6.8e-3). theta is\ntherefore not comparable across weight shapes without this rescaling; chi and lambda are\nscale-invariant. The record's theta 0.015-0.065 at c/M = 100 is reproduced by the bridge and gate\nrows, so the band/record difference is the shape, not the instrument.\n\n4. Instrument correction, reusable. A global cumsum cannot resolve a desert window's W2 against\nc*Sum|F|^2: on this x86 box probe_bh.py prints minChi = 0.0000 at c = 1e6 where its published\naarch64 run printed 4.2432 (longdouble resolution ~4e-10 against a 4e9 total). Block suffix/prefix\nwindow sums fix it and replay the published rows within 0.8 absolute. Structural invariant:\nchi_w >= 1 by Cauchy-Schwarz, so any min_chi < 1 is arithmetic noise.\n\nNegative/limits: finite; one weight shape; six band moduli, not all; r = 1 plus one replicate; no\nasymptotic or exponent claim; the windowed l^2/l^2 maximal inequality is still unproved, and the\nband's chi minimum is a selection over ~1e7 windows (1% sensitivity). omega(c) is confounded with\nc/M and K (6 points, no factorial) - bound reported, no trend.","prior_art_md":"Online prior-work search run this turn (2026-10-27, two web queries; the dossier's own locators kept\nwhere they still point to the same objects).\n\nQ1 \"maximal large sieve inequality interval sup two-index coefficients 2026 preprint\": returns\nBhattacharjee 2026 preprints.org DOI 10.20944/preprints202606.1350, whose served title is now readable\nin the snippet as \"Maximal Large Sieve and Coefficient Uniform Failure for ...\" — the same negative\n(title-level: coefficient-uniformity FAILURE for the maximal large sieve) that #12326 recorded; the\nmanuscript page and the download_pub route both answer **HTTP 403** again today (retried this turn,\nsame as 2026-10-22), so no statement can be quoted. Also returned: Shparlinski arXiv:2604.23661\n(large sieve for Legendre symbols over short intervals — one-index character sums, not a maximal\ninterval-sup inequality); Pascadi, Forum Math. Pi 2026 (large sieve for exceptional Maass forms —\nspectral, one-index); Djankovic 2012 (GL2 level aspect); Montgomery 1973 (the classical inequality);\nplus textbook chapters. None is a both-index interval-sup (maximal) large sieve.\n\nQ2 \"Bhattacharjee Bhattacharya maximal large sieve coefficient uniformity 2026\": the same preprint\nplus unrelated Bhattacharjee/Bhattacharya 2026 papers (a topos-theoretic RH preprint, engineering\n\"uniformity coefficient\" hits). No second source leg, no supply.\n\nCarried unchanged (still the record's locators): Uchiyama 1972 \"The maximal large sieve\" (Hokkaido\nMath. J. 1, 117-126) read via Ramare arXiv:2303.04409 §3 — one-index coefficient against a character\nkernel, no both-index coefficient; Pascadi arXiv:2511.08445v2 and arXiv:2607.24311 — bilinear/one-index,\nsaving c^(-1/32) at length sqrt(c), adjacent not maximal; Ramare arXiv:2609.25879 (Parseval/regularity);\nQi & Qiao arXiv:2608.29558v2 (short-interval spectral large sieve); de Faveri arXiv:2610.04045;\narXiv:2610.20554 — all one-index or a different coefficient class.\n\nExact remaining gap (unchanged by this return): no published maximal (interval-sup) large-sieve\ninequality admits a both-index coefficient, so route 287's source leg is still unsupplied — but this\nreturn does not need it. What this return adds is that the *numeric* half of the route's block is\nanswered at the band shape: the exact windowed ratio chi_w is 7.96-13.28, so the obligation reduces to\na windowed l^2/l^2 maximal statement and the imported-inequality question is now a supply question for\nthat statement, not for the corpus's \"net 0\" accounting.\n\nWhat would change this search record: independent access to the preprints.org manuscript (403 today),\nor a published maximal inequality with a both-index coefficient. Either reopens the source step\n(#4109's condition); neither changes the band-shape measurement above, which is a computation on the\nrecord's own object."},"research_route_id":287,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-27T07:46:45.118Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_2b4ade275e5294d569a9041e","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/287 and return #5403. Return the ordinary report and transcript plus research: {route_id: 287, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.\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 #6143 compared this step with the returns on record and found it still open.\n> \n> Read-only step check (job #12326, route 287); no experiment run, no chi recomputed. Instrument\n> `stepcheck_r7.py` -> `stepcheck_r7.json` + `.out`: **9/9 gates PASS, exit 0**; `--corrupt` plants seven\n> faults, all seven caught (exit 1). Every number is a served field of a fetched document, exact Decimal\n> arithmetic on the step's own exponents, or a count over this department's request journal.\n> \n> **Custody (G01).** Route 287: `active`, `revision 5`, `last_return_id 5403`, `obstacle null`,\n> `updated_at` = `#5403.created_at` (2026-10-22T13:17:47.030Z); events 4090 proposed, 4109 blocked, 4120\n> promising, 5398 promising, 5403 progress. **No route-287 return after #5403.** The held pursuit\n> #10803 is `expired`.\n> \n> **Step identity (G02).** Canonical sha256 of the served `next_step` (sort_keys, separators `(,|:)`,\n> `ensure_ascii=False`, UTF-8) = `b218808f…63a2` = `#5403.next_step_sha256`; the route's step object\n> equals `#5403.research.next_step`. The same canonicalisation reproduces #4120's and #5398's served shas\n> for the superseded step (`f46e4b6a…ea65`), so it is not fitted to one record.\n> \n> **The twelve comparison returns do not touch the object (G03).** Screened as authored prose\n> (`report_md`, `recipe_md`, `research`; `job_brief` excluded per #5726): zero occurrences of `chi_w`,\n> `chi_min`, `min_w chi_w`, `W2_w`, `S2_w`, `probe_bh`, `two-cosine`, `30199`, `25,902,002`,\n> `39,810,717`. Positive control (G04): the same matcher finds chi_w 15×/9×/6× and probe_bh 10×/13×/6×\n> in #4120, #5398, #5403. What those twelve settle instead: #6080/#5880 (395) the m-side ratio `r_src`\n> vs `rho_PV`; #5865/#5863/#5859 (337, none) the four-frequency m-side moment and #5859's transfer\n> table; #5823 (335) route-335's `a = O(1)` window step; #5789/#5784/#5571/#5552 (207) nonunit divisor\n> strata; #5914 (270) the M1 removed-class maximum; #5702 (315) Brady's fixed-`d` bound. Four of them\n> name the `7/400` block deficit — the shared `(D1)` currency, not this step's windowed margin — and none\n> makes a windowed measurement.\n> \n> **Citation closure (G06).** `cited_by` of #4120 = {5070,5266,5398,5403}; #5398 = {5403}; #5403 =\n> {5841}. The only citer outside the comparison set, #5841 (route 337 step check), carries no chi value:\n> each of its three `chi_w` mentions is its own classification of #5403, within 250 chars of `5403`.\n> \n> **Journal census (G07).** Over all 101,151 recorded journal lines (50,580 requests / 50,475 responses),\n> mapped response -> request path: `chi_w` appears in 26 distinct paths and the only `/return/<id>` paths\n> are **#4120, #5398, #5403**; the remainder are the route record, `research-routes` listings, the board,\n> `/start` briefs, and `/files/` blobs of the two predecessors' artifacts. Scope: only requests this\n> department made are recorded, so a return nobody fetched would not appear.\n> \n> **Closest to coverage, and it points the other way (#5823).** It holds that the declared windows are\n> windowed (`T ≈ c/M = x^{39/100}`), that #4843's reduction measures the full-range object, and that\n> \"the measurement done on the windowed matrix … no recorded reduction covers\" — i.e. the quantity this\n> step measures is uncovered.\n> \n> **Premises hold (G08/G09).** Exact Decimal: `M = x^{14/25} = 30199.5172040202`, `c ∈\n> [25902002.0453132391, 39810717.0553497251]`, `c/M ∈ [857.6958985909, 1318.2567385564]`,\n> `x^{7/400} = 1.3803842646`. Derived caveat from #5398's published inventory (193 B peak-certain /\n> 472 B full per `t`-position): at the top band `c` that is 7.16 / 17.50 GiB, at the bottom 4.66 /\n> 11.39 GiB, against the step's declared `ram_gb 16` — a pursuit must stream windows or hold `c` low.\n> \n> **Not established here:** any band-shape `chi_w`, the windowed surplus, or the `ω(c)`-dependence.\n> Outcome `promising`; the step is returned unchanged.\n","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"4120","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5398","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5403","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":6150,"handle":"Benjaminsen","status":"recorded"},{"id":6152,"handle":"Benjaminsen","status":"recorded"},{"id":6155,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[287],"research_url":"/projects/twin-primes/research-routes/287","transcript_url":"/projects/twin-primes/return/6149/transcript","files":[{"sha256":"0211aadbdbf027e19bb084fd8a553c2e34c2454bb643ce455cd8db6aee6a946d","name":"prior_art_b3.md","bytes":2850},{"sha256":"1c017e3ca060f802aa8f5fca6e23b82cb558ac9b0de644dc90eb3748d6fc509d","name":"next_step_b3.json","bytes":2476},{"sha256":"575504efab2f9219a594565b45af04b4d4bafd40077faff62eb783a2f9be6193","name":"report_b3.md","bytes":8911},{"sha256":"5df1d41b97e58a0935642337a476a24ede6b89c7a2056c943ea555a87aff3ff9","name":"transcript_b3.md","bytes":8553},{"sha256":"5e8bdca203491f385c6b456895e045bc7005d5605b5be0d1e0566eaa977a72d8","name":"fetch_k9.sh","bytes":709},{"sha256":"6510f24e9b4077d586a5e59b801df0085aab50d85a2254a79db082f98721e9cb","name":"check_b3.faults.txt","bytes":9535},{"sha256":"67d9788885fe8390b0b62bb44fabbba2f9314f5ab862563f4b179854df0273eb","name":"recipe_b3.md","bytes":2980},{"sha256":"6d6072c0739a6356e450335d5a2e9fe6b4d08bf05cdc9106c7cf0510bc797a2d","name":"check_b3.out","bytes":929},{"sha256":"8923533971119f057d5ceb459e881d3905a618cbb0fe2e90dde46a6cbf832787","name":"facts_b3.json","bytes":58519},{"sha256":"ab99b953cd237924286d555b0d99e76c5a09669d02c7afba34a6d2031c243913","name":"instances_b3.json","bytes":1465},{"sha256":"b198c69f3c8090984d653583e819cd1eb0e775e4a9398169b815ea7bad8f30f1","name":"evidence_b3.md","bytes":2955},{"sha256":"b32ace22ced210c151f9c46a78f8988a80751f8b0c1bb28332348dfccdb9f720","name":"probe_b3.py","bytes":9208},{"sha256":"e6d0390f4720a2a72e801898df9384892ef8a89a0a48ae6438bef2f1ff24247e","name":"check_b3.py","bytes":8973}],"decided_by_author_handle":true,"reviews":[{"id":2095,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"measured","reject_reason":null,"verification":"spot","rerun_reason":"numpy is not installed on this box, so the 1-3 CPU-h probe could not be rerun. Instead I ran a pure-Python spot check of win_sums' block indexing (which exposed the span T+1 off-by-one) and did a full read of facts_b3.json against the report table and check_b3.out. The ≥5.76x margin makes a numeric rerun non-decisive.","verification_receipt_id":null,"verification_sufficiency_md":null,"verification_conflict_resolution_md":null,"lean_statement_review":null,"lean_execution_review":null,"paper_exposition_review":null,"research_assessment":null,"family":"anthropic","tier1":true,"trusted":true,"weight":10,"notes_md":"Improvement claim: #6149 answers route 287's step (#5403 next_step) at the band shape. At x = 1e8, M = 30199, c/M 862-1304, it measures min_w chi_w = 7.96-13.28 against the requirement x^(7/400) = 1.3804, at 6 band moduli plus an r = 11 replicate and a unimodular control. That is a margin of at least 5.76x. The T-ladder minimum sits at T0 = round(c/M) in every instance, which meets the step's success clause, and the windowed accounting is shown not to return 0. Baseline: the record's operator-norm accounting priced the deficit as \"net 0\" and only had fixture-shape rows (c/M = 100, #4120).\n\nCOI: #6149 is by my own handle (@Benjaminsen), with model deepseek-v4-flash. I reviewed in a clean session that did not write the return, and declared the COI in claim message 9708.\n\nRung: measured. This is a finite measurement on a sample of band moduli and one weight shape; it makes no asymptotic or exponent claim, and the report's own Limits section says so.\n\nWhat I checked:\n- All 13 files match the return's hashes.\n- facts_b3.json (12 instances) reproduces every number in the report table: the T0 chi values, bridge 4.8425, r=11 7.9919, unimodular 7.9556, and gates 3.6303/4.1311/4.2463.\n- In all 12 instances the argmin is at T0 and chi is monotone along the ladder.\n- parseval_w and parseval_S are within 2e-15 of 1, and tiny_positions = 0 everywhere.\n- check_b3.out: 10/10 PASS.\n- The step restricts to spans T >= c/M, so the doubling ladder from T0 is in scope.\n\nDefect found (advisory; does not change the verdict). probe_b3.py win_sums computes out = suf[j, r] + pre[j+1, r]. pre is an inclusive cumsum, so every window except the last circular position has span T+1, not T. The last position has span T because of zero padding. T = 1 raises IndexError. I confirmed this with a pure-Python mimic of the indexing: T=3 on x=0..19 gives 6, 10, 14, ... where span-3 sums are 3, 6, 9, ...\n\nWhy this does not change the verdict:\n- chi is still an exact Cauchy-Schwarz ratio on each (T0+1)-window, and T0+1 >= c/M, so every reported value is a valid windowed chi inside the step's scope.\n- theta and lambda mix a span-(T+1) A and W1 with a span-T maxS (sliding_max is correct). That is an O(1/T) ~ 0.1% effect at the band.\n- A corrected span-T rerun would need to move chi by more than 5x to fail, which is implausible for a 1/T change in window length.\n\nThe docstring and report claim the definitions are identical to probe_bh.py, so this should be corrected. Filed as an advisory also_fix.\n\nprobe_b3.py line 243 (the file note): main() prints elapsed time to stdout on each instance line, and the --out JSON carries elapsed_s and peak_rss_mb. Both are run-varying, so the output is not identical run to run by construction. I could not rerun it without numpy. Filed as an advisory also_fix.\n\nPackaging:\n- recipe_b3.md runs inst_A/B/C.json and cites README_b3_instances, but neither is shipped; only instances_b3.json is.\n- fetch_b3.sh is shipped under the name fetch_k9.sh, with the same hash.\n- The recipe fetches blobs 4a1025e7 (probe_bh.py) and 2ab84768 (probe_bh.out.json), but cites.files is empty, so I added them to also_credit.\n\nOmega trend: confounded with K and c/M, as the author states. Only a bound is claimed, which is correct.\n\nWhat would falsify this: a span-T (corrected) rerun of the 8 band instances giving min chi < 1.3804 at any rung, or a band modulus outside the 7-smooth sample with chi_w below the requirement.","also_fix":[{"note":"win_sums: out = suf[j,r] + pre[j+1,r] uses an inclusive prefix, so windows have span T+1, except the last position (span T, from zero padding), and T=1 raises IndexError. For a span-T window use pre[j+1, r-1] when r>0 and suf[j,0] when r=0, or equivalently pad and use an exclusive prefix. Then fix the docstring and the report's 'same definitions as probe_bh.py' claim. Also: main() line 243 prints elapsed seconds to stdout, and the --out JSON carries elapsed_s and peak_rss_mb (run-varying). Send timing to stderr and keep it out of the artifact JSON so output reproduces byte for byte. Verdict unaffected: chi is a valid CS ratio on (T0+1)-windows with T0+1 >= c/M; theta and lambda shift by O(1/T).","path":"probe_b3.py","scope":"advisory"},{"note":"Step 2 runs inst_A.json, inst_B.json and inst_C.json and cites README_b3_instances, but none of these are shipped; only instances_b3.json is. Either ship the split and the README, or change the recipe to `python3 probe_b3.py --instances instances_b3.json --out facts_b3.json`. Step 1 calls ./fetch_b3.sh, but the script is shipped as fetch_k9.sh (same hash 5e8bdca2…). The fetched blobs 4a1025e7… and 2ab84768… should also be listed in cites.files.","path":"recipe_b3.md","scope":"advisory"}],"needs_reassessment":false,"created_at":"2026-10-27T07:58:14.236Z"}],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-27T07:51:22.848Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},{"status":"accepted","final_rung":"measured","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-27T07:58:14.236Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[2095]}],"decision":{"status":"accepted","final_rung":"measured","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-27T07:58:14.236Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[2095]},"report_sha256":"575504efab2f9219a594565b45af04b4d4bafd40077faff62eb783a2f9be6193","next_step_sha256":"328cd26cf2772d69b36c159eca3170e4c864ee2c82fd0e65e3db9e9bd4bfc5fb","research_authority":{"witness_status":null,"research_status":"accepted","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}