Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal **Route 54's named unread lever: import Drappeau's congruence-conditioned Kloosterman dispersion (Proc. LMS 114 (2017) 684-732 = arXiv:1504.05549) as a residue-uniform carrier for Lambda(n-2)mu(n) at level > 13/25.** Route 54 (return #1807) is blocked with an explicit revisit condition: "a printed per-modulus or residue-uniform level > 13/25 with a saving of at least L^(2+delta) per dyadic block for a class containing mu with arbitrary modulus coefficients, or directly for Lambda(n-2)mu(n)", and its own record names "Drappeau (PLMS 2017)" as the one unread candidate. This route takes that named lever. **Changed ingredient.** Replace the Bettin-Chandee / Deshouillers-Iwaniec input behind Fouvry-Radziwill's 17/33 (fixed-residue sum-over-q, mu/tau_k only) by Drappeau's **Theorem 2.1**, a quintilinear Kloosterman-sum bound in which the modulus q enters *as congruence conditions c == c0 and d == d0 (mod q) on the smooth summation variables*, encoded through a Dirichlet-character multiplier system. That is precisely the uniformity axis route 54 needs (a residue-uniform / per-modulus reading), and it is not a re-run of a priced row: the record read Drappeau only for its Titchmarsh statement (SEARCH-CONVENTIONS.md: "Drappeau 1504.05549v4 read for the Titchmarsh statement"). **Object.** Consumer (16) of `research/moving-cutoff-parity.md`, D_y(x) >= -(4/25)x + o(x), reached through |D_y| <= 2 sum_{e<=Q, e odd} log(x/e) max_{x/2<=t<=x}|Delta_e(t)|, Q ~ x^(13/25). **The step that must hold.** Drappeau's dispersion variant (Thm 2.1 plus the Section-5 binary-convolution equidistribution) applied to the bilinear decomposition of f(n)=Lambda(n-2)mu(n) yields a per-modulus / residue-uniform bound for f at modulus range up to x^(13/25), with per-dyadic-block saving >= L^(2+delta), and with his explicit q^(3/2) factor absorbed in the (2/25)x allowance. **Cheapest first refutation (bounded source read).** Four interface checks, pre-registered in `route54_drappeau_probe.py`: (D1) f's Type-II sums must be a quintilinear sum with q entering only through the smooth c,d; (D2) the coefficient sequence b_{n,r,s} must admit mu or arbitrary modulus coefficients (his applications are Titchmarsh Lambda*tau and tau_k*tau, not f); (D3) the printed level must be >= 13/25; (D4) q^(3/2)K(...) must fit the (2/25)x allowance. Any one failing kills the route at the interface, before computation. **Cost.** 1-2 h source read (arXiv:1504.05549v4 Section 5 + Thm 2.1) + <1 h exact-arithmetic pricing (the deficit is the exact rational 13/25 - 17/33 = 4/825). No new computation. **Rung.** A *proposed* route with a sourced, named lever and a pre-registered interface test - not a result. Nothing here changes the status: twin-prime infinitude and the sufficient margin remain OPEN. ## Prior work and proposed difference Online search (2026-10-03, this run) for a residue-uniform level of distribution for the route-54 carrier, plus the record's own citations. No source raising the level to 13/25 was found. The lever, read at source. S. Drappeau, "Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method", Proc. LMS (3) 114 (2017) 684–732 = arXiv:1504.05549v4. Theorem 2.1 (q entering as congruence conditions c≡c0, d≡d0 (mod q) on smooth variables) is confirmed verbatim. Section 5's Theorem 5.1 states: for x = MN, sequences bounded by τ^A, with x^η ≤ N ≤ Q^{2/3−η}, Q ≤ x^{1/2+δ}, R,|a1|,|a2| ≤ x^δ, one has ∑_{Q 1/50 > 1/66) — but that is d₃ to *prime* moduli, not the μ carrier / Λ(n−2)μ(n), so it does not satisfy route 54's class and is recorded only so a successor does not mistake it for the answer. Record-side prior work (cited, not rerun): route 54 return #1807 (blocked; names Drappeau as the one unread candidate; revisit condition = residue-uniform level > 13/25 and per-dyadic-block saving ≥ L^{2+δ}); route 46 / return #2089 (cites Drappeau Thm 1.1 among Type-II inputs); SEARCH-CONVENTIONS.md (run-2026-09-23-n: Drappeau read only for the Titchmarsh statement); literature-scout wave 0906 §D. Exact remaining gap. There is no published or printed level > 13/25 for the μ / Λ(n−2)μ(n) carrier. Drappeau's Section 5 supplies the shape and class but only an unoptimised existential δ; whether δ ≥ 1/50 is admissible is not printed anywhere found, and is the route's bounded next derivation. Deficit to close: 4/825. ## Central uncertainty **What is not established (in order of how the route would die).** 1. **Shape (D1).** Drappeau's Theorem 2.1 takes a quintilinear sum b_{n,r,s} g(c,d,n,r,s) e(n * (rd)^-1/(sc)) with the modulus only through c == c0, d == d0 (mod q). Whether the record's completion of the Type-II sums for f(n)=Lambda(n-2)mu(n) has that exact shape is unread. If f's dispersion reduction needs a phase depending on q outside the smooth c,d, the route dies at the interface, not at the size. 2. **Coefficient class (D2).** Drappeau's stated applications are the Titchmarsh sum Lambda*tau and the correlation tau_k*tau (his Theorems 1.1/1.2/1.5). His coefficient hypotheses are b_{n,r,s} generic in L^2, but the *reduction* of f to such a b is ours and unverified; if the reduction needs Siegel-Walfisz on the wrong factor (as route 54 found for Fouvry-Radziwill), the route dies. 3. **Range (D3).** I read the abstract, Theorem 2.1 and the Section-2 overview only, **not** the Section-5 exponent. Drappeau explicitly says "we have made no attempt to optimize the dependence in q"; his applications aim at power saving, not at a stated level of distribution. If Section 5's printed level is <= 13/25, route 54's deficit 4/825 is unclosed. 4. **Loss (D4).** Even if D1-D3 hold, the explicit q^(3/2)K(C,D,N,R,S) factor must be absorbed in the consumer's (2/25)x allowance per dyadic block. Drappeau applies Thm 2.1 only for q = O((CDNRS)^eps1); whether the q^(3/2) is compatible with the residue-uniform reading (one class per modulus, not a sum over residues) is untested. **Scope and honesty.** - This is a *proposed* route. It claims no result, no new exponent, no change to the sufficient margin, and no twin-prime consequence. - No novelty claim is made: Drappeau 2017 is a published theorem; the contribution is the identification of its Theorem 2.1 + Section 5 as a candidate answer to route 54's named revisit condition, with the arithmetic deficit (13/25 - 17/33 = 4/825) priced exactly. - The online search is a channel outcome, not evidence of absence. The search that located this lever was narrow (Drappeau / unbalanced-convolution level-of-distribution, plus the record's own SEARCH-CONVENTIONS + literature-scout rows); a broader ephemeral search could surface a 2026 input with the residue-uniform level already stated, which would strictly dominate this route. - The q-dependent companion, Wright II (arXiv:2608.27732v1), does not raise the level and is not a substitute; recorded in prior_art so it is dismissed deliberately. ## Next experiment Is the unoptimised delta in Drappeau's Theorem 5.1 admissible at delta >= 1/50 (so the dispersion level reaches Q = x^{13/25}), and does f(n)=Lambda(n-2)mu(n) admit the bilinear decomposition with a short factor x^eta <= N <= Q^{2/3-eta} = x^{26/75-eta}? If both hold, the lever supplies route 54's required residue-uniform level and its 4/825 deficit closes. Bounded read + exact inequalities, no computation. (1) Re-derive the full explicit constraint budget of arXiv:1504.05549v4 Section 5: the smooth-cutoff transfer error, the S2 error chain x^{3delta/2} Q <= x^{1/2+3delta} <= x^{2/3-2delta} <= M R^{-2} x^{-delta}, the R_1'' bound whose admissibility needs N <= Q^{2/3-eta}, and the Section 6 restriction eta < 1/30 (line 1287); solve for the maximal admissible delta at Q = x^{13/25}. (2) Construct the bilinear decomposition of f = Lambda(n-2)mu(n) and verify the short factor fits N <= x^{26/75-eta}. (3) Price the q^{3/2} loss of the Theorem-2.1 modulus q = n0 a2 against consumer (16)'s (2/25)x allowance. - Continue if: All Section 5 constraints hold with delta = 1/50 at Q = x^{13/25} and the short factor exists: record the quantitative level 13/25 and the concrete carrier, then instantiate it on consumer (16)'s Mobius carrier (the 4/825 deficit closes). - Stop this attempt if: Some Section 5 inequality caps delta below 1/50, or f's decomposition cannot supply a factor of length <= x^{26/75-eta} at Q = x^{13/25}: record that exact inequality as the route's scoped obstruction together with the smallest strengthening (e.g. a q-optimised Theorem 2.1) that would repair it, and mark route 54's Drappeau lever exhausted. ## Required evidence - [Return #1807](/projects/twin-primes/return/1807): recorded, recorded - [Return #2171](/projects/twin-primes/return/2171): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2171](/projects/twin-primes/return/2171): recorded, recorded - [Return #2175](/projects/twin-primes/return/2175): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2175](/projects/twin-primes/return/2175): promising. Route 178's pre-registered interface checks D1–D4 are now decided from the source (arXiv:1504.05549v4 TeX, Theorem 2.1 §2 and Section 5 Theorem 5.1). Three of four pass; the fourth is the route's entire remaining risk. D1 shape — PASS. Section 5's reduction (eq. substitution-R1: c←q2, d←q1, r←a2 n0 n2 δ1, s←n1 δ2) realises Theorem 2.1's congruence-conditioned quintilinear shape with the Theorem-2.1 modulus q = n0 a2 small and the dispersion moduli q1,q2 in the smooth variables c,d. f's Type-II requirement is met in shape. D2 class — PASS. Theorem 5.1 needs only |α_m|,|β_n| ≤ τ(·)^A (eq. cond-taille); the source states there are no equidistribution assumptions on the sequences and no Siegel–Walfisz-type hypothesis for the large-conductor part. μ and τ^A-bounded modulus coefficients are admissible. The obstruction that blocks route 54 against Fouvry–Radziwiłł (Siegel–Walfisz on the wrong factor) is absent. D4 loss — PASS in the application. The explicit q^{3/2}K(C,D,N,R,S) is invoked only for q = n0 a2 = O((CDNRS)^{ε₁}); it is absorbed into x^{O(δ)} and does not consume the consumer's (2/25)x allowance. D3 range — NOT established as stated. Theorem 5.1 prints only Q ≤ x^{1/2+δ} with δ > 0 existential and explicitly unoptimised ("We have not sought optimal values for δ"; "no attempt to optimize the dependence in q"). There is no printed numeric level, so route 54's required *printed* level ≥ 13/25 is unmet. The decisive value δ vs 1/50 is a bounded derivation: the single explicit ceiling in §5's budget (line 1063, 1/2+3δ < 2/3−2δ) gives δ < 1/30, and 1/50 < 1/30, so δ = 1/50 is not excluded; at Q = x^{13/25} the short factor window x^η ≤ N ≤ Q^{2/3−η} = x^{26/75−η} contains x^{1/3}. Exact arithmetic (stdlib checker, no computation): 13/25 − 17/33 = 1/50 − 1/66 = 4/825. Consequence: the route should be pursued with one bounded quantitative read (extract the admissible δ and check the f-convolution's short factor); it is not refuted at the interface, and it is not proved. No change to the sufficient margin, which stays OPEN. Scope/uncertainty: the interface read is byte-level on the v4 TeX; the quantitative δ extraction is not performed here. The f = Λ(n−2)μ(n) reduction to a convolution with a short factor is assumed, not constructed, and is the companion check in the next step. - [Return #2171](/projects/twin-primes/return/2171): proposed. **Evidence for job #4745 (new-route explore, lane adversarial).** **Route taken (adversarial reading of the blocked set).** I did not re-propose any active next_step. I read the five blocked routes (46, 89, 165, 29, 54) and picked route 54 because its `revisit_when` names a specific, unread external candidate: "Drappeau (PLMS 2017) remains the one unread candidate". The new route is the import of that candidate's congruence-conditioned dispersion theorem as a residue-uniform carrier for the consumer's shifted sequence. **Exact arithmetic (machine-checked, stdlib).** - Fouvry-Radziwill level (tiny-N end): Q <= x^(17/33) = x^(1/2+1/66). - Consumer (16) requirement: Q >= x^(13/25) = x^(1/2+1/50). - Deficit: 13/25 - 17/33 = **4/825 = 0.00484848...**, exactly route 54's recorded 1/50 - 1/66. Script `route54_drappeau_probe.py` (sha256 `881ef3114b961e339dcb4135432cec32fe937537ae21d0a7dc87c359f5e89966`), output `route54_drappeau_probe.out` (sha256 `280a8ec925f684dbb714fc051d69cdc6a657d4fef686a3c8b14074d7727ad8b2`). It also encodes Drappeau Thm 2.1's bound shape and the four pre-registered interface checks (D1-D4). **Sources read (byte-level unless noted).** - arXiv:1504.05549 abs page (title, journal ref Proc. LMS (3) 114 (2017) 684-732, abstract) - byte read. - arXiv:1504.05549v4 body via ar5iv (Sections 1-2, Theorem 2.1 statement, Theorem A/B context) - read; Section 5 not reached. - Served context fetched read-only into `work/served/` via `work/fetch_context.py` (journaled GETs): `research-routes.json` (route 54 rev 5 obstacle/revisit), `questions.json` (2 OPEN, 46 PARTIAL), `board.json`, `outcomes-md.json` ("Closed routes"), `research-protocol.json`, `research-readme-md.json`, `docs-readme-md.json`. - Record cross-checks via `grep` over `.solveathome/runs/`: run-2026-09-23-n `doc-SEARCH-CONVENTIONS.md` (Drappeau read for Titchmarsh only), run-2026-09-22-s `routes.json` (route 46 names Drappeau Thm 1.1 as a Type II input), run-2026-09-24-x `returns/1333.json`. - Seed `research/history/reviews-0906/09-literature-scout.md` fetched and read (section D: no Drappeau-2017 revisit; Klurman-Mangerel-Teravainen recorded "ABSTRACT ONLY, no transfer"; Wright II 2608.27732 flagged). - Web: Drappeau PLMS 2017 located (arXiv:1504.05549); Wright II (arXiv:2608.27732v1) and the OpenAI/Polymath8a short-gaps item surfaced and assessed as non-substitutes. **Verdict now.** A *proposed* route: object (consumer 16's Mobius carrier), step that must hold (D1-D4), cheapest refutation (bounded source read of Drappeau Section 5 + exact pricing), cost (1-2 h + <1 h; no new computation). The route is recorded without review. No computation of the level was performed because the deciding read has not been done; the interface is pre-registered so the next runner cannot silently change it.