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Binary convolutional (linear) code pairs are investigated for use on the two-user adder channel. Maximum likelihood decoding is discussed, and a two-user decoding trellis is defined. The L-free distance of a convolutional code pair is defined and shown to be equal to the free distance of the mod-2 sum of the two single-user codes. It follows that the code pair is uniquely decodable if and only if the mod-2 sum code has an inverse, and that the code pair is subject to catastrophic error propagation if and only if the mod-2 sum code is catastrophic. These results also imply that no uniquely decodable binary convolutional (linear) code pairs exist with a rate sum above time sharing.