I can only go by the latest estimates of real world performance differences that I could find.  If those are inaccurate, then I will leave it to you to compose some letters to those fine people among the infamous "they" who are so grievously in error and correct them. 

It looks like no one's gonna take on OP's bet, haha ... XD

Their next console could indeed be weaker than the PS4 ...

Small form factors and low power consumption aren't exactly what I'd call recipes for a high performance platform. There's a reason why PC gamer's purchase 800+ watt PSUs ...

Given other external factors such as low transistor price scaling in the near future the probability that the WII U would be weaker than the PS4 ranges from 65% - 85% ...

They're working with western console designers this time around. I have a feeling we will see Snes-64 power hungry Nintendo next gen. They can't afford to do otherwise. Still, power is the last thing a real game designer looks at. Keep that in mind. It's always what you do with it and if it matters what you've done with it. Yoda moment.... With wii u they crafted a console with slightly better last gen graphics but with a 180$ controller. Assuming that they will drop the pad, the question becomes what will they do with it? Now that they dont have to use a overpriced controller, whatever technology they decide in will most likely be nothing less than the best.

It's really up to timing. If the Console comes out in the next 2 years then It might be stronger unless nintendo tries the cheap rout and opts for a really cheap GPU. But I can't see nintendo going with something Weaker then the Jaguar. Since AMD has come out and said they are making Pumas for a 'consumer' the CPU side will atleast be stronger. With DDR4 out Nintendo could go that rout but I doubt they will put 8GB of DDr4 in one box. Maybe around 4 or so. 

My real conclusion is that It could possible be an x86 APU Puma. Probably with something like 2 to 4gb of DDr4 and a CPU with around as many shaders and such as a PS4. Faster CPU, Faster Ram, and On Par GPU would make Nintendos Next console ahead of the PS4 but only slightly. 

OT: Depends of when they plan to release their next console

If they release in 2016 i can see you being right cause PS4 will probably be costing something like $300 until there and there is no way Nintendo would make a 300+ machine.

Nintendo wants the Wii crowd and for that, they need a mass market price.

Their next console will be significantly more powerful than the PS4

The next console will be less powerful than the PS4 because

1.) It will be using the same components as the next handheld, just souped up. So it'll be able to probably run games at 1080p, but they'll still look similar to Wii U games (much like the GameCube > Wii situation).

2.) I don't think Nintendo wants graphics any better than the Wii U. The games are already expensive enough to make that you can bankrupt a studio with even PS3/360 level visuals. Nintendo games look great on the Wii U as is, I think they're quite content with that level of visuals and don't have much interest in making a COD or Destiny or Gran Turismo or NBA 2K killer where graphics and realism matter. For Mario Kart? Something even just double the Wii U with more RAM would give you a gorgeous looking game.

What? *looks below*

Oh.

If you win the bet, nintendo is in for an extremely rough next-gen.
Theyre too late for the jump to this processing level, just as they were with the wiiu getting to roughly ps3/360 level, only difference is, the jump from wiiu to roughly-ps4 level is much less than 360-xbo or ps3-ps4, and significantly less than wii-wiiu.

If I were you, i'd hope Nintendo held out till microsoft and sony released theirs, and actually put something out around the same performance level, preferably with the same architecture too.

Lol, not anymore they don't.

When the Wii U was announced they promoted it as a machine where you can get all the games, including the "hardcore" games.