If you deposited and withdrew BTC from Mintpal, those transactions are permanently recorded on the blockchain. It doesn't matter what transactions took place on Mintpal's internal database.
You sent BTC from Address A (which you can prove control of). You received a higher amount of BTC at Address B (which you can prove control of). The difference is your BTC profit, but what matters for taxes is the USD profit. Each trade on Mintpal is taxable, and the net of those gains determines your taxes due.
The fact that Mintpal no longer exists just makes it harder to disprove the records you provide for your Mintpal trades. That's why any good faith effort to pay taxes on the gains made between Address A and Address B is probably good enough. You made 4 BTC profit, which had a USD equivalent when you bought back BTC. That's your taxable gains.
When you cash out an amount of any relevancy, you must prove the origin. There is no real way to prove that the addresses where you deposited and withdrew from, are from Mintpal or from a drug dealer or anything else, because Mintpal doesn't exist, so there's no way to link them properly into the exchange, unless you can prove me otherwise, because all we got is some addresses on the blockchain now, without the context of the website. I don't know about the US, but in other countries you MUST prove the origin is not ilicit, and if they don't like the evidence, you risk losing your money because they would confiscate it or tax it to ridiculous levels.
So again, you send 1 BTC to Mintpal.
You make 4 BTC from trading it against some altcoin.
You send back your BTC to your wallet.
You end up with 5 BTC.
Now prove these addresses belong to Mintpal. And no, you can't prove ownership of the deposit address, because you don't control the private keys of a deposit address of an exchange, so you can't sign them.
But you
can show the origin. It's right there on the blockchain. Even if Mintpal is gone, why can't you claim the coins are from Mintpal? Why can't you claim trading records based on that? Tax authorities are interested in taxable income. You're paying that, yes? By paying taxes on cryptocurrency trading income from a defunct non-KYC exchange, you are going far beyond what most people are doing.
You may not be able to prove that the addresses belong to Mintpal, but if they truly did, I wouldn't worry about it. It can be easily determined by blockchain analysis companies (or possibly even Walletexplorer) that they did. You can sign a message from the address that you sent 1 BTC from. You can sign an address that held 5 BTC withdrawn from them.
How can they definitively link addresses together? Personally, I am extremely careful about linking wallet addresses together, even if it means paying higher fees to consolidate outputs.
Well, if you are going to sell years worth of signature campaign earnings, they will ask where it came from and they will see the address is linked to your account where you posted at with a simple google search, then they could browse your posting history and track all of your receiving addresses for the signature campaigns.
That's why you should probably report income that is easily trackable. This makes it very important to avoid linking wallet addresses together. You should consider how you spend UTXOs and how that links your BTC holdings together on the public ledger.