# How to determine if a point is just a point or a valid public key?

In ECC, specifically over finite fields, in my mind there must be other points that exist that still yield $$y^2 \bmod p=x^3 + ax + b \bmod p$$ to be true but are never used because the Generator Point (or base point) never "lands" on that point before reaching the order and effectively starting over. How can we calculate if a point is actually part of the order (not sure if thats the correct term) and not just a point that satisfies the equation?

in my mind there must be other points that exist that still yield $$y^2 \bmod p=x^3 + ax + b \bmod p$$ to be true but are never used
Actually, that's not true if the order of the curve is prime; examples of such curves are P256 and Sec256k1. In those curves, every single point can be expressed as $$xG$$ for some integer $$x$$.
Now, this is (usually) not true for curves with a cofactor > 1; in those curves, we generally work in a prime-sized subgroup; there will be points that are 'missed'. To determine whether a point $$H$$ we've been handed is such a point, one way (which works with the curves we use in cryptography) would be to compute $$qH$$ (where $$q$$ is the size of the prime subgroup) - if that is not the neutral element, then $$H$$ cannot be generated by the generator.