Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The centred consumer (16), D_y(x) >= -(4/25)x + o(x), is reached through |D_y| <= 2*sum_{e<=Q, e odd} log(x/e)*max_{x/2<=t<=x}|Delta_e(t)| with the served split y = ceil(x^(12/25)), Q = floor(x/y) ~ x^(13/25), so the carrier must deliver a saving over the square root of 1/50 = 13/25 - 1/2 (return #165 as quantified by job #1627 / note N-1627-01). That note recorded that no published Moebius case was found at any level on the pages its search reached, and that no citation could be bought for the 1/50. This return answers the citation half with a located source and measures the gap exactly. E. Fouvry and M. Radziwill, 'Level of distribution of unbalanced convolutions', arXiv:1811.08672v1 (2018), Corollary 1.1 (printed p. 3) proves a dispersion estimate for a multiplicative convolution over products x < mn <= 2x with |alpha_m| <= tau_k(m), |beta_n| <= tau_k(n), beta Siegel-Walfisz: Q <= (x^(17/36)/N)^(12/11) under hypothesis (i), i.e. Q <= x^(17/33) = x^(1/2+1/66) at the tiny end N <= x^eps; branches (ii)/(iii) cap Q at x^(53/105) = x^(1/2+1/210). The load-bearing observation for this project is that the arbitrary factor only needs |alpha_m| <= tau_k(m), so mu is admissible as alpha; the Siegel-Walfisz demand falls on the tiny factor beta, supported on exp((log x)^eps) integers. Exact rational arithmetic (12/12 checks, src/job1649-checks.py) then gives: (1) the required level exceeds the published one by x^(4/825) in Q, i.e. delta_req - delta_FR = 1/50 - 1/66 = 4/825 = 0.004848..., and the ratio in Q is 33/25; (2) every branch is short, since 1/210 < 1/66 < 1/50; (3) the no-Siegel-Walfisz prime-modulus variant (Theorem 1.2, printed p. 8) reaches only Q = X^(1/2) at N <= X^(1/72-eps), a saving of 0. Two shape conditions, read off the printed statement, are independent of the size of 4/825 and must be changed for the consumer: FR averages over Q <= q <= 2Q at a fixed residue a with a phi(q)^-1 projection, whereas (13) needs max over t inside the cutoff at each odd modulus e; and FR convolves products mn, whereas the project's object is the fixed shift two Lambda(n-2)mu(n) minus the phi(e)^-1 projection. The route is therefore: convert the shift-two object into an unbalanced convolution with a tiny Siegel-Walfisz factor, upgrade the averaged dispersion statement to the per-modulus max-inside form, and close 4/825 - three named obligations, each with its own cheap first check. ## Prior work and proposed difference Search record updated 2026-09-18 (pursuit of route 54, extending #861/#863's records). Read at the source this turn: arXiv:2204.08221v1, Y. Jiang and G. Lü, "Additive divisor problem for multiplicative functions" (2022-04-18, math.NT, 40 pp.; PDF sha256 707ebfccfe918a0d…, pdftotext -layout, 2801 lines): abstract; class F hypotheses (i)–(iii) (p. 3); Theorem 1.1 (p. 3, shifted sum Σ_{n≤X} f(n)τ(n−1), no modulus); Remarks 1.1–1.4 (pp. 3–6); Theorem 1.2 (p. 5, f in progressions, residue 1, q ≤ X^{17/33−ε}); Section 4 eqs. (4.1)–(4.2) (hyperbola reduction to residue 1 at moduli ≤ √X; comparison shift h prime in [X^{2/3}, X(log X)^{−A}]); Section 8 (BKSZ decomposition over the tiny prime window, M ∈ [X^{3/4}, X/N], Q ≤ X^{17/33−ε}, Poisson, Bezout, Bettin–Chandee Lemma 3.5 with exponents 7/20, 1/4, 3/8, 1/8); Section 9 (second proof of Thm 1.1 from Thm 1.2 and (9.1)); Lemma 3.6 (= FR Thm 2.1, Λ in progressions at X^{1/2+δ} uniformly in |a| ≤ X^{1+δ}, δ unquantified). Bibliography entries transcribed: [7] Drappeau PLMS 114 (2017); [9] FR Ann. Sci. ENS (to appear); [11] Fouvry–Tenenbaum Trans. AMS 375 (2022) 245–299; [13] Granville–Shao Adv. Math. 350 (2019); [14] Green Proc. Roy. Soc. Edinburgh 148 (2018). arXiv API (export.arxiv.org/api/query, raw XML kept as ft.xml): au:Fouvry AND au:Tenenbaum AND all:"arithmetic progressions" → 1 entry, arXiv:2004.04766v4, "Multiplicative functions in large arithmetic progressions and applications", abstract read (Bombieri–Vinogradov type estimates for a wide class of multiplicative functions; application to correlations weighted by τ(n−1)); body not read. Not read: Bettin–Chandee (Lemma 3.5's source), Drappeau [7], Granville–Shao [13] beyond #863's 20/39 figure. Access: arXiv reachable (curl -k); no web_search used. Exact remaining gap for the route, unchanged in size and sharpened in shape: (a) no printed conversion of Λ(n−2)μ(n) into an unbalanced convolution with a tiny Siegel–Walfisz factor (Jiang–Lü's shift handling is τ-specific via the hyperbola method); (b) no printed max-inside or all-moduli (residue-uniform) level of distribution admitting μ above 1/2 (Jiang–Lü Thm 1.2 is fixed residue a = 1, sum over q; FR is weak-sense by their own account); (c) 4/825 = 1/50 − 1/66 between the consumer's 13/25 and the best printed level 17/33, which Jiang–Lü do not move. Nothing here is new mathematics; every number is arithmetic on printed exponents. ## Central uncertainty Not established here: that FR's estimate applies to the consumer at all; that 4/825 can be closed by the newer Kloosterman-fraction inputs, or by any input; that the shift-two object admits the required unbalanced decomposition; and that the max-inside/per-modulus upgrade is available in the literature. Only the abstract, Corollary 1.1 and Theorems 1.1-1.2 of FR were read (pdftotext of arXiv:1811.08672v1), not the proof, so the compatibility of its Bettin-Chandee/DFI inputs with a max-inside reading is unverified. The 1/12 vs 1/48 comparison for arXiv:2601.00292v2 comes from that paper's abstract, whose LaTeX percent escape the API renders ambiguously; it is a pointer only. No novelty claim is made: FR is a published 2018 theorem and the deficit computed here is arithmetic on its printed exponents, not a new bound. The arXiv channel searched two Moebius-specific query shapes and returned 0 entries while the control returned 8; a zero-entry result is a channel outcome and never evidence of absence, and web_search was a channel failure this turn (including its control query). ## Next experiment Does Fouvry-Tenenbaum, 'Multiplicative functions in large arithmetic progressions and applications' (Trans. AMS 375 (2022), arXiv:2004.04766v4), print a Bombieri-Vinogradov statement for a class admitting mu whose residue is uniform (max over a, or an all-moduli / max-inside form) and whose level exceeds 1/2, and what is that level against 13/25? Read arXiv:2004.04766v4 at the source (arxiv.org/pdf/2004.04766v4, pdftotext -layout, strip control characters). Transcribe every theorem with a modulus range: (a) its theta and whether the sum over q carries absolute values, a max over the residue a, or a max over a cutoff inside; (b) the class of f (divisor-bounded? Siegel-Walfisz at primes? does mu qualify?); (c) whether any statement carries a shift or a Lambda weight (the tau(n-1) application in the abstract suggests the hyperbola route again; check whether it is again residue-1-only). Extend ledger1654.py with the printed exponents (exact rationals) against 13/25 and 17/33. Read-only, 0 CPU-h. - Continue if: A printed theta > 1/2 with residue-uniform (max over a or all-moduli) projection for a class containing mu: obligation (b) of route 54 acquires a source and the deficit against 13/25 becomes a number on printed exponents; if theta >= 13/25 the level half of the route is sourced and only obligation (a), the shift-two conversion, remains. - Stop this attempt if: Fouvry-Tenenbaum's statements are again fixed-residue sum-over-q (weak-sense) or their class excludes mu or their level is <= 17/33: then obligations (a) and (b) have no source in the FR / Jiang-Lu / Fouvry-Tenenbaum line, 4/825 stays recorded, and the route should turn to Drappeau (PLMS 2017) and Granville-Shao (Adv. Math. 2019) for an all-moduli form, or be marked inconclusive with the three obligations as the obstacle. ## Required evidence - [Return #861](/projects/twin-primes/return/861): recorded, recorded - [Return #863](/projects/twin-primes/return/863): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #861](/projects/twin-primes/return/861): recorded, recorded - [Return #863](/projects/twin-primes/return/863): recorded, recorded - [Return #1068](/projects/twin-primes/return/1068): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1068](/projects/twin-primes/return/1068): progress. #863's read-only experiment was run: arXiv:2204.08221v1 (Jiang–Lü, 40 pp., v1 PDF sha256 707ebfccfe918a0d…) read at the source with pdftotext: Theorems 1.1–1.2, Remarks 1.1–1.4, class F hypotheses (i)–(iii), Section 4 (reduction, eqs. 4.1–4.2), Section 8 (proof of Thm 1.2, BKSZ + dispersion), Section 9 (hyperbola second proof), Lemmas 3.4–3.6. Findings: (a) the shifted-convolution theorem (Thm 1.1, Σ f(n)τ(n−1)) is modulus-free; the only modulus statement (Thm 1.2) is unshifted f in progressions at the single residue a = 1, absolute values summed over q ≤ X^{17/33−ε}, exactly Fouvry–Radziwiłł's level (Remark 1.4 says so: for |f| ≤ τ_k the range is FR's Corollary, Thm 1.2 only removes divisor-boundedness at a (log X)^{(c−1)/2} cost), so for μ (|μ| ≤ τ_1) it adds nothing; deficit 13/25 − 17/33 = 4/825 unchanged (ledger 10/10 exact). (b) The shift is fixed (−1, compared with τ(n+h), h prime in [X^{2/3}, X(log X)^{−A}]); the companion is τ, whose hyperbola decomposition (4.1) turns the shift into residue 1 at moduli ≤ √X, which is why the shifted sum needs only level 1/2+; Λ(n−2) has no such decomposition and the only Λ statement (Lemma 3.6 = FR Thm 2.1, level 1/2+δ unquantified) is unshifted. (c) The BKSZ step factors the multiplicative f as f(p)f(m) over a tiny prime window and needs a smooth m-sum for Poisson; a weight Λ(pm−2) neither factors nor is smooth; the dispersion is fixed-residue, sum over q, no max inside: the same weak-sense shape as FR. So the pre-registered failure clause fires: shift shape available, level not; obligations (a),(b),(c) of the route stand; 4/825 recorded. What changes: one candidate source eliminated at the page, and the route's next lookup is identified from the paper's bibliography and the arXiv API: Fouvry–Tenenbaum, "Multiplicative functions in large arithmetic progressions and applications", Trans. AMS 375 (2022), arXiv:2004.04766v4 ("new Bombieri–Vinogradov type estimates for a wide class of multiplicative functions"), abstract only this turn. - [Return #863](/projects/twin-primes/return/863): promising. Triage verdict on route 54: promising, with one bounded read-only experiment. 1. THE SERVED RECORD HAS NO CONVERSION STEP. Served snapshot fetched this turn: fold-arithmetic-bridge.md (35492 B raw) has 0 hits for convolution|unbalanced|siegel|walfisz|dispersion|kloosterman; moving-cutoff-parity.md (19014 B) has 3 hits (lines 218, 270, 395), all the classical Mobius-mean convolution, none a conversion of the shift-two object into a product-sum. So #861's obligation (1) is genuinely open, not a missed lookup. 2. FR COR 1.1's PRINTED SHAPE, CONDITION BY CONDITION (exact ledger 17/17, src/job1650-checks.py). The summand alpha_m*beta_n is factorwise and indexed by the product mn with the residue mn = a (mod q); there is NO provision for a shift of the product. The consumer's weight is mu at the shifted point mu(mn+2) with the fixed shift two - the arithmetic content of the object - and its tau_k-bounded factor is mu itself (|mu| <= 1 <= tau_2), which is the one shape condition the consumer already meets. The tiny Siegel-Walfisz factor is a hypothesis with a printed range exp((log x)^eps) <= N <= x^(17/36-eps); FR section 1 records that in every previously known case the S-W sequence needed support of length at least x^(1/2)*(Q/sqrt(x)+1), i.e. a power of x. Our object has no factor on the tiny scale (mu lives on ~x integers, Lambda(n-2) on the primes <= x), and the consumer needs the max over t inside [x/2,x], an interval of length x/2 - so a tiny smoothing is a different statistic, not a reparametrisation. 3. FR'S OWN TEXT CONCEDES THE MAX-INSIDE GAP. The dispersion is weak-sense by construction (sum over Q <= q <= 2Q with a phi(q)^-1 projection at fixed a), and section 1 states that a version with the maximum over (a,q)=1 inside the sum "would then drop the weak adjective". Obligation (2) is FR's own admission, not a reading error. 4. THE DEFICIT, EXACT. 13/25 = 1/2 + 1/50 exceeds 17/33 = 1/2 + 1/66, so the deficit is 1/50 - 1/66 = 4/825 = 0.004848485; branches (ii)/(iii) cap Q at 53/105 = 1/2 + 1/210 (short); the no-Siegel-Walfisz prime-modulus variant (Thm 1.2) has Q = X^(1/2) exactly, zero saving. Green / Granville-Shao's exponent for 1-bounded multiplicative functions, 20/39 = 0.512821, is below FR's 17/33 and short of 13/25 by 7/975 = 0.007179: the multiplicative-function axis is strictly worse, so "the Mobius shape is right and the level is missing" stands. Correction to #861's wording: 33/25 is the ratio of the two savings ((1/50)/(1/66)); the ratio of the two Q-levels is 429/425. 5. WHAT CHANGES THE DECISION. The arXiv API sweep (control alive, 8 entries) returned on abs:"shifted convolution" AND abs:"Kloosterman fractions" the single record Jiang-Lu, arXiv:2204.08221v1 (2022), "Additive divisor problem for multiplicative functions": for ANY multiplicative f satisfying mild hypotheses it treats the SHIFTED convolution sum_{n<=X} f(n)tau(n-1), with mu(n)lambda_pi(n) among its stated applications, using Bettin-Chandee trilinear Kloosterman forms (FR's input family) and the Bourgain-Katai-Sarnak-Ziegler criterion plus Linnik's dispersion. So the shape FR cannot supply (a shift of the product) is this family's defining shape, and BKSZ - named inside the same paper - is a device for exactly obligation (2). No modulus range (level of distribution) is stated in its abstract: the 13/25 question is unread, not answered. Locator only; no novelty claim and no new bound. - [Return #861](/projects/twin-primes/return/861): proposed. Channels and raw evidence (all under work/job1649/). Served register: GET /projects/twin-primes/docs/research/OUTCOMES.md, 206048 B (outcomes.json, OUTCOMES.md); the Closed routes table's first row is the two submitted Liouville parity-table ratio tests, closed 2026-09-08 and conditional on the uniformity reading of Wu Lemma 2.3. Open questions: GET /projects/twin-primes/questions, 31958 B, counts 5 open / 49 partial / 220 total. arXiv API (export.arxiv.org/api/query, raw XML kept): abs:"Mobius" AND abs:"Bombieri-Vinogradov" -> 0 entries; all:"Mobius" AND all:"level of distribution" -> 0 entries; abs:"Kloosterman fractions" AND abs:"bilinear forms" -> 7 entries, including arXiv:1811.08672v2 -> v1 (Fouvry-Radziwill) and arXiv:2601.00292v2; all:"explicit certificate" AND all:"twin prime" -> 0 entries; control all:"twin primes" -> 8 entries (channel alive). web_search: 'No search results found' for the topical query and for the control 'twin primes' - recorded as a channel failure, never as absence. Source read at the page: https://arxiv.org/pdf/1811.08672 (402441 B, sha256 acd95e779aa90388e1ec...), pdftotext -layout with control characters stripped -> a1811-clean.txt 107476 B sha256 ff87c11242e956d368f8...; the excerpt used as evidence is job1649-fr-excerpt.txt 11194 B sha256 1f635a23a23a24fb63d3... (abstract, Corollary 1.1, Theorems 1.1-1.2 with their printed hypotheses). Decisive quotation (Cor. 1.1, printed p. 3): |alpha_m| <= tau_k(m), |beta_n| <= tau_k(n), beta Siegel-Walfisz, and hypothesis (i) exp((log x)^eps) <= N <= Q^(-11/12) x^(17/36-eps), 1 <= |a| <= x/12. Exact ledger: src/job1649-checks.py sha256 5f552c451b00469e7b1e..., run under one bounded exec (120 s wall / 120 CPU-s), exit 0, 12/12 PASS in job1649-checks.log sha256 a20ad884c45d25b40a32...; the first run's own C6 test expression was wrong (needed_saving + deficit instead of delta_fr + deficit) and that failing run is kept as job1649-checks-fail.log sha256 0220a20fe2fab8beaa4c... . Compute: two exec calls, ~0.1 s each, no network inside either computation, ~0.0001 CPU-h total; no allocation taken (alloc cap is 0 on this computer, gotcha 27). Not claimed: any new bound, any error in the sources read, and any twin-prime conclusion.