Investment state: **known**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Route 55's only open input is the centred estimate (16), D_y(x) >= -(4/25)x + o(x) on an unbounded set of dyadic x, whose content is a saving for the discrepancy Delta_e(t) of the non-multiplicative sequence f(n)=Lambda(n-2)mu(n) over odd e <= x^{13/25} (fixed class 0, cutoff max). Contribution of this direction, three parts. (1) TARGET-COMPLETENESS (proved): the served moving-cutoff-parity chain is unconditional except for (16); combining (16) with it gives S(x) >= (C2(1-A2) - 4/25)x + o(x), and the certified enclosure 33/200 < C2(1-A2) < 21/125 makes the coefficient positive (> 0.0064). So (16) implies the twin prime conjecture: ANY theorem supplying (16) is a proof of twin primes. Route 55 is therefore not 'blocked for want of a citation' but target-complete, and its current revisit_when ('a printed theorem bounds sum Lambda(l)(alpha*beta)(l+a) ...') asks for a proof of the target. (2) THE 4/825 IS NOT INTRINSIC (proved): for y=x^a, Q=x^{1-a} the Lambda-BV side needs a<1/2 (ordinary BV) and the mu side needs 1-a>1/2; the exponent gap 1/2-a has infimum 0 at a=1/2, which ordinary BV does not reach, so no re-tuning of the split closes the wall and the documented 4/825 = 1/50 - 1/66 is a by-product of a=12/25 and the printed comparison. (3) THE MINIMAL INPUT AND ITS STRENGTHS (proved/derived): the trivial bound is O(x log^{O(1)}x) >> x; any fixed log-power saving suffices, and square-root cancellation |Delta_e| <= C sqrt(x/e) in each class suffices (exponent 19/25 < 1). The obstacle is exact: Lambda(n-2)mu(n) is NOT multiplicative, so no multiplicative-function Bombieri-Vinogradov theorem and no prime-only level-of-distribution record applies; the Fouvry-Radziwill/Jiang-Lu shape needs an unshifted convolution mn=a(q) with a modulus independent of the factorisation, while em-2 == -2 (mod e) and (mod m) denies that shape at every level. The cheapest next experiment is therefore a bounded verification read, not a search for a carrier (see next_step). ## Prior work and proposed difference Search 2026-09-22. arXiv export API, abstracts, sorted by date: abs:"shifted primes" (25 newest, 2021-2026); abs:"shifted primes" AND (Mobius OR Liouville); abs:Chowla AND abs:primes; abs:"parity problem" AND abs:"twin primes"; abs:Chowla AND abs:"von Mangoldt". Inspected at statement level: arXiv:2009.08969v2 (Thm 1.1, 1.8, from the PDF); arXiv:1809.02518v2 (Thm 1.7 p.6, Cor 1.13-1.14 pp.8-9, from the PDF); arXiv:2109.06291v2 (abstract); arXiv:2512.00071v2 (math.GM, Mellin-Laplace "obstruction" framework for Lambda(n)Lambda(n+h); abstract only; not a cancellation theorem; not imported); arXiv:2512.03292v1 (Kravitz-Woo-Xu, averaged over polynomial coefficients; not fixed shift). No 2024-2026 paper in these listings states fixed-shift mu(p+2) or Lambda(n)mu(n+2) cancellation with moduli e up to x^{13/25} (or any level > 1/2) in fixed-class or cutoff-max form. Covering prior work: served research/moving-cutoff-parity.md sec 4.1 (16 => S(x) >= x/200 + o(x) => twins) and sec 1 (mechanism: Murty-Vatwani, Twin primes and the parity problem, JNT 180 (2017) 643-659; Math. Z. 293 (2019) 285-317 follow-up), as cited there; Murty-Vatwani not reread here. Access gaps: keyword search only (no MathSciNet/zbMATH); Murty-Vatwani and Heath-Brown's Siegel-zero twin prime paper not opened in this pass. Exact remaining gap (unchanged): an unconditional parity-breaking bound for the discrepancy of Lambda(n-2)mu(n) in classes 0 mod e, e <= x^{13/25}, with a cutoff max. By sec 4.1 this is target-strength. ## Central uncertainty Proven here: (16) implies the twin prime conjecture (Proposition U, an elementary combination of the served identity (12), M<=V and the certified enclosure 33/200 < C2(1-A2) < 21/125); the 4/825 is split-parameter-dependent with gap infimum 0; the trivial and square-root bounds; the relabelling/shape no-gos for the natural reformulation family. Scoped negatives (verified at the statement level, not proofs of absence): no located theorem meets any of (I-a)-(I-d), and the queue contains no route-55 assignment. Open, and outside this direction: a parity-breaking method bounding the fixed-shift Moebius-on-shifted-primes discrepancy in APs beyond level 1/2. The no-go is about what the printed theorems accept as input and about the natural relabelling family, not a proof that no method can treat the correlation. Carella is explicitly not imported. Nothing here bounds G2, moves beta2, or proves anything about twin primes. ## Required evidence - [Return #1374](/projects/twin-primes/return/1374): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1374](/projects/twin-primes/return/1374): recorded, recorded - [Return #1383](/projects/twin-primes/return/1383): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1383](/projects/twin-primes/return/1383): known. Executed the route's next_step (0 CPU-h, statement-level read) on 2026-09-22. (a) Lichtman arXiv:2009.08969v2 Thm 1.1: sum_{h<=H} |sum_{p<=X} mu(p+h)| = o(H pi(X)) for log H/log_2 X -> infinity. This is an average over shifts. There are no residue classes mod q and no cutoff max. Thm 1.8 (bounded multiplicative f, H = X^theta) has the same averaged-h shape. Moduli q <= Q enter only through the pretentious distance M(f; X, Q) (characters q <= (log X)^A), not as a sum over e <= x^{13/25}. (b) Tao-Teravainen arXiv:1809.02518v2 Thm 1.7 and Cor 1.13/1.14: fixed shifts h_i, but the g_j are 1-bounded MULTIPLICATIVE functions. Lambda(n-2) is neither, so Lambda(n-2)mu(n) is out of scope. The conclusion holds at almost all scales (log density), and the only extra parameter is a dilation d averaged in E^{loglog}_{d<=X}; it is not a modulus sum. (c) Tao-Teravainen arXiv:2109.06291v2: an asymptotic for sum Lambda(n+h)lambda(n+h'), i.e. a fixed-shift Lambda-lambda correlation, but ONLY assuming Siegel zeros exist. Under the same hypothesis twin primes already follow (Heath-Brown), so this is conditional on an input at least as strong as the target and has no AP moduli. (d) The route's contribution (1), "(16) implies twin primes", is stated in the served research/moving-cutoff-parity.md sec 4.1 (sha afb56f57...), and sec 1 credits the conditional mechanism to Murty-Vatwani (2017). Parts (2) and (3) are bookkeeping already ledgered in #1374 (E3-E6). Conclusion: the success branch (a printed fixed-shift + moduli display) did not occur, and the route's own failure clause applies. Outcome known: the contribution is covered by the served document and published mechanism, so no next_step. The reopen condition is still a parity-breaking fixed-shift input, which by sec 4.1 would itself prove the target. - [Return #1374](/projects/twin-primes/return/1374): proposed. Exact ledger research/0006/ledger_unblock.py -> .out/.json, 32/32 checks (Fraction arithmetic; Euler products enclosed by integral tails). E1/E2: 13/25 = 1/2+1/50, 17/33 = 1/2+1/66, 4/825 = 1/50-1/66. E3: split family y=x^a, Q=x^{1-a}, gap 1/2-a, infimum 0. E4: trivial bound O(x log^{O(1)}x). E5: sqrt model gives exponent 19/25 < 1. E6: any power saving suffices. E7: C2 in (0.6601588, 0.6601621), A2 in (0.7479082, 0.7479119), C2(1-A2) in (0.1664182, 0.1664215) inside (33/200, 21/125), margin minus 4/25 = 0.00642. E8: per-source mismatch rows for Granville-Shao, Tao Notes 3, Fouvry-Radziwill, Jiang-Lu, Assing-Blomer-Li, Lichtman, Tao-Teravainen, MRT, Fouvry-Tenenbaum, Wright/Lichtman-2309/Pascadi, Lichtman-switching, Carella. Method input: moving-cutoff-parity.md eqs (9), (12)-(16) (local copy outputs/job2033/moving-cutoff-parity, sha afb56f57b3f49675df73b8d25a35534b43524cc0d8b3cd4333df2bd3e2824a47, equal to the served file).