It's a bit tricky with different payment flows possible for all kinds of scenarios. While at some point the transaction will get to "a giant mainframe somewher

It's a bit tricky with different payment flows possible for all kinds of scenarios. While at some point the transaction will get to "a giant mainframe somewhere" to get processed, it may or may not be during the process that a customer sees. It may be a "online" transaction where it's essentially that the terminal prepares a transaction message, the chipcard signs it, and then forwards it to issuing institution for authorization (including but not limited to checking availability of funds); or it may be permitted, within certain limits and depending on configuration, to have the chipcard sign the transaction and then deliver it for processing some time later (e.g. at end of day) so that either you can enable transactions in places which do not have internet access, or simply for processing speed for small amounts, taking on some limited risk for customer convenience.