Called Rogers 
Figured I would try something different last night. I tried to put the
SMC WBR14T-G firewall on the internet instead of the Netopia. The SMC is stupid, but it does seem to log more about what it is trying. I let it bang away with DHCP requests for a while but it never seemed to get anywhere, so I returned the Netopia to the front of the stack and reset both it and the modem. The internet returned.
Immediately afterwards I called Rogers about the lousy internet service. Of course, since it was up, they said that there was nothing wrong. I showed the guy the www service graphs, indicating the long gaps in service; this seemed to impress him that I knew more or less what I was doing. The end result: no notes about this area of service, no notes on my account, everything is currently working, so call back when it isn't working. Uh ok, fine.