Well, Quad core meaning 4 cores on one CPU instead of 4 CPUs is being pedantic. XD For all intents and purposes it's like having 4 CPUs. More efficient to have 4 cores on one CPU with hyperthreading and all that, but, yeah.
If GFX doesn't mean much, they should make it mean something. Quite a lot of untapped power they could use there [but probably they mean because the GPU in the gamecube is pretty low end].
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Nah, it doesn't work like that at all. Kaillera isn't open source, but there is no difference on Kaillera between any emulator. Kaillera is just a front end for the socket crap. The way netplay works is by sending controller button inputs, that's it. The way a game is synced is by how the game recieves those packets. You can hotswap the Client for another one, no difference really.
The way to fix the Dolphin netplay is to make it forefully lag when it's behind. See, what happens now is if one game isn't steady 60FPS, and the other is, it'll desynch because it's not getting the 60FPS to lag. In SSB64, any lag on one side turns into lag on the other. Probably the best way to acheive this is to put the packet reading stuff at the same place as the interrup that deals with FPS or whatever. If it doesn't get the signal, it'll stall the game until it does.
Button delay is precalculated before the game starts, you can see it if you use a GOOD Kaillera Client [me and MattNF changed the client, same DS because the "glue" is all the same, but it gives more information like the Delay]. So, if you have a giant spike during loading and it's 12 frames of delay, you need to reload the game to get it to more reasonable button delay [smoothing adds button delay].
That's more or less why some people may choose to add button delay. Because if the emulator does the netplay right, it'll lag if a signal is taking too long.