How does one prove that they sent bitcoins to an unspendable wallet? Proof of burn? How does one prove that they sent some BTC to an address that no one has the private keys for? Is there an address that one can send to? Basically proof of burn.
Essentially you must send coins to an address that is provably unspendable. There are many ways to do this.


Put the coins in an OP_RETURN output. These outputs are provably unspendable because the consensus rules prevent it.
Send coins to any script which resolves to false.
Since addresses are encoded hashes, start with a hash, of which it would be nearly impossible to find the pre-image, e.g. assume a script hash of 0000000000000000000000000000000000000000. Encode that into a valid address:


bs58 -c 050000000000000000000000000000000000000000
31h1vYVSYuKP6AhS86fbRdMw9XHieotbST


This address has actually had Bitcoin sent to it, but never from it: https://blockstream.info/address/31h1vYVSYuKP6AhS86fbRdMw9XHieotbST