Poseideon β’, the necessary intercalary month. As we know, the first day of the Year is the Noumenia of Hekatombaion, which must fall always after the Summer Solstice (cf. in particular Phot. Lex. 201; Anecd. Bekk. 247.1; Suda s.v. Εἰσιτηρία. For the italian readers, we suggest you to check this book for more references: https://hellenismo.wordpress.com/2014/08/27/calendario-religioso-tradizione-ellenica-religious-calendar-hellenic-tradition/). According to the sources, the intercalary month has to be added after Poseideon α’, and thus we make because otherwise, if we not insert now the intercalary month, the next Noumenia of Hekatombaion will be not after, but before the Summer Solstice, that is not absolutely possible.