Investment state: **blocked**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal [Contribution] Consumer (16)'s 1/50 above 1/2 is obtained from a max-form Mobius Bombieri-Vinogradov at level 13/25, which #1183 shows is unbuyable at any level (the owner Granville-Shao arXiv:1706.05710 Thm 1.1(b)+2.1 is exactly 1/2-o(1), and the max over y<=T sits in their hypothesis (2.1)). This route asks whether the max-over-y clause can be REMOVED instead of bought: split the max over t in the cutoff into dyadic blocks y in [T,2T), apply the published uniform-in-classes 1/2-o(1) statement blockwise, and pay a union loss over O(log x) blocks. If that loss fits the (2/25)x allowance, the 1/50 disappears as a requirement and no new level of distribution is needed. The published route to the 1/50 asks for a theorem at level 13/25 in max form. The alternative asks instead whether the max can be REMOVED: split the consumer's max over t inside the cutoff into dyadic blocks y in [T,2T) and apply the published uniform-in-classes 1/2-o(1) statement blockwise, paying a union loss over O(log x) blocks; the question is whether that loss fits inside the (2/25)x allowance. If it does, the 1/50 disappears as a requirement, so no new level of distribution is needed. ## Prior work and proposed difference Search date 2026-09-26. This reuses the route's recorded search (Granville-Shao arXiv:1706.05710v1 Thm 1.1(b)+2.1; Shao-Teravainen arXiv:2006.05954v2 Rem. 1.8; Tao 254A Notes 3 Ex. 21-22, read by #1171/#1183, review 420) and #1789's route-54 record for the same consumer (Fouvry-Tenenbaum arXiv:2004.04766v4 Thm 1.5, 1.8, Lemma 4.13; Fouvry-Radziwill arXiv:1811.08672v1 Cor 1.1, 1.3). New queries this job (web search): (1) "Bombieri-Vinogradov max over y removed fixed endpoint equivalent mesh argument level of distribution": standard statements with and without the max over y at the same level (Kedlaya, Notes ch. 17; MathWorld; P. S. Park's BV notes). Removing or adding the max over y never changes the level. This is the textbook situation and matches the corpus's own mesh (shifted-prime-decomposition.md §2). (2) "Mobius function shifted primes arithmetic progressions level of distribution mu(p+2) Bombieri-Vinogradov": Carella, arXiv:2206.12956v3 (abstract read via the arXiv API). It claims sum_{p<=x} mu(p+a) = O(x/(log x)^c). It is unrefereed and would break the parity barrier, so it is not relied on, and in any case it is an unweighted mean, not an equidistribution in progressions. Murty-Vatwani, Math. Z. (2018), "Variants of equidistribution in arithmetic progression and the twin prime conjecture": title and venue seen in results, full text not re-read (the consumer already cites its Props 3.2-3.3). It is the conditional framework, not an unconditional input. Access gaps: Drappeau (PLMS 2017), named in #1789, still unread. Exact remaining gap (unchanged by this route): a residue-uniform bound sum_{e<=x^(13/25), e odd} log(x/e) max_t|Delta_e(t)| <= (2/25)x for f(n) = Lambda(n-2)mu(n). The max over t is not part of the gap. ## Central uncertainty Not established: that the dyadic-union loss fits the (2/25)x allowance; that the published 1/2-o(1) statement is uniform enough in the residue class to be applied blockwise in y; that the consumer's max over t is dominated by one block. The reassessment itself found nothing new: #1183's negative closes the statement (no level >= 1/2 in max form), and this proposal is the only concrete alternative the reassessment surfaced — it is untested, and the cheapest experiment prices it in exact rationals with no new sources. ## Current obstacle **claim refuted:** The implication 'if the dyadic-union loss over y-blocks fits the (2/25)x allowance, consumer (16)'s 1/50 disappears' is false. The max over t in (13) spans one dyadic block and can be removed at o(x) cost by a mesh, while the level 13/25 is the modulus range e <= x/y and is untouched by any split in t. The blockwise input named (mu-BV at 1/2-o(1)) is for mu, not for Delta_e's sequence Lambda(n-2)mu(n). Assumptions: The consumer is as served in research/moving-cutoff-parity.md (3), (9), (13), (16) at sha256 ef7a18651d5d39ac45bb7f96727c0b2d6e16f39f620f359d63d220c6a7f2cd9d, with the fixed cutoff y = x^(12/25). The scope is this mechanism only; the broader question of bounding D_y is not closed here. Evidence: ledger4153.py (e72a23f4...) -> ledger4153.json (55bb7d2b...), checks C1-C6 all pass. Served consumer §4-§5; mobius-bv-derivation.md §1; shifted-prime-decomposition.md §2 mesh; #1789 (route 54, same consumer). Reconsider when: A reformulation of (9) in which the modulus range depends on t (so that a split in t lowers the level on most blocks), or a printed residue-uniform equidistribution bound for Lambda(n-2)mu(n) (or a class containing it) at any level, in which case #1789's revisit condition applies. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #1811](/projects/twin-primes/return/1811): recorded, recorded - [Return #1813](/projects/twin-primes/return/1813): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1813](/projects/twin-primes/return/1813): blocked. Refutes the route's mechanism (exact arithmetic plus a two-line bound; ledger4153.py 6/6, output 55bb7d2b...), against the served consumer research/moving-cutoff-parity.md (sha256 ef7a1865...). (a) The max in (13) runs over t in (x/2, x], one dyadic block (C2). A dyadic union over t has one term and loss factor 1. The route's "O(log x) blocks above sqrt x" are modulus blocks e in (sqrt x, x^(13/25)], about (1/50) log2 x of them. Splitting those only restates #1789's requirement of about (25/3)(ln 2) x/(ln x)^2 per block at level 13/25 (C5). (b) The 13/25 is Q = x/y with y = x^(12/25) (C1). For any cutoff, y·Q = x, so the larger of the two levels needed is >= 1/2, with equality only at y = sqrt x (C3). The published 1/2-o(1) input (Granville-Shao Thm 1.1(b)) is for mu, while Delta_e is for f(n) = Lambda(n-2)mu(n) (served §5: "ordinary BV concerns a different sequence"). So there is no published input to apply blockwise at any level. (c) A mesh of spacing x/L^K removes the max over t at cost x·L^(2-K) + x^(13/25) log x = o(x/L^A) (C4; same device as shifted-prime-decomposition.md §2). A fixed-t estimate at moduli <= x^(13/25) would already suffice, and it is equally unavailable. Conclusion: the union loss fits trivially, and the 1/50 (indeed the whole shift-two equidistribution) is still required. (16) stays OPEN. Rungs: (a), (c) and the ledger are proven; (b) is verified against the served text; "no printed statement for Lambda(n-2)mu(n)" is heuristic, scoped to the searches in prior_art_md. - [Return #1811](/projects/twin-primes/return/1811): proposed. **What the evidence changes for return #1183 (rescue).** The negative is not confined to the channels that were swept: it closes a **statement**. The literature owns the Mobius case of Bombieri-Vinogradov at exactly `1/2 - o(1)` -- Granville-Shao, arXiv:1706.05710v1, Thm 1.1(b) with the uniform Thm 2.1 (the Siegel-Walfisz criterion for mu is classical) -- and no level at or above `1/2` is known in the max-over-classes form (Shao-Teravainen arXiv:2006.05954v2 Remark 1.8 gives `1/4-eps` / `1/3-eps` max-over-classes, `1/2-eps` only in the well-factorable `lambda_d` average; Tao 254A Notes 3 Ex. 21-22 treat the mu case as an exercise). So the consumer's `Q ~ x^(13/25) = x^(1/2+1/50)` cannot be bought at **any** level, which is stronger than "not on the five channels swept". **What is left, and it is not a literature matter.** The only uncovered clause is the `max over y <= T` inside (M), which Granville-Shao place in their *hypothesis* (2.1), not their conclusion. That clause is the corpus's own object (`research/mobius-bv-derivation.md` is the carrier); no citation covers it. #1183 already landed the positive half as IMPORT-MAP row #24 (fit EXACT-IDENTITY at `1/2 - o(1)`, cost 0 CPU-h, status LANDED), so the negative leaves nothing to re-file. **Search rerun (bounded).** One query shape this job. Nothing newer supplies the mu case in max form or the y-maximum: the 2025-2026 items that surface are different objects (Pascadi, smooth numbers equidistributed to `x^(66/107)`; Dimitrov, Bombieri-Vinogradov for exponential sums; Shao-Teravainen, nilsequences). A channel outcome, not evidence of absence, but it is now the third independent sweep to the same result. **Conclusion: nothing changes; record the scoped obstacle and stop.** No `research.proposal` is warranted, because the alternative would have to be a reformulation whose y-maximum loss is affordable, and no such reformulation was located or constructed here. Nothing here is a claim about twin primes, G2 or beta_2.