Assuming that ablative case is always a semantic case (see the typical lists of its associated meanings in Latin grammars), I was wondering if Latin speakers could still assign a synchronic more or less "compositional" interpretation to the ablative phrase of the verb interest, which indicates the person to whom the thing is important (e.g., tua PARTE?). As is well-known, this appears in ablative singular feminine of the possessives mea, tua, sua, nostra, vestra. Otherwise, it is expressed in genitive case (e.g., omnium and nullius below). The thing concerned can also be expressed by ad plus accusative, as shown in the fourth example.
Tua et mea maxime interest te valere (Cic. Fam. 16.4)
Interest omnium recte facere (Cic. Fin. 2.72)
Magis nullius interest quam tua, Tite Otacili, non imponi cervicibus tuis onus (Liv. 24, 8, 17)
Magni ad honorem nostrum interest me venire (Cic. Fam. 16,1, 1)
Something similar happens with the verb refert. Cf. the diachronic explanation that relates tua to the prefix of this verb: tua res fert > tuā refert. E.g., vid. Woodcock (1959: 171): "By the apocope of final s in early Latin, this came to be pronounced rēfert. The first syllable was then thought to represent an ablative, and the last syllable of <tua> was lenghtened to agree with it": A New Latin Syntax, London: Methuen. However, I don't know how plausible this explanation (along with its alleged analogical extension to interest) is. So I was wondering if there is an alternative explanation for the ablative case in these examples.
Quid id refert tua? (Pl. Cas. 330)
Faciundum aliquid quod illorum magis quam sua retulisse videretur (Sal. Jug. 111, 1)