Numerical Verification of the Prime-Prime Lattice (PPL) Framework: Kuramoto Self-Tuning Attractors, Hierarchical Spectral Duality, and MSPP Mass Generation

We present a comprehensive quantitative verification of the Prime-Prime Lattice (PPL) theory using numerical simulations across four distinct physical dimensions. First, non-linear Kuramoto oscillator dynamics reveal that the critical line sigma=1/2 acts as a self-tuning attractor, avoiding global coherence breakdown while maintaining dynamic phase correlations. Second, finite-size scaling (N=30 -> 100 -> 10,000) proves that sigma=0.5 serves as an absolute asymptotic upper boundary in the thermodynamic limit, with statistical noise floors suppressed to 1/sqrt(N) approx 0.01. Third, spectral interval analysis under a unified sample size (N=10,000) confirms a fundamental hierarchy: bare prime modes form a structured integrable arithmetic lattice exhibiting Hardy-Littlewood gap clustering, whereas topological nodes (zeta(1/2+it)=0) undergo a phase transition to GUE level-repulsion (D=0.0275). Finally, two-dimensional field simulations demonstrate that the spatial energy density T_00 concentrates around Minimal Step Prime Pairs (MSPP) surrounding phase nodes. Formulating an effective field theory, we show that the multiplicative vacuum translates into additive phase dynamics via logarithmic projection, where the MSPP network acts as a dynamical screening jacket that generates a Yukawa-type vacuum polarization. This mechanism eliminates logarithmic divergence and guarantees a strictly convergent, finite rest mass m=E/c^2 with an exact geometric form factor E_infty* approx 1.6332.