I managed to learn a couple things about my internet connection's stability problem.

The TL;DR: my ISP is what is dropping my connection on their end of it, terminating the LCP and making my modem wait 3 to 5 minutes to reconnect. A massive and visible disruption.

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The longer explanation:

The Link Control Protocol (LCP) is what's going splat. Basically, it is what allows the modem to handshake with the ISP. Its link to the server is perfectly fine, so it has no trouble seeing my ISP's system, but that system is what is dropping my connection to the internet when it kills the LCP. It does recover on its own and reconnect after a 3 to 5 minute delay as it tries to reach my ISP and re-open the connection.

The connection drops happen on particular days, with a wide window of time (they never happen at a fixed time on those days). These are Sunday or Monday, Wednesday, Friday or Saturday. Seems like maintenance windows, doesn't it. And the times of the day it happens include 1 AM, 7 AM, 9 AM, 11 AM, 2 PM, 3 PM, 7 PM, and 9 PM. Its not a fixed time, as you can see.

For example, the most recent one was 1:50 AM this morning. The LCP went down and it didn't come back up and reconnect to the internet until 1:53 AM.

The cause of this is probably Verizon's merger with Frontier. They officially closed their merger deal in January, but they very likely began preparing their systems to take the new load in late December. They have more than 8 million Frontier customers whose IP addresses have to be added to the system; customers who still log in to the same "servers" they always had, but whose addresses now have to be handled by Verizon's backbones.

And the 3 to 5 minute dead zones once the LCP goes down is Verizon's network ignoring my modem for that time span, presumably until it sees that my prior session timed out or something. If that makes any sense.

... But I really wish this would freaking stop. Seriously. A customer is not supposed to notice any disruptions like this because they're literally supposed to be "blink and you'll miss it" events, but this is more like faceplanting.