Does the 256-bit integer secret exponent define the X, Y pair?
I usually use the following analogy to oversimplify things: The secret key is how far you walk along a known curve starting from a known point and the public point is where on the curve you wind up when you finish. If you repeat the same walk, you will always wind up at the same place. The operation is irreversible because the curve is complex, you can only figure out where you wind up by taking steps. Given a destination, you cannot figure out how to get there because you must walk in integer steps.