Obstructions to Fourfold Aggregation of Prime Phases

We examine whether independently defined aggregates of logarithmic prime phases can realize the equal-amplitude four-phase geometry of a repeated projection model. A moment criterion characterizes an equally weighted rotated quartet on the unit circle. Positive prime and von Mangoldt weighted phase measures fail this criterion at every finite cutoff containing the primes 2, 3 and 5; the prime number theorem also gives a strictly positive limiting defect throughout the open critical strip at fixed height. A density-subtracted prime-power sum has more favorable finite numerical behavior, but boundedness at even one fixed point of real part one half implies the Riemann hypothesis. Winding of the analytically continued Euler product counts enclosed zeros without determining their horizontal coordinates. A separate computation with interval arithmetic and explicit analytic majorants isolates one simple zeta zero near the first known ordinate and establishes that four raw zeta samples at radius one tenth have normalized cancellation residual greater than one fifty-thousandth. Reproducible experiments on 34 known zeros and 301 controls, supplemented by 30 nearby controls, illustrate these distinctions. The results exclude specified realizations, identify assumptions with global consequences, and leave the construction of a suitable coordinate-preserving aggregate observer open.