Resonance Functions, Collective Prime Phases, and the Critical Line

We revisit resonance functions R(s,p) linked to primes p as potential indicators of Riemann zeta function zeros. This work adds rigor by: (i) establishing uniform analytic estimates for R(s,p); (ii) providing an explicit counter-example (p=3) proving single-prime resonance fails for p>2; and (iii) proving that collective singularities of the family {R(s,p)} necessarily lie on the critical line Re(s)=1/2, thus linking collective prime behavior to the critical line.