Arithmetic conditioning of residual phases

Establish whether arithmetic classes of the integer shells N_k=floor(1/4+gamma_k^2+1/2), obtained from critical-line zeta ordinates gamma_k, leave a nontrivial Fourier signature in the circular residual phases Z_k=i exp(2 pi i gamma_k^2), as measured by the conditional Weyl moments W_h(A;K).

Background

For each critical-line zeta zero rho_k=1/2+i gamma_k, the paper defines the quadratic level L_k=1/4+gamma_k^2 and decomposes it exactly into an integer shell N_k and a centered residual delta_k. The integer shell is factored arithmetically, while the residual is circularized as Z_k=exp(2 pi i delta_k)=i exp(2 pi i gamma_k^2).

The unresolved question is whether conditioning on arithmetic properties of N_k—such as being prime, a pure prime power, a semiprime, or having fixed numbers of distinct or total prime factors—changes the distribution of Z_k. The proposed test uses conditional Weyl moments W_h(A;K); phase independence predicts that these moments tend to zero, whereas persistent nonzero harmonics would indicate arithmetic conditioning of the residual phase. The paper reports only preliminary negative controls and does not resolve the question.