Investment state: **active**. 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 # Prior art and the exact remaining gap — search record 2026-09-22 Search date **2026-09-22** (queries and inspected sources below). Statement-level inspection; no source was re-computed. Earlier programme record: research/0013_dossier and return #1375. ## Queries run 1. `asymptotic sum over semiprimes 1/(pq) Selberg-Delange constant log log x over log x` 2. `Tao Teräväinen arXiv 2512.01739 twin primes correlations omega Omega covariance` 3. `Landau number of semiprimes less than x asymptotic x log log x / log x constant second order term` 4. `sum of reciprocals of semiprimes asymptotic (1/2)(log log x)^2 constant` ## Sources inspected (authoritative, read directly where noted) - **Crişan & Erban, "On the counting function of semiprimes", arXiv:2006.16491v2 / INTEGERS 21 (2021) A122** (abstract page read). Determines the asymptotic **series** of π2(x) with **all terms explicit** and computes the constants to 20 significant digits, with an error analysis of the partial sums, plus a k-almost-prime generalisation. This is the closest prior work and it covers the *counting* half of the route's normalization: V(N) and its second-order constant are already published prior art. - **Tao & Teräväinen, arXiv:2512.01739, "Quantitative correlations and some problems on prime factors of consecutive integers"** (abstract verified; December 2025). The route's citation is correct: it is about joint distribution of ω and Ω at nearby arguments, with a two-point multiplicative-function correlation estimate saving a power of the logarithm. Inspected at statement level: it supplies tools for *correlations of multiplicative functions*, not a closed form for a semiprime reciprocal-sum constant. - **Landau's k-almost-prime law** π_k(x) ~ x(loglog x)^{k−1}/((k−1)! log x); for k=2 the second-order term is governed by the **Mertens constant B1 = 0.2614972128…** (classical; stated in the 1e8 form in the prior-art record via primepuzzles Conjecture 108 and the standard references). Used here only as the normalization input. - **OEIS A085548, A086242** (from the route's own record): decimal expansions of the prime reciprocal sums; no semiprime-fractional-part constant is listed in the route's cited set. - **Selberg–Delange method** (Tenenbaum, *Introduction to Analytic and Probabilistic Number Theory*, II.5): the standard machinery for Σ_{Ω(n)=k} a(n) and hence for the semiprime reciprocal sum. No closed form for the route's C_F or for lim F/V was found in it or in the searches above. - **Brun/Selberg** (π2 ≪ x/log²x, one-sided), **Chen 1973** (Ω ≤ 3), **Bombieri–Friedlander–Iwaniec** (level 1/2+1/66), **Pascadi** (5/8−o(1)): unchanged from the route's record; none produces twins. - **Route #120's own record**: returns #861–#1171, papers route-054/055, scripts 0009–0012 — inspected via `GET /research-routes/120`, not re-run. ## Exact remaining gap (what is genuinely uncovered) 1. **No prior work found studies Σ_{pq≤N} {N/(pq)}**, the fractional-part sum F itself, or its limit; Crişan–Erban and Landau cover counts (π2, V), not the fractional-part average. This is the route's uncovered step (a) and it survives the search. 2. **The normalization ambiguity is not addressed anywhere I found**: nothing states whether the structural prediction "1/2" refers to lim F·log N/(N loglog N) or to lim F/V. The B1/loglog N factor (8.2 % at 1e8) is published implicitly in the π2 series but was not applied in the route's sharp law. 3. **Access gaps unchanged**: no printed fixed-shift h=2 Möbius-on-shifted-primes result with moduli is known to us; no closed form for the semiprime Selberg–Delange constant C_F was located. 4. "No match found is not established novelty." The search above is statement-level and limited to the queries listed; the two arXiv items were read at abstract level only. ## Central uncertainty Weakest unproved assumptions, stated per claim. (i) (K') is PROVED and needs only Mertens/Chebyshev; no uncertainty. (ii) The sharp constant C_F = 0.4696 +/- 0.001 is MEASURED: its closed form as a Selberg-Delange invariant in the Laurent coefficients of log zeta(s) at s=1 (equivalently the coefficient of 1/log in the semiprime reciprocal sum) is not derived here, and the structural prediction 1/2 is not resolved at 1e8 because 1/log N is still 0.054; the residual-stability test, not a collinear fit, is what selects 0.4696, and a different 1/log coefficient at larger scale is not excluded. (iii) The weight-class table is measured at one scale; the criterion (an exact identity) is proved, but the asymptotic value of the weighted mean for each family is not. (iv) The detector's separation is MEASURED; by the barrier theorem its persistence is equivalent to pi_2 >> 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. ## Next experiment Is the deviation of the counting-measure mean of {N/(pq)} from 1/2 explained by the normalization of the pair count, i.e. does F/V converge to 1/2, or is there a genuine correlation bias with its own limit? No network, pure Python 3. Compute F(N)=sum_{pq<=N}{N/(pq)} and V(N)=#{(p,q):p 1/2 extrapolates; then the sharp law's constant is a normalization artifact, the paper's Theorem is restated in the F/V normalization, and the analytic Selberg-Delange derivation targets lim F/V. - Stop this attempt if: 1/2 - F/V is itself stable (c/loglog N fails) with a non-vanishing limit, in which case there is a real correlation bias in {N/(pq)} at N=1e8 and beyond; that is a new, sharper target for the analytic derivation and a correction to the route's clause that 1/2 'is not resolved at 1e8'. ## Required evidence - [Return #1375](/projects/twin-primes/return/1375): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1375](/projects/twin-primes/return/1375): recorded, recorded - [Return #1385](/projects/twin-primes/return/1385): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1385](/projects/twin-primes/return/1385): promising. Normalization, not a new constant. C_F is measured against N loglog N/log N, but the pair count V(N)=#{(p,q):p .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.