Minimal Radial Sub-Gamma Envelopes for Infinitely Divisible Random Vectors

Let X be a centered infinitely divisible random vector with finite second moment and covariance matrix Sigma. We define the radial pole C_X(t) as the smallest scale in a right sub-gamma bound for whose quadratic proxy is fixed at the true variance t^T Sigma t. A canonical directional measure and an independent Beta(1,2) multiplier give an exact variational formula for C_X. The resulting extended-valued function is positive homogeneous and is pointwise least among all homogeneous denominators compatible with the covariance quadratic form. We prove linear-map, convolution, and Levy-time rules, and show that the full family of directional remainders determines the law of X. Geometrically, C_X lies between the Minkowski functional of the moment-generating-function domain and one third of the positive support function of the Levy measure; the upper constant is sharp, and the zero set is a polar cone. The pole need not be subadditive. It is continuous on the sphere under global exponential moments and positive-definite covariance, whereas finite variance alone permits a jump from zero to infinity in nearby directions. For additive gamma-ray models, C_X equals the domain gauge and has a finite-polytope formula.