A while ago I made a transcription available of Carnap’s open letter to LeSeur, which I said could in a sense be regarded as his first publication.  Another candidate for that status is this — privately published — letter of 1914 expressing Carnap’s fulsome enthusiasm for Sweden.  It is available among the papers the Archive of Scientific Philosophy (Hillman Library, University of Pittsburgh) has put online in box 25, folder 101, pp. 28-31 (of the Carnap papers there).  Here, for convenience, is the Carnap item by itself.  Many thanks to Gottfried Gabriel for bringing this to my attention years ago.