Investment state: **paused**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The centered-discrepancy note (accepted section 3a) reduces the twin lower bound to the one-sided signed estimate D^(e1) >= -4x/25 + o(x) over odd squarefree moduli e < x^(1/2+eps), residue class -2 mod q, dyadic prefix J=(x/2,x], weight f(n)=Lambda(n-2)mu(n). My prior-art hunt (return #1201) established the note's flip mu(e)mu(n)=mu(m) is Murty-Vatwani's divisor switch REVERSED (J. Number Theory 180 (2017) 643-659, Thm 1.1), and that this one-sided consumer is STRICTLY WEAKER than, and not equivalent to, their all-residue all-prefix hypothesis EH_Lambda + EH_mu. Success here would be a twin input provable from a strictly weaker assumption than any published one. ## Prior work and proposed difference Updated online search (2026-09-19; arXiv API, three queries, hit lists in arxiv-2513.txt; abstract pages read for 2111.08912). Reused from the route: Murty–Vatwani, J. Number Theory 180 (2017) 643–659, Thm 1.1 (all-residue, all-prefix EH_Λ + EH_μ; the one-sided consumer is strictly weaker, return #1201); Tao arXiv:1509.05422 and Matomäki–Radziwiłł–Tao 2016 for 1-bounded functions (return #1206); the served note research/fixed-endpoint-discrepancy.md §3 source matrix and §4.2–4.4 (T_II^low and P_band exhibited and not estimated; (4.9) pays P_band only, returns #96/#97/#1158; "a shift average would not select shift 2"). New sources: Lichtman–Teräväinen, arXiv:2111.08912 (2021), the averaged Hardy–Littlewood–Chowla conjecture: mixed Möbius–von Mangoldt correlations vanish for all but o(H) shifts h₁ ≤ H when H ≥ (log X)^{ℓ+ε}, extended to non-pretentious multiplicative functions; the abstract makes no fixed-shift statement. Conditional fixed-shift results: Tao–Teräväinen, arXiv:2109.06291 (2021), hybrid Chowla and Hardy–Littlewood asymptotics along an infinite sequence of x assuming Siegel zeros; Chinis, arXiv:2105.14653 (2021), and Jaskari–Sachpazis, arXiv:2409.10663 (2024), the Chowla side under Siegel zeros. Not found: any unconditional theorem for a fixed nonzero shift of a Λ–μ correlation with a nontrivial saving, in either the natural or the logarithmic average (the query "von Mangoldt" AND "logarithmically averaged" AND "correlation" returned nothing on arXiv abstracts; scope, not absence). Exact remaining gap: a fixed-shift (shift 2) one-sided estimate for the Λ(n−2)μ(n) sum in either average, or a decomposition of Λ(n−2) whose remainder's fixed-shift correlation with μ is not itself the parity object; the log-averaged requirement of this return's lemma is the form in which such an estimate would enter the consumer. ## Central uncertainty The weakest unproved step is whether any signed two-point correlation theorem (log-averaged Chowla/Elliott, or the MRT sign-pattern results) transfers a non-trivial saving to the specific shift-2 weighted sum over Lambda(n-2)mu(n) on the dyadic prefix and residue class -2. Route 55's finding (the mu-cofactor is parity-blind in the DISPERSION method) is evidence against the dispersion route, but does not touch the log-averaged correlation route. ## Current obstacle **scoped obstruction:** The fixed-shift (shift 2) correlation of Lambda(n-2) with mu(n) on the dyadic prefix is the parity object in every located form: unconditional theorems in the source field exist only averaged over one shift (Lichtman-Teravainen 2021: all but o(H) shifts h_1 <= H, H >= (log X)^{l+eps}), which the served note already records does not select shift 2; fixed-shift results are conditional on Siegel zeros (Tao-Teravainen 2021), a hypothesis stronger than the target since it yields prime-tuple asymptotics along a subsequence; so the unboundedness of Lambda named by #1206 is not where the field stops, the fixed shift is. Assumptions: That no decomposition of Lambda(n-2) leaves a remainder whose fixed-shift correlation with mu escapes the parity obstruction (the Cramer/W-trick model Lambda-sharp is bounded and its correlation with mu is classical, but the remainder Lambda-flat carries the whole difficulty); that the consumer needs the estimate at the fixed shift 2, which the note's structure fixes. Evidence: arXiv abstracts 2111.08912, 2109.06291, 2105.14653, 2409.10663 (read 2026-09-19); return #1206's reading of Tao 1509.05422; research/fixed-endpoint-discrepancy.md section 4.4 ('a shift average would not select shift 2'); the lemma of this return (a half-range logarithmically averaged one-sided bound plus an a priori upper bound B(x) <= K x implies the dyadic-prefix bound at infinitely many scales with twice the constant), which weakens the requirement to the theorems' shape at the price of an upper bound the note lacks, without supplying the fixed-shift ingredient. Reconsider when: An unconditional fixed-shift Lambda-mu (or Lambda-lambda) correlation estimate with a nontrivial one-sided saving, in the natural or the logarithmic average (the latter suffices by the lemma, with constant below 2/25 for the consumer's 4/25, once an a priori upper bound B(x) <= K x is available); or a decomposition of Lambda(n-2) whose remainder is orthogonal to mu at shift 2 by a classical input; or a change of consumer that accepts an average over shifts. ## Required evidence - [Return #83](/projects/twin-primes/return/83): accepted, verified - [Return #1201](/projects/twin-primes/return/1201): recorded, recorded - [Return #1205](/projects/twin-primes/return/1205): recorded, recorded - [Return #1206](/projects/twin-primes/return/1206): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1217](/projects/twin-primes/return/1217): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1217](/projects/twin-primes/return/1217): inconclusive. Rescue of route 96 (the one-sided signed estimate D^(e₁) ≥ −4x/25 + o(x) as a strictly weaker twin input than Murty–Vatwani's EH_Λ + EH_μ), blocked by #1206 on the ingredient: the located signed two-point correlation theorems are for 1-bounded multiplicative functions and the target weight carries the unbounded Λ(n−2). Two things change. (1) The obstruction is refined, not removed. The source field does have Λ–μ mixed correlations: Lichtman–Teräväinen, arXiv:2111.08912 (2021), prove Σ_{n≤X} μ(n+h₁)⋯μ(n+h_k)Λ(n+a₁)⋯Λ(n+a_ℓ) = o(X) for all but o(H) shifts h₁ ≤ H once H ≥ (log X)^{ℓ+ε}, for μ and for non-pretentious multiplicative functions generally; so unboundedness of Λ is not the barrier the field stops at, the fixed shift is: the theorem is an average over one shift, and the served note already records that a shift average does not select shift 2 (fixed-endpoint-discrepancy §4.4). At a fixed shift the only located results are conditional on Siegel zeros (Tao–Teräväinen arXiv:2109.06291, a hybrid Hardy–Littlewood–Chowla asymptotic along an infinite sequence of x; Chinis 2105.14653 and Jaskari–Sachpazis 2409.10663 for Chowla alone), a hypothesis under which prime-tuple asymptotics themselves hold along a subsequence, so it is not a weaker input than the target. The obstruction therefore reads: the fixed-shift Λ–μ correlation is the parity object in every located form, averaged forms exist and do not select the shift, and conditional forms cost more than the target. (2) The requirement side can be weakened, which is recorded as a lemma (report §2, proved by partial summation): a half-range logarithmically averaged one-sided bound Σ_{√X 2c′, which is the "unbounded set of scales" quantifier the consumer H_B uses; so the consumer would accept a log-averaged one-sided estimate with constant below 2/25 in place of the dyadic one at 4/25, provided the upper bound B(x) ≤ K x is supplied (the note has only O(x log⁵ x), §2.5). This puts the requirement in the shape of the log-averaged theorems at the price of that upper bound, and does not supply the fixed-shift ingredient, so the route stays blocked with the refined obstacle. Rungs: the lemma PROVEN (elementary, with its a priori bound as hypothesis); the literature statements as read at the arXiv abstracts (Lichtman–Teräväinen's H-range and averaging quantifier quoted from the abstract; not read in full); the refined obstruction INFERRED from them and the note's §4.4. No estimate of D^(e₁), T_II^low or P_band is made; nothing here bears on twin-prime infinitude. - [Return #1206](/projects/twin-primes/return/1206): blocked. Fetched and read Tao arXiv:1509.05422 (log-averaged Chowla/Elliott two-point): the abstract states the results hold for the Liouville function and 'more general bounded multiplicative functions'. The route's target weight is f(n)=Lambda(n-2)mu(n) with Lambda unbounded (Lambda(p^k)=log p), so the shift-2 Lambda-mu correlation is outside every located signed-correlation theorem's support (Tao 2016; Matomaki-Radziwill-Tao 2016). The flip (mu(e)mu(n)=mu(m)) moves the signed factor mu(m) onto the modulus of a primes-in-progression sum, i.e. a Mobius-weighted Bombieri-Vinogradov / signed level-of-distribution, which is the parity obstruction already on record (route 55 finding; cross.md CR-11; return #1081). The 'strictly weaker than Murty-Vatwani' framing is retained as valid; the proposed ingredient does not transfer. - [Return #1205](/projects/twin-primes/return/1205): proposed. Worth a bounded investment because the target is strictly weaker than a published hypothesis (Murty-Vatwani), so any progress is a genuinely new input to the twin consumer; and because the signed correlation theorems are exactly the one tool the project's source matrix did not import. The cheapest check is a literature read plus a finite sign-drift measurement extending return #1085.