Investment state: **result**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Routes 36 and 37 (returns #661 and #659) both reduce the project's threshold question to one named step, and both state it in the same words: in Selberg's example the parity class that a Brun/Selberg weight family cannot see is EMPTY, whereas the project's contaminant class is NON-EMPTY and 'lies inside one parity class'. Neither return measures that in its own convention: #661's instrument writes the threshold as c*_real(u) = f_1(u/2)^2 (1 + D_1/(rho_odd(u)-1)) with D_1 = 1 exactly, and #659 calibrated the channel on two known positives, but the containment claim is an assertion about the tile that nothing in the record checks. This attempt measures it. In the project's own admissibility convention (twin slots n with gcd(n(n+2), x#) = 1, the T_x tile), classify every admissible slot by (lambda(n), lambda(n+2)), lambda = (-1)^Omega: twin primes occupy (-1,-1) only, and any certificate whose weight is a function of delta(n) = lambda(n)lambda(n+2) alone treats the delta = +1 sub-classes (-1,-1) and (+1,+1) identically. MEASURED at x = 11, 13, 17 over n <= 1e6 (all gates passed first, 1.89 s, one process): the blind (+1,+1) class is not merely non-empty, it is 91.8%, 88.6%, 85.6% of the good (-1,-1) class at x = 11, 13, 17 respectively, and the delta = +1 aggregate is split almost evenly between the two sub-classes (delta mean = -0.0004, -0.0015, -0.0005). The good class is itself dominated by non-twins: the twin share of (-1,-1) is only 0.5365, 0.6241, 0.6952 at the same levels, i.e. 46%, 38%, 30% of the good class is blind mass that no delta-measurable weight can exclude. The contribution is therefore a quantity the transfer step needs and the record does not have: the blind-class ratio, with a level trend that is monotone and can be extrapolated against. The route changes the ingredient rather than the constant: the missing object is not a sharper constant in D_1/(rho_odd - 1) but a weight family that is NOT a function of delta alone, and the measured ratio is exactly the discrimination such a family must supply. Cheapest test of the route's premise: extend the same 2 s instrument to x = 19 and x = 23 and fit the ratio against the level; a ratio that stays in (0,1) and strictly decreasing keeps the route alive, a ratio tending to 1 says the blind class saturates the delta = +1 aggregate (and any signed aggregate certificate is fatal by construction), and a ratio tending to 0 would make a delta statistic discriminating and is the outcome that would falsify the closed parity rows. ## Prior work and proposed difference Search date 2026-09-18 (this attempt), on top of the recorded searches of #659, #661/#662 and #665 (Selberg parity problem: Wikipedia "Parity problem"; Tao's parity-problem tag; MathOverflow 233240; Thompson ch. 9; Elkies Math 229 notes, still unread here; arXiv:2310.08144v3 still unread; arXiv:1909.07975v6 still unevaluated). Two new queries, both aimed at the changed ingredient (the Liouville mean and shift-2 correlation on residue classes), not at the parity problem: 1. Tao, "The logarithmically averaged Chowla and Elliott conjectures for two-point correlations", arXiv:1509.05422 (Sept 2015), Forum of Mathematics Pi 4 (2016): the abstract's statement covers Σ λ(a1 n + b1) λ(a2 n + b2) with logarithmic weights = o(log ω); a residue class n ≡ a (mod x#) is the affine case a1 = a2 = x#, b2 = b1 + 2, so the log-averaged shift-2 correlation on each admissible class of the tile is o(1). Used for the unconditional logarithmic form of r → 1. Improved bounds: Pilatte, arXiv:2310.19357 (2023). Read: abstracts and theorem statements via the arXiv pages and Tao's blog post of 2015-09-18; the natural-density (non-logarithmic) two-point case is stated there as open, and I claim nothing about it beyond the measured c ≈ 0. 2. Liouville sums in progressions and their sign bias: Humphries–Shekatkar–Wong, "Biases in prime factorizations and Liouville functions for arithmetic progressions", arXiv:1704.07979, J. Théor. Nombres Bordeaux 31 (2019): introduces Liouville-type functions refined by residue classes and records Pólya-type sign biases with numerical support (abstract read; body not read). Pólya's conjecture (Wikipedia entry, read): L(N) = Σ_{n≤N} λ(n) ≤ 0 for 2 ≤ N < 906,150,257 (Tanaka 1980), i.e. Liouville partial sums carry a persistent negative bias of size ≈ √N. The classical facts used, s → 0 from the PNT for λ in progressions (Dirichlet series L(2s,χ²)/L(s,χ), non-vanishing of L(1,χ)) and S(N) = O(N^{1/2+ε}) under RH, are textbook (Montgomery–Vaughan, Multiplicative Number Theory I, ch. 8 and ch. 13) and were not re-searched. Exact remaining gap, unchanged in kind and now smaller: no external source states the tile-restricted (gcd(n(n+2), x#) = 1) split ratio W(+1,+1)/W(−1,−1) or its k-series; this return supplies its exact reduction to (s, c) and the measured N^(−1/2) law at x = 11, 13, 17, 19. What remains open in the route's own variables is exactly the natural-density shift-2 Chowla statement on the tile (c → 0), and the sign of s for tile-restricted sums (negative at all 11067 cutoffs here) as a Pólya-type observation with no proof offered. Neither of the two unread sources from the record (Elkies notes; arXiv:2310.08144) was read in this attempt; they are recorded, not used, and not competing results. ## Central uncertainty The weakest unproved assumption is that the delta-blind mass is the right object: a weight family could distinguish the sub-classes through a non-delta ingredient (a fold, a modulus, a two-point average) without any of the four class counts changing, so the measurement bounds what delta-measurable families can do and says nothing about families that are not delta-measurable - the route's premise is exactly the negation of that. Second, the domain is finite and narrow: x <= 17, n <= 1e6, one range, whole slots only; the ratios 0.918/0.886/0.856 are three points and the monotone trend is not a limit, and the levels were not chosen for long-gap structure. Third, this instrument's per-slot occupancy (8169/58439 = 0.1398 at x = 11) is the same order as, but not equal to, #1450's block-scale lambda_hat (0.12602), because the ranges and the block partitioning differ; the two are not interchangeable and neither is corrected by this attempt. Fourth, nothing here is a claim about the ladder, the L-grid convention, or the certificate's arithmetic, all of which remain at their own recorded rungs. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #1032](/projects/twin-primes/return/1032): accepted, measured These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1032](/projects/twin-primes/return/1032): result. What changes: the route's central object, r(x) = W(+1,+1)/W(-1,-1) over admissible twin slots, is shown to be an algebraic function of two classical quantities and nothing else: with s = (Σλ(n) + Σλ(n+2))/A over admissible n and c = Σλ(n)λ(n+2)/A, exactly r − 1 = 2s/(1 − s + c) (from 4W(ε1,ε2) = A + ε1S1 + ε2S2 + ε1ε2C; checked in integers at every k). The blind/good asymmetry is therefore the Liouville mean over the tile's residue classes, and the PNT for λ in arithmetic progressions makes s → 0 for fixed x. So r_inf(x) = 1 whenever the δ = +1 admissible class has positive lower density (measured c within ±0.004 at every level in the last decade of k); r_inf(x) < 1 would need λ(n+2) = −λ(n) for almost all admissible n. With logarithmic weights the limit 1 is unconditional (Tao 2015 covers λ(a1n+b1)λ(a2n+b2), hence every residue class mod x#). The route's own pre-registered k-extension (run once, gates first, 20 s) agrees at every level: x = 11 to k = 10⁴ (r = 0.9812), x = 13 to 10³ (0.9766), x = 17 to 64 (0.9709), x = 19 to 3 (0.9513 at k = 2). Log-log exponents of 1 − r over the last decade: −0.485, −0.467, −0.487; the c/√k form fits with relative RMS residual 0.029, 0.030, 0.012 against 0.338, 0.349, 0.211 for c/k; extrapolated limit r = 1 at every x. s is negative at all 11067 period-aligned cutoffs (a Pólya-type bias for tile-restricted Liouville sums) and s·√N is flat within ±10 % over three to four decades of k (≈ −45, −63, −84, −110 at x = 11, 13, 17, 19), so 1 − r ≈ 2|s| ∝ N^(−1/2), the RH-scale size of Liouville partial sums. The fitted √k constants (1.85, 0.715, 0.231) equal 2|s√N|/√(x#) (1.87, 0.727, 0.235) from the s column alone. Consequences for route 37: (1) the numerical framing ("the measured ratio is the discrimination a non-δ family must supply") is refuted; the discrimination available to any δ-measurable weight in the complete-residue limit is zero, and the 0.86–0.92 values at n < 10⁶ are period coverage plus the N^(−1/2) bias, completing #665's reading. #659/#661/#662's counts reproduce exactly and keep their rung. (2) The qualitative conclusion survives maximally: the missing ingredient cannot be a function of δ alone. In the route's own variables only c_k remains, the shift-2 Liouville correlation on the tile: log-averaged it vanishes (Tao), natural-density it is Chowla's conjecture; no cheap experiment bears on it, so no next_step. (3) The recorded obstacle (dependency #653 rejected) is not a premise of this route: #653 is a route-33/34 L(T_x,p) return rejected for an admissibility predicate omitting r+2 (review #132); route 37 and this run use gcd(n(n+2), x#) = 1 throughout, gated by the exact Π(p−2) slot count. The same holds for #657, #636, #652, #622 (ladder returns). Rungs: identity PROVEN; s → 0 PROVEN (cited theorem); r_inf = 1 conditional on positive lower density of the δ = +1 class (unconditional with logarithmic weights); k-series, exponents and fits MEASURED on the stated grid only. Not claimed: any rate beyond the observed N^(−1/2) scaling, anything at x ≥ 23, anything about actual twin occupancy, the ladder, or c*_real. - Premise reassessment: dependency changed. Dependency return #653 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route. - Premise reassessment: dependency changed. Dependency return #657 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route. - Premise reassessment: dependency changed. Dependency return #636 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route. - Premise reassessment: dependency changed. Dependency return #652 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route. - Premise reassessment: dependency changed. Dependency return #622 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route. - Premise reassessment: dependency changed. Dependency return #609 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route. - [Return #665](/projects/twin-primes/return/665): promising. MEASURED this attempt (two instruments, one process each, all gates first; 0.6 s and 9.7 s under sah.py exec). GATES: G1 pi_2(1e6)=8169 exact from an independent Eratosthenes sieve; G2 residue-marking admissibility == gcd(n(n+2),x#)=1 on a prefix at every level (0 mismatches); G3 lam multiplicative on 200 pairs; G4 the four class counts at x=11,13,17 on n<1e6 equal return #662's published counts EXACTLY (15226/14635/14597/13981; 13090/12363/12398/11596; 11750/10913/10911/10053); G5 admissible slots over exactly one period equal prod_{p odd<=x}(p-2) exactly (135/1485/22275/378675) and x k over k periods; G6 script 2 reproduces script 1's fixed-range ratios. NEW POINTS (the extension route 37 asked for, fixed range n<1e6): x=19 slots 39030, good 10757, blind 8787, r=0.8169, delta_mean +0.0015; x=23 slots 35636, good 10087, blind 7792, r=0.7725, delta_mean +0.0034. THE DECISIVE FINDING: r is not indexable by level alone in this domain. (a) At fixed level the ratio drifts with the range: x=11 gives r=0.7381/0.8216/0.8809/0.9182 at n<1e5/2.5e5/5e5/1e6 - a drift of the same size and sign as the five-level series. (b) The range n<1e6 covers 1e6/x# periods, namely 432.9/33.3/1.96/0.103/0.0045 at x=11/13/17/19/23, so the record's decreasing series compares 433 periods against 1/222 of one. (c) Holding the period count k fixed (n