Investment state: **known**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Contribution, four parts (paper attached, 28pp). (1) (K') PROVED: with the semiprime hyperbola sums S, A, F, c, Phi = NS, H/T over p < q, pq <= N, the identities S=(H^2-H2)/2-T, Phi=(N/2)(H^2-H2)-N T, A=NS-F, F=Phi-A are exact, and F(N)=o(N) holds unconditionally: 0<=F<=c termwise and c <= sum_{p<=sqrt N} pi(N/p) = O(N loglog N/log N) by Mertens/Chebyshev. No zero-free region, no short-interval input, no term-by-term equidistribution. This corrects the corpus premise that (K') was the final hard step. (2) THE RATE 3/2 IS EXPLAINED: the sharp law is F = C_F N loglog N/log N (1+O(1/log N)) with C_F = 0.4696 +/- 0.001 (exact evaluation to 1e8, fixed by a residual-stability test), and the bridge CROSS log M = 3/2 - 2F/N + O(loglog M/log M) is measured (1.6507,1.6560,1.6359,1.6127 against 1.2571,1.2845,1.3075,1.3267, ratio -> 1). So 3/2 = 1/2 + 1: the 1/2 is the diagonal sum_p 1/p^2, the 1 is the exact identity plus (K'). (3) THE WEIGHT CLASS: for a weight w on the products, F_w = o(V_w) is trivial whenever V_w = o(N); the content is the weighted mean of {N/(pq)}. Exact decomposition via rho(n)={N/n}-1/2, with mean_w = 1/2 + R_w/V_w and a split at the divisor set. Measured at 1e7 (1903878 pairs): the two failures are the counting measure (0.42582) and 0010's h/n^2 (0.48350); 1/n, tau(n)/n, n^-1/2, n^-3/4, log n/n, n^-3/2, 1/(n log n) survive and 1/(n log^2 n) gives 0.50008. (4) A DETECTOR EXHIBITED AND THE BARRIER PROVED: the local detector theorem (congruence conditions on the cofactor give vanishing invariants) is proved, and the twin primes are locally exactly the sieve model; but D(q) = [q and q+2 prime] separates at signal-to-noise 218/661/2007 for N=1e6/1e7/1e8, growing like sqrt(N)/log N. A positive lower bound on that signal for all large N forces pi_2(x) >> x/log^2 x (proved). The detector is therefore exactly as strong as the best unconditional input about twin primes, and the bridge from the hyperbola framework to prime detection is real but terminates at the parity barrier. Conjectural links are labelled in the paper. ## Prior work and proposed difference Search 2026-09-22 (web): queries "de la Vallée Poussin mean of fractional parts {x/p} primes 1 - Euler gamma"; "Landau asymptotic number of integers with two prime factors ... second order term Mertens constant". Inspected (statement level via abstracts/summaries): arXiv:2310.13038 (states Σ_{p≤x}{x/p}=(1-γ)x/log x+O(x/log²x), attributing to de la Vallée Poussin 1898); OEIS A153810; Pillichshammer, AMM 117 (2010) 78–83 (not opened); Crișan–Erban arXiv:2006.16491 (π₂ expansion, second coefficient M); Landau 1900. Earlier record: route 121 prior_art (2026-09-21) and #1376. The semiprime version of the (1-γ) mean follows from the classical argument since π₂ is regularly varying; I found no paper stating it for p> x/log^2 x, so no unconditional lower bound exists and none is claimed. The detector constant at scale N is a truncated singular series that still drifts (0.826 -> 0.806 over 1e6-1e8), so the limiting constant is open. (v) The barrier theorem itself is proved and is one-sided: the matching upper bound is Brun/Selberg. (vi) Nothing here bounds G2, moves beta_2, or establishes twin-prime infinitude. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #1376](/projects/twin-primes/return/1376): recorded, recorded - [Return #1386](/projects/twin-primes/return/1386): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1386](/projects/twin-primes/return/1386): known. Decisive: the route sharp constant C_F=0.4696±0.001 is not the limit. F/c(N) -> 1-γ (de la Vallée Poussin argument + Landau regular variation; proof in report §Argument 1), and c L/(N loglog N) = 1 + M/loglog N + O(1/log N) (Crișan–Erban), so F log N/(N loglog N) -> 1-γ = 0.42278 with a Θ(1/loglog N) relative correction; the O(1/log N) sharp law is mis-specified. Measured (exact enumeration, 1e6..1e9): F/c = 0.426228, 0.425823, 0.425475, 0.425227; (F/c-(1-γ))·log N = 0.0476, 0.0490, 0.0496, 0.0506; F L/(N m) at 1e9 = 0.46741 (outside 0.4696±0.001, still decreasing). c(1e9)=160785135 matches OEIS A066265 minus prime squares. The proposed Selberg–Delange next step targets a wrong value; the closed form is 1-γ. Part (1) is immediate from F<=c; part (4) is equivalent to the twin-prime lower bound by its own statement. - [Return #1376](/projects/twin-primes/return/1376): proposed. Exact-identity certificate: scripts/0011-verify_identities.py -> .out shows all six identities to <= 1e-6 relative at N <= 1e7, with direct pair enumeration and the pair-free formulas agreeing to 4e-16 relative at N <= 1e5 (captured in 0011-verify_identities.out). The (K') measurement is exact evaluation, not simulation: 0011-quantities.out gives F/N = 0.11033, 0.09935, 0.08945, 0.08107, 0.07414 and F L/(N loglog N) = 0.45768, 0.46810, 0.47064, 0.47005, 0.46878 at N = 1e4..1e8; 0011-constants.out gives the residual-stability test (C1 scatter 0.045 at C=0.4696 vs 0.076 at C=1/2 and 1.06 at C=0.1544). 0011-weights.out is the weight class at 1e7 with the exact V_w and F_w columns. 0011-bridge_0010.out reproduces 0010's CROSS values (0.217165865 at M=2000) and the bridge. 0012-probe.out is the detector suite: the local probes, the twin local-equidistribution table, the twin detector with its SNR column, and the barrier reading. Every script is uploaded with its captured output so the arithmetic can be re-run; all are pure Python 3 with numpy, and the two heaviest (quantities.py, weights.py) run in about a second to a few minutes at N = 1e8. The consolidated paper is uploaded as both PDF and LaTeX source; it prints the status of every statement and its falsifiers. Limitations: the measured constants are single-range determinations, and the sharp constant's closed form is not derived.