Public land is just that. Public. That is OWNED BY THE PUBLIC, not the government. Provided he broke no laws in digging the hypothetical box I say it's his to do with as he pleases.
I'm finished debating you. We will have to agree to disagree. Good luck digging your septic tanks and cisterns. As you follow other leads I hope you strike it rich one day. Since you're quite the upstanding, law abiding citizen, I am certain you will be able to post about it here. Or write a book about it. That would be cool.
Oh, and please see my post above yours. If he was trespassing all bets are off.
I am not sure what you mean by Public land. Is a public beach an example? It is owned by some sort of entity as all land is. We have a barrier island reachable by bridge. You can walk on and detect on the beach along the entire length of it. There are places along it that are part of a public beach. None of it is a state park but some of it is county parks. There are places where the beach has eroded and the private property owners lot lines extend into the Gulf of Mexico. There are three different cities within this island that own parts of the beach. It can get kind of confusing at times.
The OP (original poster) asked about a chest of coins on a beach in Florida. I know of someone that found a large clump of coins from a chest on a beach and turned them in and they were never seen again. If you found a chest on a beach and were to ask and keep asking anybody that you can think of I am sure that someone will decide that you can't have it. If it is in a location and depth that you can't dig without everybody knowing about it you better have a plan. Was this chest found by accident or did something lead you to that exact spot and make you think a chest was there? In other words was it a premeditated find? I am not trying to be difficult on here but the extenuating circumstances always have an effect on the outcome of anything that involves the law of the land.