Lime, LOL. My rage wasn't directed towards you at all, and I shouldn't have raged in general, its just stuff like pushes my buttons. I don't like when someone passes the buck, you know?
Anyway, the IP that the router gets from the modem is called the "CPE IP" or customer premise equipment. For most ISP's using a standard cable modem, this IP is dynamic and changes often. How often it changes depends on alot of things. First, each IP on a DHCP network has what is called a lease, or time the DHCP server allows that IP on the network without a renewal. On most network it's 24 to 48hours. The DHCP server in the 4500 is set to 1440 min, or 24 hours by default. When that time ends, the computer must talk to the DHCP server, and ask for permission to renew the lease. The server asks around, sees if anyone has picked up the IP between the time it expired and the time the computer asked for a new one (or if it has to reject it for other reasons, but that gets complicated) then it renews it. Since most computers ask for a new IP right when the lease is up, the server has no reason to deny the request. Complicated stuff comes in because the server could be backed up, or the node may have changed, etc. To put simply, sometimes your DHCP IP lasts 24 hours..sometimes it can last years. Usually atleast a couple of weeks.
Now, one shouldn't confuse the IP the computer gets from the DHCP server "through" the modem, and the IP the modem itself gets assigned. That is also dynamic, but with a much longer lease time. I highly doabt that IP would change often enough to cause any issues. Also, it shouldn't be confused with the static IP's the router itself can hand out using it's own DHCP server, and not the one in the ISP.