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We address the well-known problem of determining the capacity of constrained coding systems. While the one-dimensional case is well understood to the extent that there are techniques for rigorously deriving the exact capacity, in contrast, computing the exact capacity of a two-dimensional constrained coding system is still an elusive research challenge. The only known exception in the two-dimensional case is an exact (however, not rigorous) solution to the -run-length limited (RLL) system on the hexagonal lattice. Furthermore, only exponential-time algorithms are known for the related problem of counting the exact number of constrained two-dimensional information arrays. We present the first known rigorous technique that yields an exact capacity of a two-dimensional constrained coding system. In addition, we devise an efficient (polynomial time) algorithm for counting the exact number of constrained arrays of any given size. Our approach is a composition of a number of ideas and techniques: describing the capacity problem as a solution to a counting problem in networks of relations, graph-theoretic tools originally developed in the field of statistical mechanics, techniques for efficiently simulating quantum circuits, as well as ideas from the theory related to the spectral distribution of Toeplitz matrices. Using our technique, we derive a closed-form solution to the capacity related to the Path-Cover constraint in a two-dimensional triangular array (the resulting calculated capacity is ). Path-Cover is a generalization of the well known one-dimensional -RLL constraint for which the capacity is known to be .