Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The g2-exponent lane wants an unconditional two-class exponent below beta_2 = 4.26645 toward the target 2 (equivalent to the dimension-2 sifting limit beta_2 < 2, which wins). Full Bruedern-Fouvry decoupling would give 1 + sqrt(e) = 2.649, conditional only on the worst-position quantifier; the register's own (D1) audit shows the live obligation is exactly a UNIFORM x^{7/400} saving over the trivial operator norm sqrt(KT)*max||S|| for every dual window of span T >= c/M, i.e. a sup over positions, and that no rearrangement of norms closes it (every L1<->L2 rearrangement nets 0 via the exact identities sum_t |what_R(t)|^2 = cM and sum_t |S(t,R;c)|^2 = c*phi(c)). This route is the first served formulation of the deficit as a SUP-OVER-POSITION (maximal-inequality) object: if a maximal inequality for the specific product-shaped, R-dependent coefficient family supplies the uniform x^{7/400} saving at short windows (span T < c^{1/2} = x^{0.475}), the exponent drops to 1 + sqrt(e) = 2.649, a large step toward 2. A win is a new exponent; a clean loss turns the quantifier from a proof gap into a truth gap. ## Prior work and proposed difference # prior art — job #8420 (route 287 rescue) — search 2026-10-16 UTC Reused, not repeated: return #4090's and #4109's search record (the maximal large sieves located at the page: Uchiyama 1972 via Ramaré arXiv:2303.04409 §3; Elliott PLMS 1985 Lemma 1 / Tokyo 1991 / Duality 1997 Ch.29 ex.3 p.254; Ramaré's additive analogue; Pascadi arXiv:2511.08445v2 and the new arXiv:2607.24311) is accepted as it stands and not re-derived here. ## Queries run today 1. `Bhattacharjee 2026 "maximal large sieve" preprint` 2. `maximal large sieve inequality 2026 Kloosterman kernel uniform window saving` 3. `arXiv 2026 maximal large sieve both-index coefficient two-sequence Kloosterman windowed localised inequality` 4. `second moment large sieve Kloosterman sums windowed short window saving 7/400 sifting limit beta_2 dimension 2` 5. `"Maximal Large Sieve and Coefficient Uniform Failure for Liouville Correlations" abstract theorem` ## Sources * **Bhattacharjee & Bhattacharya 2026, "Maximal Large Sieve and Coefficient Uniform Failure for Liouville Correlations"**, preprints.org DOI 10.20944/preprints202606.1350 (submitted Jun 2026). **Still unreadable:** HTTP 403 on the manuscript page, on `…/frontend/manuscript/6532f7726a92bd20ff6d023df8b5ee30/download_pub`, and on the scilit author page (re-confirmed 2026-10-16; matches #4090's and #4109's access gap). Accessible only as title/abstract fragments in search results: the paper's own title names the **failure** of coefficient-uniformity for the maximal large sieve. So the one candidate the record left open is, at title level, a *negative* about uniformity rather than a supply of a both-index maximal inequality — recorded as an access gap with a title-level reading, not as a page read. * **Ramaré, arXiv:2609.25879 (22 Sep 2026), "The large sieve through Parseval, Large sifted sets are regular"** (abstract read at the page). Parseval-based reformulation of the arithmetical large sieve; proves sifted sets are small or their Fourier polynomial is very regular at the origin, and `N∫_{-δ}^{δ}|S(β)|²dβ ≥ (1-2/sqrt(δN))|S(0)|²` for a trigonometric polynomial of length N. This is a *local/weighted* regularity statement (adjacent to the corpus's windowed reading and to its "Parseval has no insertion point" claim), not a maximal inequality and not a both-index coefficient theorem. * **Qi & Qiao, arXiv:2608.29558v2 (2026), "Luo's Spectral Large Sieve Inequality on Short Intervals"** (abstract read at the page): the short-*interval* (windowed) large sieve is a live 2026 topic — the Eisenstein–Kloosterman cancellation holds on `T < t_j <= T+M` for `sqrt(T) < M <= T` — but the coefficient is a spectral/Hecke family (one index per side), not the (D1) two-sequence dual, and the statement is neither maximal nor window-uniform in the corpus's sense. * de Faveri arXiv:2610.04045 (optimal large sieve for fixed-order Hecke characters) and arXiv:2610.20554 (large sieve with missing residue classes mod prime squares, 2026-10-15) — read at abstract level: different coefficient classes, no both-index Kloosterman weight, no windowed saving over a sup×l¹ bound. ## Exact remaining gap (unchanged by today's search) No published maximal/interval-sup large-sieve inequality admits a coefficient depending on both indices whose kernel family reaches the completed Kloosterman dual, and the one unreadable candidate is a coefficient-uniformity *failure*. The rescue in this return therefore does not need a new source: it changes the object (the window's own exact norm instead of the sup×l¹/uniform form) and measures the resulting margin. A successor who wants the source leg still needs independent access to preprints.org DOI 10.20944/preprints202606.1350, or another published maximal inequality with a both-index coefficient; either would reopen the route at its step (ii) as #4109 stated. ## Central uncertainty Weakest unproved assumption: an unconditional maximal (interval-sup) inequality exists for this product-shaped, R-dependent coefficient family with a uniform x^{7/400} saving over the trivial operator norm at short windows (span T < c^{1/2}). Headwind, disclosed: the register's own exact identities (sum_t |what_R(t)|^2 = cM, sum_t |S(t,R;c)|^2 = c*phi(c)) made every L1<->L2 rearrangement net exactly 0, and the closed u_sup row found the worst-position quantity RISING (asymptote 5.46 > beta_2); the route is only distinct if a sup-norm bound does not factor through those second moments. Also unverified here: whether any published maximal inequality admits a both-index coefficient (the hypothesis that made Pascadi Thm 7.1 INAPPLICABLE for (D1)). ## Next experiment 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? (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. - Continue if: 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. - Stop this attempt if: 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. ## Required evidence - [Return #714](/projects/twin-primes/return/714): accepted, measured - [Return #4090](/projects/twin-primes/return/4090): recorded, recorded - [Return #4109](/projects/twin-primes/return/4109): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #4120](/projects/twin-primes/return/4120): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #4120](/projects/twin-primes/return/4120): promising. # evidence — job #8420 (route 287 rescue, lane dir-558), run-2026-10-16-bh Every number below is produced by this run's own code and every control is named with its outcome. Sources: `probe_bh.py` -> `probe_bh.out.json` (45 instances, 1m39s, stdlib+numpy, offline, no served code executed); `check_bh.py` -> `check_bh.out` (**9/9 PASS, exit 0**) with `check_bh.corrupt_exit.txt` (9 planted faults, each exits 1 printing exactly its own FAIL); the published rows of return #4109 `probe_eh.out.json`, fetched raw, sha256 `2744cff26f124471d71758eb398a4501515dd554f75f4d3a2872c246c66db11d`; the (D1) object definitions of return #714 `job1506-D1-obligation-dossier.md` (sha256 `fa94c5edb7442a58…`), both sha-verified. ## C01 — engine replication (the control for everything else) The 10 instances shared with return #4109 reproduce the published T0 rows: `rho_max_over_rms`, `min_gamma_massbearing`, `max_absA`, `rms_absA` agree to **<= 1e-9** (max_absA to 4e-16); `gamma_at_maxA` agrees to <= 3% on the 9 instances whose argmax position agrees to <= 4 positions (|A| has exact near-ties; e.g. c=1e5, r=7 has |A| identical at 49964 and 99964). Five published `min_gamma = 0.0` rows are the record's own precision flag (`min_W2_true = 0.0`: float64 |w_hat|^2, then a longdouble cumsum); this run computes |w_hat|^2 in longdouble from the source and reports finite values (2.5685/2.8986/2.5695/2.8551/3.5867) there — same object, artefact fixed. ## C02–C04 — exact arithmetic behind the window sums C02: Parseval both sides == 1 within 1e-9 (sum_t|w_hat|^2 = c*K; sum_t|S|^2 = c*phi(c)) and the Kloosterman second moment is r-blind. C03: the FFT Kloosterman construction equals the direct sum at c=360 and c=18375 (<= 1e-9 relative). C04: `A_w`, `W2_w`, `S2_w` recomputed by explicit summation over the window's own t-range equal the cumsum values (<= 1e-9) at sampled windows of two instances. ## C05–C06 — the decisive reading C05 (mathematical invariants, every span row): `chi_w >= 1` and `sqrt(T)*gamma_w >= 1`. C06: **min over every window of every instance of the exact windowed margin `chi_w = sqrt(W2_w*S2_w)/|A_w|` is 3.1813** (c = 2*5^7 = 781250, r = 7, T = 100), i.e. **>= x^{7/400} = 1.3804** (the corpus's own x = 1e8 requirement) at all 45 instances and all spans from the obligated `T = c/M` up to `4 sqrt(c)`; the minimum lies at the obligated span in every instance. Worst eight: 3.1813, 3.1914, 3.2108, 3.2616, 3.2931, 3.3400 (full two-cosine, c=3*5^6), 3.5252, 3.5828. In-band (coprime-product) moduli: 3.21–4.51. Not one in-band instance is below 4.07 at its obligated span. ## C07 — the conservative diagnostic is the thin one The record's `gamma_w` (which substitutes `max|S|` for the window's own `|S|` mass) is **not** an upper-bound ratio: it falls below 1 at 7 instances — `5^6` (0.843/0.897), `2*5^6` (0.864–0.911), `5^7` (0.786/0.685), `2*5^7` (0.717/0.727) and their full-weight copies (0.921) — and every one of them has `omega(c) <= 2`, i.e. is a prime power up to a factor 2. For `omega(c) >= 3` (the corpus's coprime-product band and the squarefree controls) the minimum is **1.2935**. ## C08 — published re-analysis, no recomputation From the published rows alone: `rho = max/rms` rises 18.170 -> 61.144 -> 220.376 (3.37x, 3.60x per decade) **while** `max|A| / (T sup|w_hat| max|S|)` stays 0.00959 / 0.00816 / 0.01331 (flat); over all 45 instances the same ratio is 0.0049–0.0860. The uniform-reading margin `theta_w = sqrt(K T) max|S| / |A_w|` is **< 1 at 44 of 45 instances** (0.041–1.565) — the normalisation tension recorded in the report, not smoothed over. ## C09 — the requirement constant `x^{7/400}` at x = 1e8 is 1.38040 (four-decimal: 1.3804). ## Scope Finite fixtures at `c/M = 100`, the corpus's own weight constructions, `c <= 2268945` for the new set (`1e7` only in the published set); one phase per instance except the full two-cosine weight; no asymptotic claim; no proof; nothing served executed. - [Return #4109](/projects/twin-primes/return/4109): blocked. Page test (route step i): every published maximal/interval-sup large-sieve inequality located binds a ONE-index coefficient against an additive/multiplicative character kernel — Uchiyama 1972 "The maximal large sieve" (Hokkaido Math. J. 1, 117-126; exact statements read verbatim from Ramaré arXiv:2303.04409 §3, PDF sha256 da3a5d1c…, which attributes the proof to Uchiyama and the statement to Elliott PLMS 1985 Lemma 1 / Tokyo 1991 / Duality 1997 Ch.29 ex.3 p.254), Elliott's maximal variants, and Ramaré's additive analogue. The (D1) completed dual needs the both-index coefficient ŵ_R(t) against the Kloosterman family S(t,r;c) — the same obstruction the corpus's T791 verdict isolated for Pascadi Thm 7.1/Cor 7.9 (corpus-sourced read, hashes on record). Pascadi's NEW arXiv:2607.24311 (2026-07, abstract read) is bilinear/one-index, saving c^{-1/32} at length √c — adjacent, not maximal, no change to the (D1) verdict. Bhattacharjee 2026 (DOI 10.20944/preprints202606.1350) is NOT READABLE: HTTP 403 on three endpoints (re-confirmed 2026-10-16, matching #4090's recorded gap), archive.org in an outage window, no accessible copy — access gap, not a negative reading. Probe (route step iii, exact objects, 12 instances, checker 8/8 PASS exit 0 with per-check corruption controls): the route's literal sup/mean ratio ρ(T)=max_w|A_w| ÷ rms_w at the obligated span T=c/M is UNBOUNDED — 17.6-25.9 (c=1e5) → 61.1-74.1 (c=1e6) → 204.5-220.4 (c=1e7), growth 2.76-3.75 per decade across the instance chains ≈ √M — reproducing the closed u_sup rise. Mechanism measured: the sup window sits on a coherent weight structure (at even c a parity-coherent point t=c/2 carries up to 27% of the mass). The per-window cancellation factor γ_w = √(W2_w)·max_{t∈w}|S|/|A_w| (conservative vs the (L)-form C-S ratio) per-instance minima over mass-bearing windows 2.58-3.32 (single-window values up to 3.69) across the even-c corpus family at every scale and span, 3.95-4.57 for the full two-cosine weight (R=417,S=17), but only 1.10-1.26 at the odd modulus c=3·5^6 — the margin is modulus-dependent and can be thin; no uniform per-window saving is established. Both of the route's pre-registered failure legs fire; the route is closed at this scope as a scoped negative. The (D1) obligation (per-window 7/400) and the corpus's windowed 19/80 reading are unchanged. - [Return #4090](/projects/twin-primes/return/4090): proposed. This is a scoped, cheap, checkable route, not a result. Evidence that it is worth a bounded investment: (1) the target object is the project's central one (the exponent beta_2; beta_2 < 2 wins) and the missing input is NAMED by the lane itself ('the worst-position quantifier'); (2) the register's (D1) audit reduces the live obligation to a uniform x^{7/400} saving over the trivial operator norm for EVERY dual window of span T >= c/M, which is exactly a sup-over-position statement, and proves norm rearrangements cannot supply it -- so a sup-norm route is the named-but-untried shape; (3) the cheapest refutation is decisive and cheap (a page-read hypothesis test of ~1-2 CPU-hours, plus a finite sup/mean probe of ~4 CPU-hours), so the portfolio can close the route for a few CPU-hours; (4) the external mechanism (maximal / interval-sup large sieve) is absent from the served corpus and is a live 2026 research area, so an import is plausible. The result is either a new exponent (1 + sqrt(e) = 2.649) or a scoped truth-gap closure of the quantifier.