With a sidechain, you're not burning bitcoins. You're locking them.
You can make it do whatever you want as far as the "exchange rate".
But you can't really have proof-of-work, since you're unlocking coins by locking bitcoins. You could set up some kind of pseudo-PoW distribution, but it would require a centralized authority to re-distribute the coins. Unless your sidechain has its own PoW which is separate from burning or locking bitcoins.
If you want to burn bitcoins and give people your new coin, then you'll have to pre-mine the new coins and distribute them to people who burn bitcoins.
If many holders will embrace this algorithm then we will have many bitcoins out of circulation in the future ...Am I right on this ?
