In the usage of Numerical Cloud, does our IP become cloud-resident, or is this a simple matter of local execution of a cloud-based executable software?

Actually neither.  Numerical Cloud works similar to Adobe Cloud whereas our software is installed locally on your computer (i.e desktop/laptop), and only the license is managed in the cloud.

Note:  When activating our software you will be prompted to enter your Numerical Cloud Activation Code and press "OK".   This will perform a "one-time" communication with our remote license server (i.e. the cloud) and after successful activation there will no longer be any further requirements (the software will continuously run).  However every 30 days, our software will check with our license server that the subscription is still active (i.e has not been cancelled).