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What is incompatible between these two features
Nothing.
and why can they not both be implemented?
They can. You just need to get a developer to code it up, and then get an overwhelming majority of users to agree to use your code.
Why not BU, Segwit, sidechains and lightning network?
Because there isn't any support behind it yet. If anyone has coded that up yet, then I'm not aware of it.
Is it just a power play thing?
That's one way to describe it. Another would be that it is a political battle.
Core refuses to add Unlimited to their code because some Core developers feel that Unlimited is a bad idea.
Unlimited refuses to add SegWit to their code because some Unlimited developers feel that SegWit is a bad idea.
Nobody else has combined the two and convinced enough people to use their code yet.
Or are there technical reasons?
No.
Also, please do not attack me if you see this as an attack on "your side". I know this has become a religious discussion. Please keep it technical.
You'd have been better off making this thread "self-moderated" so you could delete posts from those that fail to respect this request.