The visibility of the transactions in the blockchain enables anyone and everyone to confirm the validity of the ledger. Cryptonote does not have this visibility, therefore the validity of the transactions cannot be confirmed, which means there is no independent way of verifying that the implementation or the protocol have not been broken.
It could be broken, we dont know and we have no way of showing that it is not being exploited right now.
That's not true. You can verify Monero's coinbase as you do in Bitcoin.
Yes, but not the rest of the chain unfortunately.
Not sure what you're talking about, even when CT is implemented, you can still validate the coin total.
"[–]fluffyponyzaXMR Core Team 3 points 2 months ago
So, having only the blockchain data, is it possible to mathematically prove that all blocks inside followed the rules?
Yes - read gmaxwell's write-up on CT and you'll see that amounts can still be verified.
To illustrate it as simply as I can: imagine if every transaction input was 1000 XMR. But, using ring signatures, you mix your real input of 55 XMR with a bunch of other ones that adds up to 1000 XMR. Analysing the blockchain we can still verify that it adds up to 1000 XMR, but we can't tell which value is yours."