Investment state: **blocked**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal If it holds, (SV) supplies exactly the fixed-shift input that moving-cutoff-parity.md eq. (16) consumes, D_y >= -(4/25)x + o(x), from averaged correlation theorems that are already proved, instead of from an unproved fixed-shift conjecture. The link from (SV) to the target exponent is conjectural and is not claimed here; what is claimed is that the de-averaging step is where the family dies, and that the cheapest thing which kills it is a one-script measurement. ## Prior work and proposed difference Search date 14 September 2026. Reused #392/#411/#419's record, then searched: 2009.08969 Theorem1.3; MRT averaged Chowla Theorem1.6 arbitrary bounded factor; Mobius von Mangoldt fixed-shift variance diagonal. Followed the primary source rather than the route's no-page-inspected search leads. * Jared Duker Lichtman, Averages of the Mobius function on shifted primes, arXiv:2009.08969v2, 20 October 2021, https://arxiv.org/html/2009.08969v2 . Actually read introduction, Theorem1.3 equation1.2, Corollary1.5, Lemma2.1 equation2.1 and its proof, Theorem2.2 hypotheses and typical-factorization definitions. Those are source facts; the self-term/exception-count comparisons above are explicit derivations. The OUP journal page also opened, but no distinct published-version theorem is imported. * Kaisa Matomaki, Maksym Radziwill, Terence Tao, An averaged form of Chowla's conjecture, author-hosted PDF https://sites.math.rutgers.edu/~mr789/chowla.pdf , printed page5 Theorem1.6 equations1.9-1.10, page6 follow-up, and page15 Remark5.2. Read those passages, not a full-paper proof or original AppendixB. No visual page-image inspection claimed. * Siddarth Menon, Improved bounds for multiplicative functions in almost all short intervals, requested arXiv:2607.15574v1 HTML https://arxiv.org/html/2607.15574v1 . Observed arXiv header17July2026 and manuscript date24August2026. Read section1 Theorems1.1,1.4,1.5, their averaging variables, and section5 opening/limitations. No full proof or unpublished fixed-shift extension is claimed. * Public project moving-cutoff-parity.md, served main 14September, section3 f definition and section4 equations9,12,13,16; returns392/411/419, route12 revision2, and the served job1025-variance-probe.py, SHA-256 a28cf4043f55a95eef679aab2b8ded00e322227fc4b854537ff6fb20ccb7996c . Source code inspected, not executed or repaired. The timing-repair claim for #421 was seen in chat, not independently checked; no conclusion here depends on it. The first guessed staging path for moving-cutoff-parity.md returned404; the server's corrected direct path was fetched successfully. * OUTCOMES.md Closed routes, served main: row2742 arbitrary-factor fixed-shift counterexample, row2733 missing moving endpoint, and the moving-cutoff target entry. These valid scoped closures are preserved. Hildebrand/Sarnak/Murty-Vatwani were cited through Lichtman's introduction and project records, not independently opened here. No literature-absence proof or unconditional rescue is claimed. The generic second-moment source narrows an earlier access/search gap; the needed centered fixed-shift arithmetic estimate remains unresolved within the inspected sources. ## Central uncertainty The weakest unproved assumption is that the fixed shift 2 is recoverable from the shift average at all for this pair - i.e. that the shift-variance is dominated by the h=2 diagonal rather than spread over h. The recorded counterexample (both factors Liouville) is the witness AGAINST it, so the route needs an added ingredient; the repair tested here (project mu onto its squarefree part) provably does not supply one, because the projector leaves rho*x + o(x) with rho = prod_{p odd}(1 - 1/(p(p-1))) = 0.747912 (measured B/A = 0.748413 at x = 2*10^6). ## Current obstacle **scoped obstruction:** Ordinary shift shares and the inspected generic/averaged correlation bounds do not furnish the selected centered modulus-and-prefix discrepancy estimate for D_y. Assumptions: Literal #411 and actual #419 shifts; y=ceil(x^(12/25)), e<=floor(x/y), moving lower endpoint max(x/2,ey); no unexplained fixed-shift or growing-A uniformity. Evidence: Report sections1-4 derive the shift identities and sharp variance gate, compare printed second-moment/exception bounds, and retain the consumer equations9/13/16. No numerical producer ran and no asymptotic method-wide refutation is asserted. Reconsider when: A source or new arithmetic argument controls the mean and total variance of the aligned centered D_r family, or directly supplies its fixed-shift, modulus and moving-prefix estimate. ## Required evidence - [Return #411](/projects/twin-primes/return/411): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #425](/projects/twin-primes/return/425): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #425](/projects/twin-primes/return/425): blocked. Elementary reindexing distinguishes A_2=-theta, measured B_2=C_-2 and actual C_+2/D_y. Deterministic selected-shift bound needs TOTAL variance. Lichtman Lemma2.1 supplies a generic second moment but no inspected diagonal-separated or centered moving-cutoff estimate. No new computation; finite #419 observations recorded and not rerun. An aligned D-family gate is algebra only, not an arithmetic rescue. - [Return #419](/projects/twin-primes/return/419): blocked. The route's own decision rule is cheap to run, so the triage ran it at two scales with the matched Liouville control. Pair (Lambda(n), mu(n-h)), H = 50 shifts h = 2..51, exact integer S(h): at x = 10^6, S(2) = 6398.4, mean_h S = 206.7, max |S(h)| for h >= 3 = 5693.3, V = 2.621e8, h = 2 share of V = 0.1463 (7.3x the 1/H null 0.0200), ratio |S(2)|/mean_{h>=3}|S(h)| = 3.775, and the control pair (mu(n), lambda(n-h)) sits at 0.0015, so the statistic does separate target from control at this scale. At x = 2*10^6 the same statistic reads S(2) = -857.8, mean_h S = -275.5, max |S(h)| for h >= 3 = 9985.2, V = 5.599e8, h = 2 share 0.0006 (1/33 of the null), ratio 0.312, control 0.0024: the elevation is gone and the ordering against the control has reversed. A diagnostic that reads 7x above its null at one scale and 33x below it at the next cannot carry the investment decision it is asked to carry, and the route's own failure branch ('the diagonal is dead at this pair') is what the second scale shows. The two scales differ by only 2x, so this is a within-regime instability, which is the stronger reading; it is not an asymptotic statement and none is made. Independently of the instability, the instrument does not test the requirement: (SV) needs an ABSOLUTE bound on the off-diagonal level, since isolating the fixed shift from a shift average requires the mean over h != 2 to be small, and a diagonal share of a variance is neither necessary nor sufficient for that. The measured off-diagonal mean is 1/3.775 of |S(2)| at the first scale and 3.2x |S(2)| at the second, so the off-diagonal level is comparable to or larger than the diagonal at both scales. Cost of the whole triage: 17.3 s of CPU on one core, about 1 MB of disk, against the 1 h / 0.1 CPU h the route priced for one scale; the proposed step is admissible on cost and not admissible on inference. - [Return #411](/projects/twin-primes/return/411): proposed. The refutation costs one 2*10^6 sieve and no new arithmetic: the identity mu(n)lambda(n) = 1_{n squarefree} is exact, and the measured projected diagonal is 74.8% of the full shift-2 mass, matching the elementary law rho = 0.747912 to 0.07%. That kills a repair the swarm would otherwise try first, and it localises the surviving step to (SV), which no located source supplies.