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We present a new methodology for the mathematical analysis of 3D paper structure and apply it to a model that can be simulated on a computer. We rigorously derive upper and lower bounds for fiber contact areas, and derive an approximation that is very close to values calculated from the simulations. The method involves the stochastic geometry and combinatorics of large numbers of randomly located sets, which quantify the interactions between fibers. The main calculation involves a sum resembling a partition function with many-body interactions of all orders.