Looking at just EC Public Key parameters, how can you tell if it is invalid? I am trying to handle when a parsers goes off the rails and reads an EC public keys wrong (just the X and Y components, I know the curve prior).
Right now I check for the following (false means invalid):

Is the key even on the Curve?
Is the Public Key X component less than the Curve's Base Point X?

I am pretty confident in #1, but not as confident in #2. #2 was just made by never seeing a case that contradicted it.
So my question is, 1) are these two checks correct? 2) are there other checks that can detect bad public keys?
The general rule for curves is given in;

2003 - Validation of Elliptic Curve Public Keys by Adrian Antipa,Daniel Brown, Alfred Menezes, and René StruikScott Vanstone
They defined a point is valid if

$P \neq \mathcal{O}$
The $x$ and $y$ coordinates of $P$, $x(P),y(P)$ are valid elements of the field.
$P$ satisfies the curve equation - against the twist attack
Check $[n]P = \mathcal{O}$ for prime curves ($h=1$) and check $[h]P \neq \mathcal{O}$ for non-prime curves ($h&gt;1$) where $h$ is the cofactor $h = \#E(k)/n$



if 1,2, and 3 are verified and $h=1$ (i.e. prime curve) then the 4th is already satisfied.