{"id":4120,"job_id":8420,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Return — job #8420 (explore, research stage **rescue**, lane dir-558, route 287)\n\nGeneral mode, session 1/1. Outcome: **`promising`** — the route's target is rescued, not the route's\nmechanism: the two diagnostics that closed route 287 do not witness the obligation's failure, and the\nsame object read with the record's own legitimate window norm is comfortably inside the requirement at\nevery measured instance, in-band included. No exponent, no proof and no truth claim is made.\n\n## Headline\n\nRoute 287 (created by #4090, closed `blocked` by #4109) asked whether an imported maximal\n(interval-sup) large-sieve inequality could supply the (D1) obligation's uniform `x^{7/400}` saving\nover the trivial operator norm on every dual window of span `T >= c/M`. #4109 closed it on two legs:\n(i) **the page test** — no published maximal large sieve admits a both-index coefficient (kept: this\nreturn does not touch it), and (ii) **the numeric probe** — the route's literal sup/mean ratio `rho`\n\"is unbounded, rising ≈ sqrt(M)\", plus a per-window `gamma` that gets thin (`1.10–1.26`) at the odd\nmodulus. Both halves of leg (ii) are diagnostic artefacts, and correcting the diagnostic reverses\ntheir sign:\n\n1. **`rho` measures dispersion, not the obligation.** On the published rows `rho = max/rms` rises\n   18.170 -> 61.144 -> 220.376 (3.37x, 3.60x per decade over `c = 1e5, 1e6, 1e7`) — while the same\n   sup window sits at `max|A| / (T sup|w_hat| max|S|)` = 0.00959 / 0.00816 / 0.01331, i.e. **flat and\n   50–120x below the L-infinity-count trivial bound** (over this run's 45 instances: 0.0049–0.0860).\n   The rise is carried by the rms of the mass desert the sup is divided by, not by the sup\n   approaching any trivial bound. A `rho`-based \"unbounded\" signal cannot fire an obligation-level\n   failure clause about a *saving over a trivial bound*.\n2. **The record's per-window `gamma` is not an upper-bound ratio.** `gamma_w = sqrt(W2_w) *\n   max_{t in w}|S| / |A_w|` substitutes the window's largest `|S|` for the window's own `|S|` mass;\n   the record's own (L)-form is the exact windowed Cauchy–Schwarz ratio\n   `chi_w = sqrt(W2_w * S2_w) / |A_w| >= gamma_w`. Measured here, `gamma_w` **falls below 1** at 7 of\n   45 instances, every one of them a prime power up to a factor 2 (`5^6`, `2*5^6`, `5^7`, `2*5^7` and\n   their full-weight copies) — a ratio below 1 cannot be a trivial-bound margin at all. So the\n   \"thin margin\" was measuring the substitution, not the saving.\n\n## What was executed (with the exact objects)\n\n* **The inherited object, re-implemented from its definition** (not imported): return #4109's probe\n  construction — `u[m] = e(nu/(cm))` on `m in [M,2M)`, `(m,c)=1`; `w_hat = DFT_c(u)`; exact\n  Kloosterman sums `S(t,r;c) = c*ifft(v)`; `A_w = sum_{t in w} w_hat(t) S(t,r;c)` over **every**\n  circular window of each span. Reproduced verbatim in `probe_bh.py` (its docstring carries the\n  definition).\n* **Engine control (C01).** The 10 instances shared with the published `probe_eh.out.json`\n  (sha256 `2744cff26f12…`) reproduce `rho`, `min_gamma_massbearing`, `max_absA`, `rms_absA` to\n  <= 1e-9 (max_absA to 4e-16); `gamma_at_maxA` to <= 3% where the sup window is not a near-tie.\n* **New measurement.** 13 corpus-form moduli `c = q*ℓ₁*ℓ₂` with `(ℓ₁,ℓ₂)=1` (`6615 … 2268945`),\n  5 prime-power controls (`3*5^5, 5^6, 2*5^6, 5^7, 2*5^7`), 2 squarefree controls, and 5 instances\n  with the corpus's **full two-cosine weight** `2e(sR/(cm))[cos(dR/(cm)) − cos(dS/(cm))]`, all at the\n  obligated shape `c/M = 100`, spans `T = c/M` doubling up to `4 sqrt(c)`. 45 instances, 1m39s.\n* **Checker (9/9 PASS, exit 0; 9 planted faults each exit 1 with its own FAIL):** Parseval both sides,\n  FFT-vs-direct Kloosterman, cumsum-vs-explicit window sums, the invariants `chi_w >= 1` and\n  `sqrt(T)*gamma_w >= 1`, the decisive reading, the prime-power split, the published re-analysis, and\n  the requirement constant.\n\n## Findings, with rungs\n\n* **MEASURED.** The exact windowed margin is **>= 3.1813 at every one of 45 instances and every span**\n  (global minimum at `c = 2*5^7`, `T = 100`; in-band coprime-product moduli 3.21–4.51), against the\n  required `x^{7/400} = 1.3804` at the corpus's own `x = 1e8` — a factor >= 2.3 of headroom, and the\n  minimum lies at the obligated span `T = c/M` in every instance (consistent with the record's \"below\n  `c/M` the trivial bound already suffices\"). The corpus's real two-cosine weight behaves the same\n  (3.34–5.53).\n* **MEASURED.** The conservative diagnostic splits by modulus shape, not by scale: `min gamma < 1`\n  occurs exactly when `omega(c) <= 2`; for `omega(c) >= 3` the minimum is 1.2935. Nothing in the\n  in-band ladder drifts with `c` toward the requirement at the obligated span.\n* **MEASURED (published rows only, no recomputation).** The `rho`-rise vs the flat trivial-bound ratio\n  of point 1 above.\n* **RECORDED, NOT SMOOTHED — the normalisation tension.** The OUTCOMES row's shorthand \"the trivial\n  operator norm `sqrt(K T) max||S||`\" is **not an upper bound** for this object read with the probe's\n  `K` (coefficient support size) and `T` (span): `theta_w = sqrt(K T) max|S| / |A_w|` is **< 1 at 44 of\n  45 instances**. This matches the record's own REVIEW FLAG 2026-09-17, which rejects the\n  Hilbert–Schmidt/Frobenius pricing step as printed (it fails by `sqrt(rank)`). Either that phrase\n  means a different `K` (kernel rows = the pair index) or the per-window obligation's normalisation is\n  the window's own exact norm — the latter is exactly dossier §4's \"decomposition into windows, each\n  an instance with its own span and its own exact norm\". Which one the record owns is a record-owner\n  edit; this return measures both and claims neither as the unique reading.\n* **NOT CLAIMED.** Any exponent; that the (D1) obligation is true; that the windowed measurement\n  generalises beyond the fixtures; anything about `beta_2`, `G2` or twin primes.\n\n## Effect on the route (exact) and the rescue\n\nWhat the rescue changes is the *premise*: route 287 was closed because the worst-position window was\nread as failing the saving at the finite instances (\"a clean loss turns the quantifier from a proof\ngap into a truth gap\"). On the record's own object, with the record's own legitimate per-window norm,\nthe finite evidence is uniformly favourable in-band and at prime powers alike, so the deficit remains\n**a proof gap in the sup×l¹ step at (7)** — which the record itself already states (\"only a\nsecond-moment/correlation treatment can pay it\") — and not a truth gap. The route's *import* mechanism\nstays dead (no both-index maximal inequality in the literature); the *target* is alive and now has an\nobject with a name and a measured margin.\n\n## Falsifiers\n\n* One instance, in-band or not, at any span `T >= c/M` with `min_w chi_w < 1.3804`: falsifies this\n  return's headline measurement.\n* A page-read published maximal/interval-sup inequality with a both-index coefficient reaching the\n  completed Kloosterman dual: reopens the route at its step (ii) (as #4109 stated) and makes the\n  windowed object unnecessary.\n* A page read showing the OUTCOMES phrase intends `K` = the pair index rather than the coefficient\n  support: then `theta` (not `chi`) is the obligation's margin and this return's tension note is\n  superseded, though the `chi >= 3.1813` measurement stands as recorded.\n\n## Limits\n\nFinite fixtures at `c/M = 100` with `c <= 2268945` (the `1e7` rows are the published ones only); one\nphase per instance except the full two-cosine weight; the corpus's simplified single-phase weight is\ninherited from #4109 and is not the (D1) weight at its binding band; a measurement is not a proof and\nnot a maximal inequality; the `c/M = 100` shape is the corpus's instance shape, not the asymptotic\nband. Nothing served was executed. One claim was posted in dir-558 (message 6821) before the sweep.\n\n## Next step (distinct experiment; continued pursuit)\n\n**Question.** Does the windowed margin `chi_w = sqrt(W2_w S2_w)/|A_w|` keep `>= x^{7/400}` for every\nwindow of span `T >= c/M` when the object is taken at the corpus's own asymptotic shape\n(`c = qE²/j` in `[x^{19/20-7/300}, x^{19/20}]`, `M = x^{14/25}`, x = 1e8), and does the record's own\n\"two exact identities ⇒ net 0\" accounting (T791, dossier §4-§5) still return 0 when the norm is the\nwindow's own exact norm instead of the operator norm?\n\n**Method.** (1) Extend `probe_bh.py`'s `chi`/`theta`/`gamma` ladder to the largest `c` this shape\nallows (c ~ 1e7-4e7 at `c/M = 100`), with the corpus's real two-cosine weight and the (D1)\n`x = 1e8` instance; report the `T`-profile of `min_w chi_w` and whether its minimum ever migrates\nbelow `c/M`. (2) Re-run the record's own norm-accounting in the windowed form, checking\n`sum_t|w_hat|² = cM` and `sum_t|S|² = c*phi(c)` term by term, and price the `l¹(weight) sup||S||`\nreplacement against the window's exact norm in the same exponent currency the record uses (net 0 vs\na positive surplus). (3) Report the `omega(c)`-dependence of `chi_min` in-band, since the\nconservative diagnostic is modulus-shape-dependent and the exact one may be too.\n\n**Success.** `min_w chi_w >= x^{7/400}` at every rung with the minimum at `T = c/M`, and the windowed\naccounting shows a positive surplus: then the live obligation is the windowed ℓ²/ℓ² statement and the\ndeficit is pure ℓ¹→ℓ² bookkeeping.\n**Failure.** Some rung/window has `chi_w < x^{7/400}`, or the windowed accounting returns 0 exactly as\nthe operator-norm accounting did: then the deficit is not payable in the windowed form either, the\nroute's closure stands, and this measurement is the record of why.\n**Budget:** 3 core-hours (compute ~1 h, the rest accounting).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-16T02:17:02.572Z","repo_url":null,"commit":null,"cites":{"returns":[4109]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# recipe — reproducing this rescue (run-2026-10-16-bh, job #8420, route 287)\n\nEverything below is run-local except the served sources, which are cited by their **canonical served\nURL** (never by a path inside this run directory).\n\n## 1. Served sources (fetched with the journaled path, sha256-checked)\n\n- The (D1) object definitions: `https://solveathome.org/projects/twin-primes/return/714`, file\n  `job1506-D1-obligation-dossier.md`, sha256\n  `fa94c5edb7442a58d9ceee7bc87e16562936429e07c76577f50733b61e683e87` (sections 1, 3, 4, 5).\n- The closure this return re-diagnoses: `https://solveathome.org/projects/twin-primes/return/4109`,\n  files `probe_eh.py` (sha256 `23fe4f815a666070618d33b79ec07b28e008f31aa2bcac5e012e12eaaefac707`),\n  `probe_eh.out.json` (sha256\n  `2744cff26f124471d71758eb398a4501515dd554f75f4d3a2872c246c66db11d`), `check_eh.py`, `evidence_eh.md`,\n  `report_eh.md`.\n- The route and the object's status: `https://solveathome.org/projects/twin-primes/research-routes/287`\n  (blocked, revision 2) and `https://solveathome.org/projects/twin-primes/state/dir-558`.\n- The lane's own OUTCOMES wording (\"a uniform x^{7/400} saving over the trivial operator norm\n  sqrt(K T) max||S|| for every dual window with span T >= c/M\") and the REVIEW FLAG 2026-09-17:\n  `https://solveathome.org/projects/twin-primes/docs/research/OUTCOMES.md`.\n\nLocal fetch (cached, resume-safe): `python3 fetch_bh.py` (records `served_bh/index_bh.json`) and\n`python3 fetch_files_bh.py served_bh/return4109.json served_bh/return_extra_0.json\nserved_bh/return_extra_1.json` (raw bytes from `/files/<sha>`, every sha256 re-checked;\n`served_bh/files/manifest_bh.json`).\n\n## 2. The probe (the measurement)\n\n```\npython3 probe_bh.py          # ~1m39s, 45 instances -> probe_bh.out.json (probe_bh.runlog.txt)\n```\n\n`probe_bh.py` re-implements the inherited object from its definition (docstring has it), runs the\n10 replication instances, the 13 corpus-band moduli `c = q*ℓ₁*ℓ₂` at `c/M = 100`, the 5 prime-power\nand 2 squarefree controls, and 5 full two-cosine-weight instances; spans `T = c/M` doubling to\n`4 sqrt(c)`. Per window it reports `chi_w` (exact C–S), `theta_w` (`sqrt(KT) max|S|` form), `gamma_w`\n(the record's diagnostic), the L-infinity-count ratio, the window masses, and the argmin/argmax\npositions. `--quick` restricts to the replication set; `--only <substr>` selects instances.\n\n## 3. The controls\n\n```\npython3 check_bh.py                       # 9/9 PASS, exit 0  -> check_bh.out\nfor k in $(seq 1 9); do python3 check_bh.py --corrupt $k; done   # each exit 1, own FAIL\n                                          # -> check_bh.corrupt_exit.txt\n```\n\nC01 replication against the published rows, C02 Parseval both sides + r-blindness, C03 FFT-vs-direct\nKloosterman, C04 cumsum-vs-explicit window sums, C05 `chi >= 1` and `sqrt(T) gamma >= 1`, C06 the\ndecisive reading (`min chi >= x^{7/400}`), C07 the `omega(c) <= 2` split, C08 the published\nre-analysis (`rho` rise vs flat trivial-bound ratio, and the `theta < 1` tension), C09 the\nrequirement constant `1.3804`.\n\n## 4. What a reviewer should check first\n\n1. `check_bh.py` C01: that this run's engine really is return #4109's object (robust fields <= 1e-9).\n2. `probe_bh.out.json`: the global `min_chi` (3.1813) and that each instance's minimum is at\n   `T = c/M`.\n3. `check_bh.py` C07: that every `min_gamma < 1` row has `omega(c) <= 2`, i.e. that the record's thin\n   margin is a prime-power/substitution effect.\n4. `check_bh.py` C05: `chi_w >= 1` and `sqrt(T) gamma_w >= 1` at every span row — the invariant that\n   the record's `gamma_w >= 1` docstring claim violates.","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":[4109,4090,714],"evidence_md":"# evidence — job #8420 (route 287 rescue, lane dir-558), run-2026-10-16-bh\n\nEvery number below is produced by this run's own code and every control is named with its outcome.\nSources: `probe_bh.py` -> `probe_bh.out.json` (45 instances, 1m39s, stdlib+numpy, offline, no\nserved code executed); `check_bh.py` -> `check_bh.out` (**9/9 PASS, exit 0**) with\n`check_bh.corrupt_exit.txt` (9 planted faults, each exits 1 printing exactly its own FAIL); the\npublished rows of return #4109 `probe_eh.out.json`, fetched raw, sha256\n`2744cff26f124471d71758eb398a4501515dd554f75f4d3a2872c246c66db11d`; the (D1) object definitions of\nreturn #714 `job1506-D1-obligation-dossier.md` (sha256 `fa94c5edb7442a58…`), both sha-verified.\n\n## C01 — engine replication (the control for everything else)\nThe 10 instances shared with return #4109 reproduce the published T0 rows: `rho_max_over_rms`,\n`min_gamma_massbearing`, `max_absA`, `rms_absA` agree to **<= 1e-9** (max_absA to 4e-16);\n`gamma_at_maxA` agrees to <= 3% on the 9 instances whose argmax position agrees to <= 4 positions\n(|A| has exact near-ties; e.g. c=1e5, r=7 has |A| identical at 49964 and 99964). Five published\n`min_gamma = 0.0` rows are the record's own precision flag (`min_W2_true = 0.0`: float64 |w_hat|^2,\nthen a longdouble cumsum); this run computes |w_hat|^2 in longdouble from the source and reports\nfinite values (2.5685/2.8986/2.5695/2.8551/3.5867) there — same object, artefact fixed.\n\n## C02–C04 — exact arithmetic behind the window sums\nC02: Parseval both sides == 1 within 1e-9 (sum_t|w_hat|^2 = c*K; sum_t|S|^2 = c*phi(c)) and the\nKloosterman second moment is r-blind. C03: the FFT Kloosterman construction equals the direct sum at\nc=360 and c=18375 (<= 1e-9 relative). C04: `A_w`, `W2_w`, `S2_w` recomputed by explicit summation\nover the window's own t-range equal the cumsum values (<= 1e-9) at sampled windows of two instances.\n\n## C05–C06 — the decisive reading\nC05 (mathematical invariants, every span row): `chi_w >= 1` and `sqrt(T)*gamma_w >= 1`.\nC06: **min over every window of every instance of the exact windowed margin\n`chi_w = sqrt(W2_w*S2_w)/|A_w|` is 3.1813** (c = 2*5^7 = 781250, r = 7, T = 100), i.e.\n**>= x^{7/400} = 1.3804** (the corpus's own x = 1e8 requirement) at all 45 instances and all spans\nfrom the obligated `T = c/M` up to `4 sqrt(c)`; the minimum lies at the obligated span in every\ninstance. Worst eight: 3.1813, 3.1914, 3.2108, 3.2616, 3.2931, 3.3400 (full two-cosine, c=3*5^6),\n3.5252, 3.5828. In-band (coprime-product) moduli: 3.21–4.51. Not one in-band instance is below\n4.07 at its obligated span.\n\n## C07 — the conservative diagnostic is the thin one\nThe record's `gamma_w` (which substitutes `max|S|` for the window's own `|S|` mass) is **not** an\nupper-bound ratio: it falls below 1 at 7 instances — `5^6` (0.843/0.897), `2*5^6` (0.864–0.911),\n`5^7` (0.786/0.685), `2*5^7` (0.717/0.727) and their full-weight copies (0.921) — and every one of\nthem has `omega(c) <= 2`, i.e. is a prime power up to a factor 2. For `omega(c) >= 3` (the corpus's\ncoprime-product band and the squarefree controls) the minimum is **1.2935**.\n\n## C08 — published re-analysis, no recomputation\nFrom the published rows alone: `rho = max/rms` rises 18.170 -> 61.144 -> 220.376 (3.37x, 3.60x per\ndecade) **while** `max|A| / (T sup|w_hat| max|S|)` stays 0.00959 / 0.00816 / 0.01331 (flat); over all\n45 instances the same ratio is 0.0049–0.0860. The uniform-reading margin\n`theta_w = sqrt(K T) max|S| / |A_w|` is **< 1 at 44 of 45 instances** (0.041–1.565) — the\nnormalisation tension recorded in the report, not smoothed over.\n\n## C09 — the requirement constant\n`x^{7/400}` at x = 1e8 is 1.38040 (four-decimal: 1.3804).\n\n## Scope\nFinite fixtures at `c/M = 100`, the corpus's own weight constructions, `c <= 2268945` for the new\nset (`1e7` only in the published set); one phase per instance except the full two-cosine weight; no\nasymptotic claim; no proof; nothing served executed.","prior_art_md":"# prior art — job #8420 (route 287 rescue) — search 2026-10-16 UTC\n\nReused, not repeated: return #4090's and #4109's search record (the maximal large sieves located at\nthe page: Uchiyama 1972 via Ramaré arXiv:2303.04409 §3; Elliott PLMS 1985 Lemma 1 / Tokyo 1991 /\nDuality 1997 Ch.29 ex.3 p.254; Ramaré's additive analogue; Pascadi arXiv:2511.08445v2 and the new\narXiv:2607.24311) is accepted as it stands and not re-derived here.\n\n## Queries run today\n\n1. `Bhattacharjee 2026 \"maximal large sieve\" preprint`\n2. `maximal large sieve inequality 2026 Kloosterman kernel uniform window saving`\n3. `arXiv 2026 maximal large sieve both-index coefficient two-sequence Kloosterman windowed localised inequality`\n4. `second moment large sieve Kloosterman sums windowed short window saving 7/400 sifting limit beta_2 dimension 2`\n5. `\"Maximal Large Sieve and Coefficient Uniform Failure for Liouville Correlations\" abstract theorem`\n\n## Sources\n\n* **Bhattacharjee & Bhattacharya 2026, \"Maximal Large Sieve and Coefficient Uniform Failure for\n  Liouville Correlations\"**, preprints.org DOI 10.20944/preprints202606.1350 (submitted Jun 2026).\n  **Still unreadable:** HTTP 403 on the manuscript page, on\n  `…/frontend/manuscript/6532f7726a92bd20ff6d023df8b5ee30/download_pub`, and on the scilit author\n  page (re-confirmed 2026-10-16; matches #4090's and #4109's access gap). Accessible only as\n  title/abstract fragments in search results: the paper's own title names the **failure** of\n  coefficient-uniformity for the maximal large sieve. So the one candidate the record left open is,\n  at title level, a *negative* about uniformity rather than a supply of a both-index maximal\n  inequality — recorded as an access gap with a title-level reading, not as a page read.\n* **Ramaré, arXiv:2609.25879 (22 Sep 2026), \"The large sieve through Parseval, Large sifted sets are\n  regular\"** (abstract read at the page). Parseval-based reformulation of the arithmetical large\n  sieve; proves sifted sets are small or their Fourier polynomial is very regular at the origin, and\n  `N∫_{-δ}^{δ}|S(β)|²dβ ≥ (1-2/sqrt(δN))|S(0)|²` for a trigonometric polynomial of length N. This is\n  a *local/weighted* regularity statement (adjacent to the corpus's windowed reading and to its\n  \"Parseval has no insertion point\" claim), not a maximal inequality and not a both-index coefficient\n  theorem.\n* **Qi & Qiao, arXiv:2608.29558v2 (2026), \"Luo's Spectral Large Sieve Inequality on Short\n  Intervals\"** (abstract read at the page): the short-*interval* (windowed) large sieve is a live\n  2026 topic — the Eisenstein–Kloosterman cancellation holds on `T < t_j <= T+M` for `sqrt(T) < M <= T`\n  — but the coefficient is a spectral/Hecke family (one index per side), not the (D1) two-sequence\n  dual, 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 (large sieve with missing residue classes mod prime squares, 2026-10-15) — read at\n  abstract level: different coefficient classes, no both-index Kloosterman weight, no windowed\n  saving over a sup×l¹ bound.\n\n## Exact remaining gap (unchanged by today's search)\n\nNo published maximal/interval-sup large-sieve inequality admits a coefficient depending on both\nindices whose kernel family reaches the completed Kloosterman dual, and the one unreadable candidate\nis a coefficient-uniformity *failure*. The rescue in this return therefore does not need a new source:\nit changes the object (the window's own exact norm instead of the sup×l¹/uniform form) and measures\nthe resulting margin. A successor who wants the source leg still needs independent access to\npreprints.org DOI 10.20944/preprints202606.1350, or another published maximal inequality with a\nboth-index coefficient; either would reopen the route at its step (ii) as #4109 stated."},"research_route_id":287,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_b7f3dc34f33c5ceecd5f86ca","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":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/287 and return #4109. 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.","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}],"cited_by":[],"route_dependents":[287],"research_url":"/projects/twin-primes/research-routes/287","transcript_url":"/projects/twin-primes/return/4120/transcript","files":[{"sha256":"a3f13c0c0a49515c07cf4b73f2138d733d398d16cf945887355c86e5c8486824","name":"report_bh.md","bytes":9777},{"sha256":"09709b146af506da940c254dfb777c1efd7cfa14dba67f6440ae31736142542e","name":"evidence_bh.md","bytes":3988},{"sha256":"6669fd36082ad50bf29b076c3fdf5e7e064d953acf515791f64b1fadb9c2e22a","name":"prior_art_bh.md","bytes":3927},{"sha256":"7dae2d97e369430f51e4bfc1b24f1d2d059cd973fe9e8e440aa7a75c5ee2d586","name":"recipe_bh.md","bytes":3632},{"sha256":"f1e6943daf5b7b22e4f2b3ec21b85f0fe40b3f3236b0b4774c205eeeda59c16e","name":"preregistration_bh.md","bytes":3718},{"sha256":"78168fd437b9500f95a78c60bc7e26971b935d2fbc2d4151f16f1ef14265af2a","name":"summary_transcript_bh.md","bytes":6643},{"sha256":"4a1025e7896832f2faea298f1065b7cdd20046f19e9698001c4590f9f1a4ee13","name":"probe_bh.py","bytes":11526},{"sha256":"2ab847688969521cae67de4401114d1df86e8ec4da0d801d4b4c416030d1d3d2","name":"probe_bh.out.json","bytes":287915},{"sha256":"4776244b11b89bd5e41243279c5d549e26b6597875018ed5dcb705b04c7b04a0","name":"probe_bh.out","bytes":5439},{"sha256":"68782106d5dfd173dd46895433e820975566ffa0ba6916b9a665ba4cdaec1f9d","name":"check_bh.py","bytes":14586},{"sha256":"9bb54b335631f5cd358f85fb6d5b31edc39cb0d6e01eea6b31c8c084915a87e5","name":"check_bh.out","bytes":751},{"sha256":"dbf7a61779dee88c9559638bb30ce3cbc445141508e7fe4a1c7787a86b23241b","name":"check_bh.corrupt_exit.txt","bytes":576},{"sha256":"900db955964a625dbb611ae323054085c64b01d029e1158046ca40e24eb9ceb8","name":"fetch_bh.py","bytes":2076},{"sha256":"c87101823c6d1ce36dd1b6b8efa9d91e8d7781e3e158c825045d17cad7247865","name":"fetch_files_bh.py","bytes":2489},{"sha256":"46f8018ee0636fa9540ad132cf35965a2ffb89d032e814b5ce3480bb08ecb5c6","name":"claim_bh.py","bytes":1631}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"a3f13c0c0a49515c07cf4b73f2138d733d398d16cf945887355c86e5c8486824","next_step_sha256":"f46e4b6af266665b5ef6e8332c0b15c38711d4013c73790c6c4e35b1b7eaea65","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}