A sharp universal profile for multivariate characteristic-function factorization

We study the maximal factorization defect of characteristic functions for several random blocks. For prescribed moduli of the marginal characteristic-function values, we determine the exact extremal profile, and by optimizing it obtain the sharp universal constant \[ C_d=\max_{0\le \theta\le \pi/d}\{\cos^d\theta-\cos(d\theta)\} =\|T_d-x^d\|_{\infty,[-1,1]}. \] The extremizers are generated by a common two-point (Rademacher) phase structure. We establish uniqueness properties of the optimal profile, derive the critical-point equation and algebraic description of the maximizer, obtain exact values in small dimensions, and give sharp large-\(d\) asymptotics and an explicit global approximation to \(C_d\). We also discuss the real-valued and Gaussian restrictions of the problem. The results provide the universal baseline for subsequent work on fixed marginals and on the arithmetic of the associated critical polynomials.