Hi Oleg,
Interesting work! Personally I like this general class of ideas. If we think about the simplest form of the idea: to enter the sidechain, you burn bitcoin, I believe its first proponent is Ruben Somsen with 'spacechains' [1]. I think you should compare your design with that? At the time he proposed it I was less positive; the obvious question/problem is that without the reverse path the 'coin' can't truly be called bitcoin, it's a different thing. And I also thought, it's hard to convince people a one way transfer is worth it (yeah, that's really vague; who knows; it's more just intuition, less fact). More recently, I put together a PoC of what I called "hodlchain" [2] which, apart from fancy theorizing about time value of money aspects, is basically just 'spacechains but with CLTV/fidelity bond style time locks instead of burn'. I liked this version because I saw it as an unlock of dormant value: the hodler gets rewarded for the positive externality of their behavior without having to change it.
With regard to your minting and supply as per your Section 3: yes I wrestled with that a bit in 'hodlchain' too. The reason I didn't like your minting schedule: fixed per epoch minting --- is that it doesn't feel like the fairest option for the ordinary user: they're forced into an auction/market that they have to make a judgement about whether they're going to get good value or not. But in holdchain I had another 'TVOM' variable to play with (r) due to using locking. With burn, I guess your choice must be right: fixed minting lets the other side (burn) vary according to market's assessed value of flame; if you had a fixed rate of flame:btc in minting then if the market doesn't like it, the system doesn't work (your consensus relies on the minting).
Also, it's unfair of me to compare: you're actually trying to build *consensus* from the burns; I wasn't bothering with that. That makes your design intuitively seem like it must be the correct one: proof of sacrifice of value ~= proof of work. I guess it's 1-layer-down PoW in the sense of, it has PoW security assuming bitcoin's value and consistency, so, assuming Bitcoin's PoW security.
I won't comment on the detailed design of the flame-sidechain itself. It looks pretty interesting :) I think this 'it's not bitcoin' thing is the main thing that puts people off, though a counterpoint is, systems can get quite some interest if they have a speculative element from mining for new/minted coins. The problem with that angle is there are lots of other places to put your bitcoin into another system, speculatively, with all kinds of new features. And it's not hard to get your bitcoin back when you want to via swaps.
Cheers,
AdamISZ/waxwing
[2] https://github.com/AdamISZ/hodlchain