Strict Monotonicity of the First Positive Steklov Eigenvalue on Eccentric Spherical Shells in Every Dimension

For d ≥ 2, consider the region between two nested spheres of fixed radii, with the inner sphere translated a distance ε from the outer center. We prove that the first positive eigenvalue of the Steklov problem, with the spectral boundary condition imposed on both spheres, decreases strictly with ε throughout the noncontact interval. This settles the planar monotonicity conjecture formulated by Nigam and establishes its higher-dimensional counterparts. In dimension two we use a double-center holomorphic expansion and a pair of positive compact coefficient operators. For every d ≥ 3, solid Gegenbauer harmonics yield analogous axisymmetric and transverse operators, with a positive rank-one monopole correction. Weighted angular-mode comparison for d ≥ 3 and a direct matrix comparison for d = 2 identify the first positive eigenvalue as the unique level-one crossing of an associated spectral radius. The first eigenspace has dimension d in the concentric case and dimension one for every nonconcentric configuration. Convergence and reconstruction estimates justify the coefficient reductions at the two boundaries.