1. A network-wide verification system comprising: a network device interface to read and write packet-forwarding rules of varied, multiple network devices comprising switches and routers of a network; a controller in network communication with the multiple network devices via the interface and to: obtain data comprising routing activity received from the multiple network devices of the network; and transmit packet-forwarding rules in real time to the network devices; computer storage in which are stored multiple network invariants expected to be upheld in valid network operations as determined by a combined effect of packet-forwarding rules across multiple network devices; and a data plane verification layer positioned between the network device interface and the controller and comprising non-transitory computer-readable storage medium storing instructions that, when executed by at least one processing device, are to: monitor packet-forwarding rule modifications in the network, including modifications initiated by the multiple network devices and by the controller; and verify validity of invariants of the network according to the stored multiple network invariants and the routing activity, to perform verification of the invariants on parts of the network capable of being influenced by a new rule modification.