Ok. I think I'm starting to piece this together. Kinda like pinging an IP address over and over until you eventually get a response.  I think it's starting to become clearer for me. Thanks a lot.

Public_N_M_E

bitcoin

Lightning Everywhere

TonyGiorgio

The offline payment solution is essentially just having LSPs on both side of the transaction (one on the sending side, one on the receiving side) and routing the payment from the sender to the sender's LSP. Then the LSPs will talk to each other to determine whenever the receiver comes back online. It's really just the illusion of offline, but payments can be "held" a lot longer without it hurting the network.