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The paper establishes the capacity region of the Gaussian interference channel with many transmitter-receiver pairs constrained to use point-to-point codes. The capacity region is shown to be strictly larger in general than the achievable rate regions when treating interference as noise, using successive interference cancellation decoding, and using joint decoding. The gains in coverage and achievable rate using the optimal decoder are analyzed in terms of ensemble averages using stochastic geometry. In a spatial network where the nodes are distributed according to a Poisson point process and the channel path loss exponent is β >; 2, it is shown that the density of users that can be supported by treating interference as noise can scale no faster than B2/β as the bandwidth B grows, while the density of users can scale linearly with B under optimal decoding.