I have been told that Kalendae is the first day of a month. However, the Latin dates — which was discussed in this other question — alone do not make this obvious. Dates are expressed by comparison to three special days every month: Kalendae, Nonae, and Idus.
Based on everything I know about the Roman calendar, it would make sense that Idus was actually the last day of a month. Or perhaps days between Idus and the following Kalendae belong to no month at all. Or maybe a month consists of the three special days and other days have to be expressed in relation to these interesting days because other days belong to no month at all. It is not obvious that every day should be part of some month.
Is it clearly stated somewhere in ancient literature that Kalendae is the first day of a month? Is there some ancient source describing the structure of a month in this sense? Evidence from inscriptions is also welcome if it sheds some light on the matter. There seems to be a very strong modern consensus that Kalendae is the first day, but I realized that I know no reason for this.