SQUARES UNDER XOR · FROM THE KIT
Every tile in a lock is a perfect square scrambled by one hidden mask; flip the mask's bits until every tile is a square and the door opens. Below each door a square nests inside a square: take its root, fold it, take the root again, down to the ground.
XOR by a constant is its own inverse, so a scrambled tile k² ^ m gives back k² the moment you XOR the right mask in; and only one mask does it for every tile at once, because a square is 0 or 1 mod 4 and rigid in its low bits. That rigidity is the dead zone: no mask touching bit 1 can ever subtract from a square.
Below the door, u = r² ^ 5 has u ^ 5 = r², a square every time, so a square can be nested to any depth by construction, and peeled by taking roots: the root of the top, folded by 5, is the next square down, until the fold stops being a square at a ground seed, 0 or 2. The locks are dealt by the indexed generator, so the same name opens the same vault on every machine.
The Square Vault · by Lars Rocha · from sitoolz: folds and indexrng.