If the modem connection is interrupted, does that make the APs dead until the modem is rebooted?
The logs in HiveManager wouldn't show an issue like this, they are more for admin functions such as tracking who made what change. We could ask the APs to log for certain issues, but this would need to be set up ahead of time. For instance, if the APs are disconnecting from the HiveManager during the outage, we could run capwap debugs to see why they were disconnecting.
To do this, we'd need to SSH in to one of the APs and run:
_debug capwap info
_debug capwap stat
_debug capwap basic
These commands are erased when the AP reboots. If you need them to endure between reboots, we'd have to build a supplemental CLI object.
Once the AP disconnects, we'd need tech data. Here is a guide that will walk you through how to get tech data via the CLI of the AP- https://thehivecommunity.aerohive.com/s/article/Collecting-Tech-Data-via-CLI
All that said, the easiest test is going to be consoling in to an AP during an outage and trying to ping google.com. If that doesn't work, trying pinging 184.108.40.206. If that doesn't work, it's likely that we can't reach the default gateway, and we'd want to check your router/modem to see why.
The APs need a connection to the default gateway. If the modem is hosting that gateway for you, that would explain what you are seeing. I would consider replacing the modem since you've narrowed the issue down to that part of the network. If you call Aerohive during one of these outages before you reboot the modem, we can confirm for you that the problem on our side is an inability to reach the gateway, or if we find that is not the issue we can troubleshoot further.