This paper is a study on the foundation of Integer Evolutionary Strategy (IES), a new heuristic technique for solving integer programming (IP) problems. Existing IESs are discretized and modified to apply the algorithms designed for continuous spaces to IP problems, which are discrete spaces, and operate based on the $\ell_2$-norm. This study adopts the $\ell_1$-norm, considers an appropriate step size, and explores an alternative method for measuring correlations on integer grids to establish a foundation for discrete search. In particular, we focus on mutation distributions for infinite integer decision variables, point out the limitations of discrete probability distributions based on uniform and binomial distributions, and focus on the truncated normal distribution (TN) and double geometric distribution (DG). We analyze their theoretical properties, including the entropy function, and propose a procedure for generating scalable correlated mutation distributions. Through extensive numerical experiments, we demonstrate that the DG distribution is more suitable for infinite integer search, and empirically verify that IESs using correlated DG mutations outperform other strategies in nonseparable quadratic IP problems. In conclusion, we emphasize that although replacing the TN distribution with the DG distribution provides theoretical justification and practical advantages, adopting the $\ell_1$-norm instead of the $\ell_2$-norm is a more significant change.