Let's consider differential object marking (DOM) along the Baltic Finnic type, i.e. one of the cases is universally used with negative verbs, and also quite often with affirmative verbs - basically, the other possible case marks a combination of things, of which affirmativeness only is one. Now, imagine constructions with contrastive gapping:
I don't eat pork but venison
Let's further assume that the DOM over time is weakened, and the more general case - the one used both with negatives and many affirmatives - turns into a more general accusative. The other case remains, but is used for whatever usages there might be.
However, we return to the form above: the case might have been used to contrast the two objects even when not enough of the usual requirements were fulfilled. So, one usage that gets tied up with it is contrast.
I don't eat pork.ACC venison.CONTR
This might extent to other roles: contrasting with subjects, locations, etc. Contrast might go even beyond negative-affirmative:
I solve problems.ACC and also cause them.CONTR
even beyond that, you can get subjects:
I solve problems.ACC and he.CONTR creates them.ACC
in both of these examples, 'and' could be more idiomatically translated into English as 'but'. However, in the language we're dealing with, 'and' might not even necessarily be marked. In a way, CONTR marks the central argument of a coordinated, contrasting VP.
It might even develop further, to get things like:
I got 99 problems.ACC a bitch.CONTR
where the first construction we saw also has an affirmative structure - i.e. we're not just contrasting verb phrases with each other, or a negative verb phrase with a contrasting object coordination, we're also doing the same for affirmative verbs.