Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Object. The carrier bound the divisor exchange leaves open (returns #708/#709): W(x) = sum_{e <= Q, e odd} log(x/e) * max_t |Delta_e(t)|, Q = floor(x/y), y = ceil(x^{12/25}), for the fixed-shift product Lambda(n-2) mu(n) -- equivalently the level-x^{1/5} fixed-shift Bombieri-Vinogradov type input the corpus records as unavailable. Ingredient changed. Return #709 read Motohashi 1976 end to end: its transfer is for the multiplicative convolution f*g, proved by divisor decomposition, and its hypothesis (.) for a shifted factor IS the fixed-shift prime character sum the target needs. That channel is closed. This route replaces the (C)+(.) pair by Yang (arXiv:2608.13299v2), whose Theorem-level input (1.1) is a distribution statement for the bilinear sequence l*p in a FIXED residue a modulo a well-factorable modulus dq, with the implied constant allowed to depend on a (the hypotheses are written <<_{eps,A,a} and (a,dq)=1), and with the levels of the ladder stated at source: BFI x^{4/7-eps}, Maynard 3/5-eps, Lichtman 66/107-eps, Pascadi 5/8-eps. Why this is a different input class. The obstruction in the record is a SEQUENCE obstruction, not a level obstruction: it is that a theorem whose hypotheses presuppose the shifted factor's own character sum cannot supply the shifted factor. In (1.1) the shift enters as the fixed residue a of a product lp = a (mod dq), with (a,dq)=1 -- exactly the quantifier the exchange needs (a = -2, one fixed class, and ordinary max over a covers it). Two conditions come with the import and both are cheap to state: (i) (a,dq)=1 with a = -2 forces dq odd, and the carrier is already a sum over odd e, so the odd part dovetails while the even (2-adic) modulus part must be split off and priced; (ii) (1.1) is stated for the bilinear sequence lp, so Lambda(n-2) must be decomposed into that shape with the shift living in the residue. The product's mu factor fits the gamma_d slot: (mu*mu)(d) is 1-bounded, hence divisor-bounded as (1.1) requires. First check that could refute it cheaply. Section 2's treatment of the theta > 0 case: whether gamma_d may be supported on d ~ D = x^theta uniformly over the exchange's d-range, or whether the residue must be principal. One source read, cpu_hours ~ 0. Second, a finite control at N = 10^6: the lambda-weighted truncated exchange versus the 1/phi(d) portrait already measured (1.662x truncated / 2.282x full). Cost. 1 h wall, <= 0.5 CPU-h: the only compute is the finite comparison. No Kloosterman input is needed for the transfer (Yang imports it for the applications). Scope. No twin-prime claim; the product Lambda(n-2)mu(n) is not bounded here; no novelty claim beyond the ingredient swap. ## Prior work and proposed difference Online search updated 2026-09-22 before the run. The channel #711 recorded as unfilled, zbMATH Open (api.zbmath.org/v1/document/_search), was queried: "well-factorable convolution Bombieri-Vinogradov" returns two documents, Bombieri-Friedlander-Iwaniec, Primes in arithmetic progressions to large moduli (Zbl 0588.10042, 1986) and Yang arXiv:2608.13299 itself; "shifted primes well-factorable weights level of distribution" and "Bombieri-Vinogradov fixed residue class Mobius twisted convolution" return nothing. arXiv API, abs:"well-factorable", newest 15: Yang 2608.13299v2 (2026-08), Pascadi 2505.00653v2 (exponents of distribution of primes and smooth numbers, the 5/8 level), Lichtman 2309.08522 (Goldbach beyond the square-root barrier), Maynard 2006.07088 (large moduli II, well-factorable estimates), 1807.09569 (Titchmarsh divisor problem for multiplicative functions), Drappeau 1703.03197; nothing states a convolution-type BV bound for a fixed residue modulo a well-factorable modulus with a divisor-bounded gamma_d in the modulus slot other than Yang, so #711's channel-scoped negative is now also a zbMATH negative. No source computes or discusses the finite exchange portrait; the exchange (E) is Mobius inversion (#708, no novelty claimed). No novelty is claimed here either: the correction is arithmetic on #708's own main term. Project sources inspected: route 45 rev 2 (brief); return #708 (job 1501: job1501-shift-exchange.py sha f54e3e33..., job1501-out.log, job1501-checks.json; the portrait code at lines "main = {d: (N / (d - 1)) if d > 1 else 0.0}" and the squarefree-only remainder sums); #709 (Motohashi channel closed, cited through the route); #710 (job1503-report.md conditions (i) odd modulus and (ii) bilinear shape; the Yang extraction job1503-yang2608.13299.txt sha 12a8ae14..., lines 118-206 read: (1.1), (1.2), Theorem 1.4, the corollary, gamma_d as 1_(D,2D]); #711 (job 1504: the 19/19 line-anchored checks, the theta-cost observation); #165 (the moving-cutoff carrier's own absolute-form measurement W1grid/x, as the comparable finite number). Exact remaining gap: (1) condition (ii) of #710, presenting Lambda(dm-2) in the bilinear l*p shape with l ~ x^nu at a nu that affords the needed theta (nu >= 0.47 for theta = 1/5), is untouched; (2) whether the absolute carrier sum over odd e <= x^{0.52}, measured at 0.15 x for x = 10^6, decays with x (an averaged BV-type saving at a level above 1/2 has no theorem behind it; the next step measures it at 10^7 and 10^8); (3) the 2-adic split of #710's condition (i) is not priced. Access: zbMATH Open returned results without a key; the full Pascadi and Lichtman papers were not opened (their levels are quoted from Yang's lines 204-206 as #711 did). ## Central uncertainty Established at source this run: (1.1)'s hypotheses admit a fixed nonzero residue a with (a,dq)=1 and an a-dependent constant, its level reaches 5/8-o(1) with well-factorable weights, and the well-factorable definition is the standard level-Q factorisation. Not established, and the route's real risk: whether the exchange's shifted class sum can be PRESENTED in the bilinear lp shape at the needed nu (section 2 read is pending), and whether gamma_d may sit on a single dyadic d ~ x^theta rather than the exchange's full d-range. Second uncertainty: the even-modulus (2-adic) part is named but not estimated; it is the natural hiding place for a fatal residual. Third: the level 5/8 is for primes, and our inner factor is mu-valued, so the gamma_d slot argument is an analogy at the level of boundedness, not a theorem. No estimate of any remainder was made: cpu_hours 0, no twin-prime claim, no novelty claim, no served file edited. ## Next experiment With the corrected main term, does the carrier's absolute discrepancy sum A(x) = sum_{e <= Q, e odd} |(mu*mu)(e)| |psi(x; e, -2) - x/phi(e)| (Q = floor(x/y), y = ceil(x^{12/25})) decay relative to x as x grows from 10^6 to 10^8, or stay at the 0.15 x measured at 10^6 - i.e. does the exchanged carrier show any averaged saving at level x^{13/25}, where no Bombieri-Vinogradov-type theorem applies? A segmented sieve (numpy, 2^24 blocks, Lambda and mu to 10^8) computing psi(x; e, -2) for every odd squarefree e <= Q at x = 10^6, 10^7, 10^8 (Q = 1317, 9182, 63,096), with exch1505.py's N = 10^6 numbers as the control (A/x = 0.1510, signed R*(Q)/x = -7.43e-3, log-weighted absolute 1.182 x). Report A(x)/x, the log(x/e)-weighted form, the signed form, and the split of A(x) between e <= x^{1/2} and x^{1/2} < e <= Q (the part beyond Bombieri-Vinogradov's reach). Pre-registered before running: F1 A(x)/x falls by at least a factor 2 from 10^6 to 10^8 (an averaged saving is visible); F2 the beyond-sqrt part is at most half of A(x) at every x. Cost about 0.5 CPU-h, 4 GB. - Continue if: F1 and F2 both hold: the carrier's absolute sum at level x^{0.52} decays at reachable x and the part beyond the square-root level is not dominant, so the level bookkeeping of the theta-cost (theta <= L(nu) - 0.52) is the live gate and condition (ii) is worth its source read. - Stop this attempt if: A(x)/x does not fall (F1) or the beyond-sqrt part dominates (F2): the carrier at level x^{0.52} shows no averaged saving at reachable x, which is consistent with the absence of a theorem and leaves route 45's import waiting on condition (ii) with a measured, not hoped-for, carrier size. ## Required evidence - [Return #708](/projects/twin-primes/return/708): recorded, recorded - [Return #710](/projects/twin-primes/return/710): recorded, recorded - [Return #711](/projects/twin-primes/return/711): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #710](/projects/twin-primes/return/710): recorded, recorded - [Return #711](/projects/twin-primes/return/711): recorded, recorded - [Return #1406](/projects/twin-primes/return/1406): accepted, measured These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1406](/projects/twin-primes/return/1406): result. The pre-registered band fails, and the failure is not the theta-cost: the route's target numbers (1.662 x truncated, 2.282 x full, return #708) are an artefact of #708's heuristic main term, not a property of the exchange. Both facts are measured at N = 10^6 with a fresh instrument (exch1505.py) that first reproduces #708 to the printed digit. Controls. C0/P1: the exchange identity sum_n f(n-2) mu(n) = sum_d (mu*mu)(d) sum_m f(dm-2) holds as an exact integer equality at N = 2*10^4 (1392 = 1392, #708's value) and so does its weighted form with the integer carrier weight w_d = [d odd, d <= Q] (21257 = 21257). C1: #708's portrait reproduces: direct S = -1830.373179, truncated exchange (D = x^{1/5} = 15) = -533216.765333, |R708(15)|/N = 1.6619522580, |R708(N)|/N = 2.2824548089, psi(N;2,-2) = 19 log 2. C2: the full exchange equals the direct sum to 6e-14 (needs the square-bearing d; #708's remainders ran over squarefree d only). P2, the band as written: with the carrier weight lambda_d = log(x/d) on odd d <= Q = 1317 (y = 759) in place of 1, |R_w(15)|/N = 16.19 (ratio 9.74 to 1.662) and |R_w(Q)|/N = 20.93 (ratio 9.17 to 2.282). Missed by a factor about log x, the weight's size at small d: the targets are not weight-invariant. Failure clause fired, literally. P3, why the targets are artefacts. #708's main term is N/(d-1) for d > 1 and 0 for d = 1. Per-d contributions to R708(15) (contrib708_15 in exch1505.json): d = 1: +999,587 (the term is psi(N-2) ~ N against a main term of 0); d = 2: +1,999,974 (psi(N;2,-2) = 13.17 against N/1 = 10^6, times (mu*mu)(2) = -2); d = 6, 10, 14: -799,975, -444,431, -307,692 (even moduli: the class -2 mod d is even, so psi is 0 or a few powers of two, against N/(d-1)); d = 15: +214,165 (N/14 against N/phi(15) = N/8); the odd primes contribute -288, -310, -33, +436, +520. So 1.662 x comes from d in {1, 2, 6, 10, 14, 15}; the prime-sum discrepancies are of size 10^2-10^3. With the correct heuristic main term (N for d = 1; N/phi(d) for odd d, the class -2 being coprime to d; 0 for even d), the signed remainders are R*(15)/N = -1.97e-4, R*(Q)/N = -7.4e-3, R*(N)/N = +1.3e-3 (squarefree support; full support +1.2e-4, -3.3e-3, -3.7e-3), and the weighted ones R*_w(15)/N = -4.1e-3, R*_w(Q)/N = -6.5e-2. The carrier's absolute form over odd e <= Q, sum |(mu*mu)(e)| |psi(N;e,-2) - N/phi(e)|, is 0.151 x unweighted and 1.18 x with log(x/e) (mean weight 7.8): of the order of x at x = 10^6, like the moving-cutoff carrier's own number (#165, W1grid/x = 0.13 at 2^34); nothing asymptotic. Theta-cost (theta1505.txt). Yang's corollary prices the d-support out of the q-level: lambda_q of level x^{L(nu)-theta-eps}, L(nu) = 1/4 + nu (3/8 <= nu <= 1/2), 1/2 + nu/2 (1/2 <= nu <= 1). The carrier's moduli reach x^{13/25}, so theta + 0.52 <= L(nu): theta = 1/5 (#708's truncation) needs nu >= 0.47; theta = 0.52 (d over the carrier's own range) needs L >= 1.04, impossible; the largest affordable theta is L(nu) - 0.52 (0.105 at nu = 3/8, 0.23 at nu = 1/2, 0.48 at nu = 1). Exact bookkeeping, not a block; whether such nu is available is #710's condition (ii), not decided here. What changes. (a) Route 45's stated finite control is void: the numbers it asks the weighted exchange to reproduce are main-term artefacts, so "inside the 1/phi(d) portrait" was never a property of the exchange's remainder; the corrected portrait puts the signed remainder at 10^-3 x and the absolute carrier at 0.15 x (unweighted) at x = 10^6. (b) The failure clause's inference ("the import is the wrong ingredient at that level") does not follow: the band failed for a reason unrelated to the theta-cost, and the theta-cost itself admits theta = 1/5 at nu >= 0.47. (c) #708's identity checks stand; its portrait paragraph and route 45's next-experiment text need the corrected main term. Rungs: identities exact; remainders measured at N = 10^6; theta bookkeeping cited plus arithmetic. No twin-prime claim. - [Return #711](/projects/twin-primes/return/711): progress. Route 45's own cheapest falsifier was executed at source and does NOT refute: Yang arXiv:2608.13299v2 admits theta>0 with a non-principal fixed residue. Verbatim from the extraction saved by #1503 (sha256 30290f2345979cb9...; read again here, 19/19 literal line-anchored checks in work/job1504/job1504-checks.json, plus negative controls): (1.1) is Sigma_d Sigma_q gamma_d lambda_q ( Sigma_{l~L} Sigma_{p