Abstract: Connexive principles trace back to the earliest accounts of conditionals, but remain difficult to formally reconstruct given their contraclassicality. This fact questions the very possibility of a semantics (1) validating all connexive and some basic conditional principles unrestrictedly (2) over a classical extensional base, (3) in a way compatible with both desiderata of the general literature on conditionals and a notion of connection. I show how this is achievable thanks to a conditional logic CX combining what I call 'minimal total choice-functional’ Segerberg frames with new semantic clauses for conditionals expressing connections at world-witnesses for antecedents. The conditionals of CX are hyper- and strongly connexive, validate identity, modus ponens and excluded middle, invalidate material implication paradoxes, antecedent strengthening, contraposition, explosion and implosion, but also simplification and distribution over conjunction, which only hold for possible antecedents. This, however, is expected given CX non-vacuist interpretation of impossible antecedents.