I think there are 2 problems with this approach:
How do you prevent the double spend problem, for example duplicate entry tickets. You would have to mark the ticket as used in a central database anyway to prevent it
What happens if the secret key material is compromised? Anyone can issue new valid numbers, etc..
Please correct me if I'm wrong.
1/ You don't have duplicates because a Feistel network assures you there is no duplicates: every input has a different output with a fixed key in every round.
2/ If the secret is compromised anyone can issue valid numbers in the same way that if your secret encryption keys for encrypting your data exposes it. The idea is that the secret key material is never compromised as it is assumed in all security cases. You should custody with the right measures based on the attack vectors you have. The custody of secrets is independent of this method.
Please let me know if you have more questions.
That's not true, we have (perfect) forward secrecy, backwards secrecy and key rotation mechanisms because we often care what happened after the key is inevitably compromised. In this case the problem makes it hard to "rotate" the keys in a meaningful way, but I'm yet to see a proof it's impossible.
The assertion you mentioned is the "what" assumption while what you said is the "how" we came to that assertion. Most probably you take my word idea in a specific sense other than I wanted to use. My answer was informal because we are here in HN and not writing a paper.