There are not mathematically is just Random. I try many theory mathematics and found some rule of puzzle. There are never complete.
If you like lottery of bitcoin. Try your luck.
Of course they are mathematical, even the keys with no exposed public keys require mathematical shortcuts to speed up the brute force, and the ones with known public keys, it is as simple as + and - , but the order of parameters and the equations makes it a bit hard.
Don't forget, everything works according to a sets of mathematical algorithms, universe functions based on equations and complicated algos.
Imagine when there is a possibility of 2 distanced sub particles interacting simultaneously with one another, a tiny elliptic curve is just a cake comparing.
Moreover, since everything is based on math, we could use the math of other fields such as astronomy, orbital trajectory, and orbital slingshot calculations are applicable here, if you knew how satellites/ spaceships use gravity to boost their speed without using any energy, you'd know that slingshot equation is the best if you tweak it a bit in order to solve the known keys.😉
I know all build with math. Hard to find the formula of puzzle but the private key is really exist.
Nice explain. Thanks.