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 I'd like to be able to provably burn like 10 sats per note I post, and have my client only display notes that have done the same - problem solved I think? 
  stacker.news works like this.

Definitely reduces spam, but questionable if this were not to reduce posts at all.

Also could be bad for newcomers. Not really appealing that either spam or you have to pay.

But probably some relays can adopt this.

I think we have space for any type of relays. Everything will have its pros and cons, and you will be able to choose. 
 I agree it will add friction for newcomers - hopefully much of that would be overcome with better clients and bitcoin payments being more ubiquitous. 

My general thought on this is any method to prevent spam that doesn't involve some kind of real-world cost is going to a tit-fot-tat battle between relays/users and spammers.

I've not considered this in any depth, but the two possible imposable real-world costs I see that are permanent spam solutions are burning sats or processing power.

Maybe new users could use processing power to start, and established users could choose to burn sats instead for convenience.