PRH | Aux | 4.6 • Twin Primes via Helson–Blur

We develop a band-limited, positivity-based framework that reduces infinitely many twin primes to two explicit analytic inputs: (i) a Gaussian-blurred pair-correlation control for the off-diagonal of the vertical convolution \(A*A\) (with \(A=-\zeta'/\zeta\)), and (ii) an identification of the simple pole at \(s=1\) of a locally filtered twin Dirichlet series with residue equal to the diagonal Hardy--Littlewood constant \(2 C_2\). Our method locks whenever a twin pair lies in the detection window and drifts otherwise; a Lyapunov-type functional couples this lock--drift on the prime side to a Reproducing-Kernel energy on the spectral side via Helson's boundary guard and a Herglotz--Nevanlinna positivity transfer ("Hilbert-Pólya via blur"). We further introduce a Hade-Hide virial flux expressing the twin deficit as a commutator with the scale generator, giving a sectorwise route to the residue at \(s=1\). The microscope applies to any fixed admissible pattern \(K=\{k_1,\dots,k_r\}\), replacing \(A*A\) by \(A^{*r}\) and the twin singular series by \(\mathfrak{S}_K\); for \(r\ge 3\) one needs the blurred \(r\)-level input. Finally, we give a log-free transfer from wide Gaussian locks to short multiplicative windows. No step uses RH.