A Scale Law Proof of Global Regularity for Critical Wave Maps

We prove global regularity for critical 2+1 dimensional wave maps with smooth data valued in compact targets. The proof uses a geometric Scale Law: R(t)^2 ∫_{B_{R(t)}} |∂φ|^2 ≤ C E0, where E0 is the conserved energy. Finite-time blow-up would require R(t)→0, which forces curvature concentration to infinity, contradicting the scale bound. This excludes self-similar and harmonic-map blow-up. The proof is PDE-free and relies only on energy monotonicity and scale invariance.