“Not your key, not your coins” cuts both ways.
Until someone derives the private key, the puzzle coins are ownerless data on-chain, yes.
"Derives the private key"?
That's basically saying "I never owned this key".
None of these arguments will ever hold in any court on this planet, simply because it is impossible to ever prove that you were the first to "derive the key", craft a TX, and having it reach a P2P node, before someone else. That shit is dealt with only after a block is mined and issue gets settled, not before.
It's one thing to hack someone's rightful assets, and a totally different thing to have 100 dudes competing over who ECDLPs first a minuscule weak key, in a mempool, over some assets that don't really belong to ANY of them until
the block gets mined and the conflict is solved.
Ownership, in a legal manner, involves proving that the private key was OWNED by you, not that it was "derived". This is done by showing that the private key was indeed a full entropy 256-bits random blob, not a lame-ass zero-filled empty blob with a few bytes at the end.
At best, it can only incriminate yourself, since "deriving" pretty much means "I cracked it, because I didn't own it".
You’re mixing up “can’t be proved” with “I don’t know how to prove it.”

Every Bitcoin Core node stores when it first saw a tx (getrawmempool true). Mempool.observer, forkmonitor.info, etc. archive those feeds. If my tx hit the network at 17:12:03 and yours appears at 17:12:27, the gap is public and cryptographically tied to the txid.
Deterministic ECDSA RFC 6979 forces each signer to use a different nonce r. Two people who really derived the key produce two distinct signatures; a copy-paste front-runner re-broadcasts the identical sig. Comparing scriptSig bytes is enough to show who computed and who plagiarised.
I can sign an arbitrary message (“Block 850 000, puzzle X, I own this key”) and timestamp that hash with OpenTimestamps or even into the blockchain days before broadcasting the spend. When the puzzle falls, I reveal the signature; the hash already on-chain nails the timeline.
Courts deal with timestamped digital evidence every day e-mails, server logs, CCTV metadata so spare us the “no court on the planet” flourish. In practice the front-runner’s best hope is anonymity, not legal theory, because the maths makes the order of discovery trivially auditable.
Since you're acting in bad faith, I certainly wouldn't want to be your friend IRL, you might steal things from me...
