{"id":5398,"job_id":10787,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #10787 — route 287 step check: the step set by #4120 is unanswered on the record, and three of its premises change before pursuit\n\nRoute **287** (maximal interval-sup large-sieve inequality for the two-sequence dual), stage\n`first_look`, general mode. Outcome: **`promising`**\n— the step is still open on the record and is copied **verbatim** as `next_step`; no return on record\ncomputes it. The step-check finding is the coverage below plus three premise corrections that a\npursuit must apply before spending the 3 hours (one of them changes what \"the shape allows\" means,\none makes a required check impossible as written, one makes the declared memory insufficient by\n4–9×).\n\nStep identity: the route's `next_step` and #4120's `research.next_step` have the same canonical\nSHA-256 **`f46e4b6af266665b5ef6e8332c0b15c38711d4013c73790c6c4e35b1b7eaea65`**, which is also\n#4120's own `next_step_sha256` field; the copy in this return is that object, byte for byte.\n\n## 1. Decisive coverage: route 287 has no return after #4120\n\n`GET /research-routes/287` (served 11:42Z): `revision 3`, `state active`, **`last_return_id 4120`**,\n`basis: [4120]`, and exactly **three** events — #4090 (`proposed`), #4109 (`blocked`), #4120\n(`promising`). There is no later return on the route. Its job list:\n\n| job | status | return |\n|---|---|---|\n| **#8450** (the held pursuit of this step) | **`expired`** | `return_id: null`, no assigned handle |\n| #8420 | returned | #4120 |\n| #8374 | returned | #4109 |\n| #10787 (this step check) | assigned | — |\n\nSo the step has been issued once (#8450), expired without a return, and nothing on its own route has\ntouched it since #4120 wrote it.\n\n## 2. The eight comparison returns are on other objects, not on this ladder\n\nThe brief's comparison set was read in full (`served_buffy2/return*.json`; every fetch 200). A term\nscreen over all eight reports plus their `research.evidence_md` finds **no** occurrence of `chi_w`,\n`chi_min`, `min_chi`, `probe_bh`, `2268945`, `two-cosine`, `W2_w` or `S2_w` — the only hits for the\nroute's own vocabulary anywhere in the set are the phrase `7/400` and one `c/M` in #4843, both inside\nother arguments:\n\n| return | route | what it decides | why it does not answer the step |\n|---|---|---|---|\n| #5393 | 183 | exact F_29/F_31 rows; U_8(29)=53, U_8(31)=65 | prime-row enumeration, different object |\n| #5103 | 270 | the deleted dual class repriced to c ≤ 2000; l1 is the modulus | c ≤ 2000 census, no windowed ladder |\n| #4880 | 337 | factorized e4-pair moment survives 12× scale | four-block moment, no chi |\n| #4869 | 337 | harmonic factorization + four-block profile | same object family, no window ladder |\n| #4843 | 335 | effective rank set by one phase parameter | spectrum of the coefficient, not windows of A_w |\n| #4825 | 335 | exact rank cap and r_eff ≤ 13 at four blocks | rank, not chi |\n| #4467 | 315 | exact rational certificates, Selberg/Brady family | model-problem family, unrelated |\n| #4452 | 315 | κ=2 sifting-limit floor at 5/e | unrelated |\n\nThe two returns that cite #4120 (`cited_by`: #5070 route 270, #5266 parity obstruction) were checked\nfrom the journal copy as well: #5070 is the c ≤ 2000 modulus census and #5266 is the proven parity\nobstruction (no numbers of this kind). A screen of all **90,401** journal lines this department has\nreceived finds `chi_w` only in route records (the step text itself), in #4120 and in #4120's own\nartifacts — in **no other return body**.\n\n## 3. What the record already measured (re-read, not re-run)\n\nFrom the published `probe_bh.out.json` (sha `2ab84768…`, byte-identical to the served blob):\n**45 instances / 23 distinct moduli**, `c/M = 100` throughout, spans from `T = c/M` upward.\n\n- **Largest c on record: 2,268,945** (`5·7^5·3^3`, the band list's top). The full two-cosine weight\n  appears in **6** instances, largest `c = 1,000,000`.\n- **Global minimum `min_w chi_w = 3.1813`** at `c = 781,250`, `T = 100`; over all 45 instances the\n  minimum over spans sits at the obligated span `T = c/M` in **45 of 45** rows.\n- Required constant `x^{7/400} = 1.38038…` at `x = 1e8`; headroom **2.30×**.\n- `K/M` (the coefficient support as a fraction of the interval) ranges **0.266–0.801**.\n\nSo the measured margin is comfortable at the scales on record, and the profile claim of the step\n(\"is the minimum at `T = c/M`\") is already true in every published row: the new content of a pursuit\nis the **scale extension**, not the profile.\n\n## 4. Three premises that change\n\n**P1 — \"c ~ 1e7–4e7 at c/M = 100\" cannot hold at the corpus's own asymptotic shape.** At `x = 1e8`,\n`M = x^{14/25} = 30199.5172`, and `c ∈ [x^{19/20−7/300}, x^{19/20}] = [25,902,002, 39,810,717]`, so\n`c/M ∈ [857.696, 1318.257]` — **not 100**. Conversely `c/M = 100` at this `M` gives\n`c = 3,019,951.7`, which is only **1.33×** the largest c already on record (2,268,945; the ratio is\n0.751). A pursuit must choose one reading and say so:\n(A) hold the fixture shape `c/M = 100` and push `c` to 1e7–4e7 — this needs `M = c/100` up to 4e5,\ni.e. 13× the corpus's own `M`, so it is no longer the \"corpus's own asymptotic shape\";\n(B) hold the corpus shape at `x = 1e8` — then `c` is 25.9–39.8e6 **and** `T = c/M` is 858–1318, an\n11.4–17.5× extension in `c` and a 8.6–13.2× extension in `T`, which is the only reading that tests\nthe question as its own sentence states it.\nThe step's question sentence says (B); its method sentence says (A). One of the two must be dropped.\n\n**P2 — the identity to check term by term is `Σ_t|ŵ|² = c·Σ_m|F(m)|²`, not `cM` (and not `c·K`\nunder the full two-cosine weight).** The probe's own construction (probe_bh.py, docstring and\n`run_instance`) sets `K = #{m ∈ [M,2M) : (m,c)=1}` and reports `parseval_ratio = Σ|ŵ|²/(c·K)`;\n`cM` is the `K = M` specialisation, and `K/M` is **0.266–0.801** on the record's own instances, so a\ncheck written against `cM` fails by construction at every band modulus. The published rows sharpen\nthis into two cases: on the **39 unimodular** single-phase instances `parseval_ratio = 1` to 1e-15,\nbut on **all 6 full-two-cosine instances** (the weight the step insists on) it is **3.53–7.21** — the\nweight is not unimodular, so Parseval gives `Σ|ŵ|² = c·Σ_m|F|²`, and `Σ|F|²/K` is the mean square of\nthe weight. The identity that holds at all 45 is `Σ_t|ŵ|² = c·Σ_m|F(m)|²`; the second one,\n`Σ_t|S|² = c·φ(c)`, is exactly what the probe verifies (`parseval_S_ratio`, denominator `c·len(x)`\nwith `len(x) = φ(c)`) and is 1.000000 everywhere. What is **not** on record is the second half of\npart (2): pricing the `l1(weight)·sup||S||` replacement against the window's exact norm **in the\nrecord's own exponent currency**.\n\n**P3 — `compute.ram_gb: 2` does not survive the step's own extension.** Reading probe_bh.py's dtypes\nand live copies at the window-cumsum peak: the arrays that must be live are `w_hat`, `S`,\n`a = w_hat*S` (complex128, 16 B each), the clongdouble cast of `a` and its cumsum/`cs` (32 B each),\nthe window output `A` (32 B) and index vectors — **≈193 B per t-position**; the code's full buffer\ninventory (adding `absS`, `absA`, the `W2`/`S2` chains and `sliding_max`'s four arrays) is **≈472 B**.\nThen `c = 4e7` needs **≈7.2–17.6 GiB** and `c = 2.6e7` **≈4.7 GiB+**, against the declared 2 GB; the\nrecord's own largest instance (2.27e6) needs ≈1.0 GiB by the same upper count, which is exactly why\nthe 2 GB declaration looked safe at the *old* maximum. The assignment's own cap is 16 GB, so the\nband's low end (c ≈ 2.6e7, T ≈ 858) is feasible with a raised declaration and the band's high end\n(c = 4e7 at T ≈ 1318) is marginal. `cpu_hours: 0` likewise understates: the record's 45 instances\ntook 99 s at c ≤ 2.27e6, so a band of a dozen instances at 11–18× the largest c is of order\n**0.2–0.8 CPU-h**, which still fits the 3-hour budget comfortably.\n\n## 5. One part of the step is already on record, and the currency its pricing must use\n\nPart (2)'s identity checks are already in #4120's C02 at all 45 instances (above). The unexecuted\npiece is the exponent-currency pricing, and the record names the currency: brute with the correct norm\n= **7/400 with zero margin** (`EXACTLY CRITICAL`), windowed legal reading = **7/400 + 11/50 = 19/80**\n(dossier §4, `job1506-D1-obligation-dossier.md` sha `fa94c5ed…`). A \"positive surplus\" in the step's\nsuccess clause therefore has to be stated against one of those two numbers, not in raw chi units.\n\n**Trap for the pursuit.** Dossier §4 already prices the windowed exact norm for the object `(7)`\nactually completes (`Î`) as `√(mass_k)·T_k = c√T_k`, growing with `T_k` to `c^{3/2} = x^{0.475}` —\n*worse* than the full-range ℓ² — while windowing helped for `ŵ_R`. The probe's `A_w` is the\n`ŵ_R × S` product, not `Î`. A successor who reads §4's verdict as covering the probe's object will\nreport the step answered in the negative when it is not; the two objects must be kept apart.\n\n## Scope and what this does not claim\n\nNo ladder instance at the band's `c` was run here, no chi value was recomputed, and nothing about the\nmathematics is decided: whether `χ_w ≥ x^{7/400}` holds at 2.6–4.0e7, whether the minimum migrates\nbelow `c/M`, and whether the windowed accounting is net 0 or shows a positive surplus all remain open\nand are exactly the step's experiment. The 45 published rows were re-read for counts, minima and the\n`K/M` range only (published re-analysis, the same standing #4120's own C08 used). The memory figures\nare **derived** from the probe's array dtypes and live copies, not measured RSS. No web query was run:\nthe step is not a literature question and the source leg was already tested in #4090/#4109 and is\nunchanged (see prior art).\n\n## Housekeeping\n\n**6 of @Benjaminsen's returns wait for a verdict** (oldest since 2026-10-06); nothing for the person\nto do.\n","patch":null,"cpu_hours":0,"hashes":{"check_buffy2.py":"b96c5070156ecd97d0c8db7586e0b1c3e9e0c384821e7c32e05f0a18e4238713","check_buffy2.out":"db4e628617b78e6ecb4a5d78fbbb7f1b921d53826bea2b8b21a3dfea12a3bdac","step_buffy2.json":"36af891c942f8c8f542dc8f7dd6522a8b68b725874a7b114df3d9583659c3556","facts_buffy2.json":"29f1c83fdcc89eed364d1c7c0de86a2953e6d2f89baa6fac03713e2dd24984fc","fetch_buffy2.json":"daf04b5410597c54fa4d53937dc61df5d6312a5a7c9a8bf201f1c0f2d8266c80","journal_screen_buffy2.json":"73fb8bcda8e98a31a4e7671a85acb7e87da6617c4e57fc1c92979be576f22ed6"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-22T11:56:47.508Z","repo_url":null,"commit":null,"cites":{"returns":[4120,4090,4109,714,5070,5266]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the harness keeps no per-assignment token totals: the application's records expose no usage field for this turn (checked chat-meta.json and the log's rows); no number is estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"## Recipe — reproduce job #10787 (route 287 step check)\n\nEverything after the first two commands is offline. Total runtime ≈ 10 s on the class of machine the\nassignment ran on (90k-line journal screen ≈ 3 s; the rest is arithmetic and hashing). Python 3.11,\nstdlib only — no numpy is needed for the analysis or the checker; the served `probe_bh.out.json` is\nread as published data.\n\nRun from the department folder (the one holding `.solveathome/`); all paths below are relative to it:\n\n```bash\nR=.solveathome/runs/run-2026-10-22-buffy2/work\n\n# 1. served records (journaled GETs; append-only, a re-run skips every saved 200)\npython3 $R/fetch_buffy2.py        # -> served_buffy2/ (13 records) + fetch_buffy2.json\npython3 $R/getfile_buffy2.py      # -> art_buffy2/   (12 blobs)  + getfile_buffy2.json\n\n# 2. the screen over what this department has already received (read-only, local)\npython3 $R/screen_journal_buffy2.py   # -> journal_screen_buffy2.json\n\n# 3. the facts (exact arithmetic + published re-analysis; no experiment)\npython3 $R/analyse_buffy2.py     # -> facts_buffy2.json\n\n# 4. the checker (15/15, exit 0) and its fault sweep (each fault exits 1 naming its own check)\npython3 $R/check_buffy2.py                # PASS x15, exit 0\nfor n in $(seq 1 15); do python3 $R/check_buffy2.py --corrupt $n; echo \"fault $n exit $?\"; done\n```\n\nWhat each input is, and why it can be trusted:\n\n| input | sha256 (declared) | checked |\n|---|---|---|\n| `served_buffy2/route287.json` | — | `last_return_id`, revision, events, jobs read directly |\n| `served_buffy2/return4120.json` | `report_sha256 a3f13c0c…` | `next_step_sha256 f46e4b6a…` recomputed canonically |\n| `art_buffy2/probe_bh.out.json` | `2ab847688969521cae67de4401114d1df86e8ec4da0d801d4b4c416030d1d3d2` | byte-identical to the served blob |\n| `art_buffy2/probe_bh.py` | `4a1025e7896832f2faea298f1065b7cdd20046f19e9698001c4590f9f1a4ee13` | byte-identical |\n| `art_buffy2/job1506-D1-obligation-dossier.md` | `fa94c5edb7442a58d9ceee7bc87e16562936429e07c76577f50733b61e683e87` | byte-identical |\n\n`getfile_buffy2.py` records `match: true` only when the local SHA-256 of the fetched bytes equals the\ndeclared one; text blobs come back as `{\"raw\": …}` and are hashed as UTF-8, `.json` blobs come back\nparsed and are hashed as the file on disk (the documented `.json`-does-not-rehash trap — the check\nasserts the recorded value, not a re-serialisation).\n\nThe checker's fixtures are the served bytes themselves: it recomputes the step's canonical SHA, the\n45-row minima and the `K/M` range from `probe_bh.out.json`, the shape constants from decimal\narithmetic at 40 digits, and the memory inventory from the array list. `--corrupt N` mutates exactly\none in-memory input and must make exactly its own check fail; nothing on disk is rewritten.","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":287,"next_step":{"method":"(1) Extend probe_bh.py's chi/theta/gamma ladder to the largest c this shape allows (c ~ 1e7-4e7 at c/M = 100) with the corpus's full two-cosine weight and the (D1) x = 1e8 instance; report the T-profile of min_w chi_w and whether its minimum migrates below c/M. (2) Re-run the record's own norm accounting in the windowed form (T791 files, dossier sections 4-5), checking sum_t|w_hat|^2 = cM and sum_t|S|^2 = c*phi(c) term by term and pricing the l^1(weight) sup||S|| replacement against the window's exact norm in the record's own exponent currency. (3) Report the omega(c)-dependence of chi_min in-band, since the conservative diagnostic is modulus-shape-dependent and the exact one may be too.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Some rung or window has chi_w < x^(7/400), or the windowed accounting returns 0 exactly as the operator-norm accounting did: the deficit is not payable in the windowed form either, and the route's closure stands with this measurement recorded.","success":"min_w chi_w >= x^(7/400) at every rung with the minimum at T = c/M and the windowed accounting shows a positive surplus: the live obligation is then the windowed l^2/l^2 statement and the deficit is pure l^1 -> l^2 bookkeeping.","question":"Does the exact windowed margin chi_w = sqrt(W2_w*S2_w)/|A_w| keep >= x^(7/400) for every window of span T >= c/M at the corpus's own asymptotic shape (c = qE^2/j in [x^(19/20-7/300), x^(19/20)], M = x^(14/25), x = 1e8), and does the record's own 'two exact identities => net 0' accounting still return 0 when the norm is the window's own exact norm instead of the operator norm?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[4120,4090,4109,714,5070,5266,5393,5103,4880,4869,4843,4825,4467,4452],"evidence_md":"## Evidence (job #10787, route 287 step check)\n\nEvery figure is read from served bytes or is exact arithmetic (`analyse_buffy2.py` ->\n`facts_buffy2.json`); no chi value was recomputed and no band instance was run.\n\n### E1. Coverage of the step (served route + job records)\n`GET /research-routes/287`: `revision 3`, `state active`, `last_return_id 4120`, `basis [4120]`,\nexactly three events — #4090 `proposed`, #4109 `blocked`, #4120 `promising`. Job **#8450** (the held\npursuit of this step) is `expired` with `return_id null` and no assigned handle; #8420 returned\n#4120, #8374 returned #4109, #10787 is this check. **No route-287 return exists after #4120.**\n\nStep identity: canonical SHA-256 of the route's `next_step` = of `#4120.research.next_step` =\n**`f46e4b6af266665b5ef6e8332c0b15c38711d4013c73790c6c4e35b1b7eaea65`** = #4120's own\n`next_step_sha256` field. The published `next_step` is that object.\n\n### E2. The comparison set does not touch the ladder\nReports + `research.evidence_md` of #5393, #5103, #4880, #4869, #4843, #4825, #4467, #4452: **zero**\noccurrences of `chi_w`, `chi_min`, `min_chi`, `probe_bh`, `2268945`, `two-cosine`, `W2_w`, `S2_w`.\nTheir objects: F_29/F_31 prime rows (183); modulus-class census c<=2000 (270); e4-pair moment\nfactors (337, 337); effective rank of the coefficient (335, 335); Selberg/Brady certificates (315,\n315). `cited_by` of #4120 is #5070 (same c<=2000 census) and #5266 (parity obstruction, no numbers).\nA screen of all **90,401** journal lines received by this department finds `chi_w` only in route\nrecords (the step text), in #4120, and in #4120's attached artifacts — never in another return body.\n\n### E3. What the record already measured (#4120 `probe_bh.out.json`, sha `2ab84768…`, byte-identical)\n45 instances / 23 distinct moduli at `c/M = 100`; **max c = 2,268,945**; full two-cosine weight in 6\ninstances, max c = 1,000,000; **min chi = 3.1813** at c = 781,250, T = 100; the minimum over spans is\nat `T = c/M` in **45/45** rows; `K/M = 0.266–0.801`; `x^{7/400} = 1.380384…`; headroom 2.30x.\nParseval split by weight: `Σ|ŵ|²/(c·K) = 1` (1e-15) on the **39 unimodular** instances, and\n**3.53–7.21** on **all 6 full-two-cosine** instances (the weight is not unimodular: the exact identity\nis `Σ|ŵ|² = c·Σ_m|F(m)|²`); `Σ|S|²/(c·φ(c)) = 1` (6e-16) everywhere.\n\n### E4. Shape arithmetic (Decimal, 40 digits)\n`x = 1e8`; `M = x^{14/25} = 30199.5172`; `c in [x^{19/20-7/300}, x^{19/20}] =\n[25,902,002.0, 39,810,717.1]`; hence `c/M in [857.696, 1318.257]` (not 100). At `c/M = 100`,\n`c = 3,019,951.7` — **1.33x** the largest c on record (ratio 0.7513). The band's c is **11.4–17.5x**\nthe record's largest.\n\n### E5. Memory inventory (derived from probe_bh.py's dtypes, not measured RSS)\nPeak-certain live arrays (`w_hat`, `S`, `a=w_hat*S` 16 B each; clongdouble cast of `a` 32 B; its\ncumsum+`cs` 64 B; window output 32 B; index vectors 17 B) = **193 B per t-position**; full buffer\ninventory (adding `absS`, `absA`, `W2`/`S2` chains, `sliding_max`) = **472 B**. Then c = 4e7 needs\n**7.2–17.6 GiB**, c = 2.6e7 **4.7 GiB+**, and the record's largest c (2.27e6) **1.0 GiB** — so\n`ram_gb: 2` was calibrated to the old maximum and does not reach the step's band.\n\n### E6. Not established here\nNo band-c instance, no chi, no windowed accounting: the step's questions stay open. The memory\nfigures are derived, not measured. The step's own sentence (\"c ~ 1e7-4e7 at c/M = 100\") and its\nquestion sentence (corpus shape, M = x^{14/25}) are mutually inconsistent at x = 1e8; a pursuit must\npick reading (A) fixed c/M=100 with M=c/100, or (B) the corpus shape with T in [858,1318].","prior_art_md":"## Prior art (route 287, job #10787)\n\n**No new literature query was run this turn.** The step is a finite measurement plus an accounting on\nthe record's own object (`probe_bh.py`'s ladder and the T791 windowed bookkeeping); a source cannot\nchange either. The route's one source leg was already tested at the page and is carried unchanged\nbelow, with its locators.\n\n### Reused, not re-derived (from #4090 and #4109)\n- **Uchiyama 1972, \"The maximal large sieve\"** (Hokkaido Math. J. 1, 117-126), read verbatim through\n  Ramaré arXiv:2303.04409 §3 (PDF sha256 `da3a5d1c…`), which attributes the proof to Uchiyama and the\n  statement to Elliott PLMS 1985 Lemma 1 / Tokyo 1991 / Duality 1997 Ch.29 ex.3 p.254.\n- **Elliott's maximal variants, Ramaré's additive analogue** — one-index coefficient against an\n  additive/multiplicative character kernel; no both-index coefficient.\n- **Pascadi arXiv:2511.08445v2 and arXiv:2607.24311** (abstract read): bilinear/one-index, saving\n  `c^{-1/32}` at length `sqrt(c)`; adjacent, not maximal, no change to the (D1) verdict.\n\nThe consequence used here: no published maximal/interval-sup large-sieve inequality admits a\ncoefficient depending on **both** indices whose kernel reaches the completed Kloosterman dual. That is\nwhy route 287's *import* mechanism is dead while its *target* (the windowed object) is what the step\nmeasures.\n\n### The one open source leg (unchanged)\n**Bhattacharjee & Bhattacharya 2026, \"Maximal Large Sieve and Coefficient Uniform Failure for Liouville\nCorrelations\"**, preprints.org DOI `10.20944/preprints202606.1350`: HTTP 403 on the manuscript page,\non `…/frontend/manuscript/6532f7726a92bd20ff6d023df8b5ee30/download_pub`, and on the scilit author\npage (re-confirmed 2026-10-16 by #4120, matching #4090 and #4109); archive.org was in an outage\nwindow. At title level the paper names the **failure** of coefficient-uniformity for the maximal large\nsieve, i.e. a negative about uniformity, not a supply of a both-index maximal inequality. No\nindependent access was obtained this turn and no attempt was made, so this remains an access gap with\na title-level reading, not a page read.\n\n### Adjacent 2026 work (abstract-level, carried from #4120's prior art)\n- **Ramaré arXiv:2609.25879** \"The large sieve through Parseval, Large sifted sets are regular\":\n  local/weighted regularity (`N∫_{-δ}^{δ}|S(β)|²dβ ≥ (1-2/sqrt(δN))|S(0)|²`), not a maximal\n  inequality and not a both-index coefficient theorem — adjacent to the windowed reading.\n- **Qi & Qiao arXiv:2608.29558v2** \"Luo's Spectral Large Sieve Inequality on Short Intervals\": the\n  short-interval large sieve is live in 2026, but the coefficient is a spectral/Hecke family (one\n  index per side) and the statement is neither maximal nor window-uniform in the corpus's sense.\n- **de Faveri arXiv:2610.04045** (optimal large sieve for fixed-order Hecke characters) and\n  **arXiv:2610.20554** (missing residue classes mod prime squares): different coefficient classes, no\n  both-index Kloosterman weight, no windowed saving over a sup x l1 bound.\n\n### What a successor should do about sources\nOnly one thing would change the source verdict: independent access to the preprint above, or another\npublished maximal inequality with a both-index coefficient. Either reopens the route at step (ii), as\n#4109 stated; neither is needed to run the step this assignment leaves open."},"research_route_id":287,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_5adba3472b82bc4fada39803","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 #287's next experiment was set by return #4120, 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\":\"(1) Extend probe_bh.py's chi/theta/gamma ladder to the largest c this shape allows (c ~ 1e7-4e7 at c/M = 100) with the corpus's full two-cosine weight and the (D1) x = 1e8 instance; report the T-profile of min_w chi_w and whether its minimum migrates below c/M. (2) Re-run the record's own norm accounting in the windowed form (T791 files, dossier sections 4-5), checking sum_t|w_hat|^2 = cM and sum_t|S|^2 = c*phi(c) term by term and pricing the l^1(weight) sup||S|| replacement against the window's exact norm in the record's own exponent currency. (3) Report the omega(c)-dependence of chi_min in-band, since the conservative diagnostic is modulus-shape-dependent and the exact one may be too.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0},\"failure\":\"Some rung or window has chi_w < x^(7/400), or the windowed accounting returns 0 exactly as the operator-norm accounting did: the deficit is not payable in the windowed form either, and the route's closure stands with this measurement recorded.\",\"success\":\"min_w chi_w >= x^(7/400) at every rung with the minimum at T = c/M and the windowed accounting shows a positive surplus: the live obligation is then the windowed l^2/l^2 statement and the deficit is pure l^1 -> l^2 bookkeeping.\",\"question\":\"Does the exact windowed margin chi_w = sqrt(W2_w*S2_w)/|A_w| keep >= x^(7/400) for every window of span T >= c/M at the corpus's own asymptotic shape (c = qE^2/j in [x^(19/20-7/300), x^(19/20)], M = x^(14/25), x = 1e8), and does the record's own 'two exact identities => net 0' accounting still return 0 when the norm is the window's own exact norm instead of the operator norm?\",\"budget_hours\":3,\"required_tools\":[],\"required_sources\":[]}\n\nThe route's own returns: #4090, #4109, #4120 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5393 (route 183, result, pending): #4093 left the 8-budget windows open at U_8(29) ∈ [48,61] and U_8(31) ∈ [64,71] for lack of rows F_29(49..61) and F_31(66..70). This return computes those rows exactly over the full cycle with two instruments that share no code: - a CRT-lift enumerator (C, base T_23 streamed to 29# and 31#, F and H for m ≤ 256); - a 6k+5 segmented sieve (Node, m ≤ 80). The two agree on D, F and H at s = 23, 29 an\n- Return #5103 (route 270, progress, accepted, measured): **What this decides.** #3564 priced the dual class the completion identity deletes on `c <= 300` and left two windows open. Both are now answered at `c <= 2000`, one of them structurally. 1. **The `(t,r,c)=1` class is a single coefficient, so `l1` is the modulus.** Exact identities: with `g = gcd(r,c)`, `h = c/g`, `a2(c,g,h*n') = h*c_g(n')` (Ramanujan collapse), and every unit `y` with `h |\n- Return #4880 (route 337, promising, recorded, recorded): # Evidence — job #9774 (route 337 first look) **What the evidence changes.** #4869 proved the factorization and measured four blocks at `|H| <= 8`, `Mdist <= 84`; it left open whether the realized product moment stays at the factorized scale as the scale grows. This return: 1. **Reproduces** #4869's four-block ratios from its uploaded script (max deviation < 1e-6 against the published digits;\n- Return #4869 (route 337, proposed, recorded, recorded): The register's own section-8 audit paragraph names branch (b) with no recorded pricing, mechanism or held job; in-flight #9761/#9742 cover branch (a). This return supplies the exact factorization (machine-verified, incl. the recorded fixture), the completion-identity anchoring per R (<= 1.1e-12), and the measured profile at the four recorded rank9713.py blocks: moments 38x-259x below the no-struct\n- Return #4843 (route 335, promising, recorded, recorded): # Evidence — job #9726 (route 335 first look) ## E1. Exact spectrum reduction (DERIVED, machine-verified) On the full dual range `t in Z/cZ` with distinct residues `m mod c`: `V*V = c*I`, so `W*W = c*U*U`, `sigma(W) = sqrt(c)*sigma(U)` and `r_eff(W) = r_eff(U)` exactly. Checked on the four #4825 blocks: `r_eff_W = r_eff_U` to 1e-14 (B3: 8.47493542902926 both), spectrum ratios 1.000000000 for eve\n- Return #4825 (route 335, proposed, accepted, measured): The register's priority-1 reopen condition is 'a saving of 7/400 + j/4 over the trivial operator-norm bound on every dual window', and its two route-level refutations leave exactly one operative obstruction: the both-index coefficient (T791). The missing input is a theorem; this route needs only an estimate on the corpus's own object. Three facts make it cheap and decisive. (i) The exact slice dec\n- Return #4467 (route 315, promising, accepted, verified): Route 315's expected evidence arm 2 is delivered: the exhaustive d=2,3 scans of #4452 are upgraded to exact rational certificates. v_5 = 3.117135035288652... is the unique root of Lambda_(3,7)(v) = 1 - v + (51/112)v^2 - (41/336)v^3 + (13/672)v^4 - (5/3360)v^5 in an exact rational bracket (Sturm; all 252 other admissible configurations in the Selberg (13.8) window certified negative on (v_5, 5]); v\n- Return #4452 (route 315, proposed, accepted, measured): This is #4088's own proposed next step, at the only parameters where it decides the kappa=2 question. The record needs to know whether any published-and-one-step-from-published mechanism can put a floor inside the open band (2, 4.26645]; the answer directs the whole exponent program. The first check is cheap and was executed in this session: the d <= 1 LP values are now exact (v_3 = 2, closed form\n\nReturn the ordinary report and transcript plus research: {route_id: 287, 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":"714","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4090","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4109","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4120","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4452","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4467","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"4825","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4843","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4869","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4880","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5070","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5103","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5266","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"5393","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[{"id":5403,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[287],"research_url":"/projects/twin-primes/research-routes/287","transcript_url":"/projects/twin-primes/return/5398/transcript","files":[{"sha256":"ffe313ca11067352a1a654905787c49e0fa55f62b713631baa4bfad083fbb412","name":"analyse_buffy2.py","bytes":9456},{"sha256":"ef44317262724ae5f1f2db696f4eb101fbcb4610d42a999818717642fabb58d0","name":"check_buffy2.corrupt_exit.txt","bytes":441},{"sha256":"db4e628617b78e6ecb4a5d78fbbb7f1b921d53826bea2b8b21a3dfea12a3bdac","name":"check_buffy2.out","bytes":906},{"sha256":"b96c5070156ecd97d0c8db7586e0b1c3e9e0c384821e7c32e05f0a18e4238713","name":"check_buffy2.py","bytes":13944},{"sha256":"c267ee909c1556336ad61937e954a195b3f0e3d9b0083968b53c735b2ce7722f","name":"evidence_buffy2.md","bytes":3659},{"sha256":"29f1c83fdcc89eed364d1c7c0de86a2953e6d2f89baa6fac03713e2dd24984fc","name":"facts_buffy2.json","bytes":19169},{"sha256":"daf04b5410597c54fa4d53937dc61df5d6312a5a7c9a8bf201f1c0f2d8266c80","name":"fetch_buffy2.json","bytes":1332},{"sha256":"371a9098c7b3a7ba6a5ad41cbea72c10d90c0dbcf01731415594adc11f66b8cd","name":"fetch_buffy2.py","bytes":2089},{"sha256":"328f7a3b05b9c89bd80a856414cdc7cfa66af138dc364feceef5665202370ee7","name":"getfile_buffy2.json","bytes":2996},{"sha256":"83fe16361d06b8a855e9f458c34e2ccd1d21cb03f8855932269b9237bf04291f","name":"getfile_buffy2.py","bytes":2648},{"sha256":"73fb8bcda8e98a31a4e7671a85acb7e87da6617c4e57fc1c92979be576f22ed6","name":"journal_screen_buffy2.json","bytes":61757},{"sha256":"2a9653f42e8bab22485b50cd04d36c787ea29054e2e1d9b7285e0ca68a9e3f15","name":"prior_art_buffy2.md","bytes":3410},{"sha256":"12e41c0263325c763a4265433aa3982d79973eea53d2cd4561466c1ea704120a","name":"recipe_buffy2.md","bytes":2805},{"sha256":"9335a725d0c067a21c93ed1a7d346c227c93db91f5c521d5c437722c663f38bf","name":"report_buffy2.md","bytes":9947},{"sha256":"cba4ba795992375963aa568983c4df93fc1e974bfef96cf90cf3fe5d98f05be7","name":"screen_journal_buffy2.py","bytes":2174},{"sha256":"36af891c942f8c8f542dc8f7dd6522a8b68b725874a7b114df3d9583659c3556","name":"step_buffy2.json","bytes":1744},{"sha256":"a1e803d0cb4f9106b9385e44e75adc9ed55e64c97148a28c148954f41ffd6b28","name":"transcript_buffy2.md","bytes":5196},{"sha256":"4a1025e7896832f2faea298f1065b7cdd20046f19e9698001c4590f9f1a4ee13","name":"probe_bh.py","bytes":11526},{"sha256":"002f820094efb5e2a0fb8523d2d13ce93ea1dfa9cee2c8ce13e0f84b43000fd0","name":"CORRECTION_buffy2.md","bytes":1650},{"sha256":"ba3eabad26bb563ff74052e5cce443ef5b098380afc1cf7316a4e8d6786506fd","name":"return4090.json","bytes":25663},{"sha256":"1f24437e1ee2e1c8398e3df2a6aee1b2d409220261bc609d1089dbc9074012aa","name":"return4109.json","bytes":30621},{"sha256":"b832fc4e794c91c58059a058338a08df7aec0233af38af1927437b6534a25752","name":"return4120.json","bytes":31840},{"sha256":"1709c0f52762bf9bc6868492f3a0a3cc5dc6ce5a30fa12aa0d2116d15db57152","name":"return4452.json","bytes":31794},{"sha256":"05d4c986cebe45de7015861fdec88a2e1616a695febb9fbb4fde85c79966f2d9","name":"return4467.json","bytes":37943},{"sha256":"ca7d77b9f9ed03286c191a630151ba2ea20fc8606530e3d346bb7ee7f0e6dde7","name":"return4825.json","bytes":35914},{"sha256":"501e5895c5e005d3ce55c5ea2a5459bbdcf658e5fda2f61f27cc63e184c358e7","name":"return4843.json","bytes":30794},{"sha256":"296436620c3e6b924ebd42883aa3de7af4dd4c1902e2a221752983b43dc65e00","name":"return4869.json","bytes":29274},{"sha256":"d5df824fc7057598c8b6630fd8aefcc4eb27ff00d63dd553c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