Quote from: bytcoin on Today at 09:54:40 AM

Yes...but what I would really like to understand is why private key 3 needed to invert the y value and private keys 1 and 2 didn't need to invert the y value.

Why do there have y values that need to be inverted and other values that don't need to be inverted?

What math or method is used to know whether or not I will need to invert the y value?

If you're using standard EC math, there no need to invert anything.
The point on the elliptic curve is generator point G multiplied by the private key. The EC multiplication formula calculates both X and correct Y and does not require additional inversion.
Looks like you're trying to calculate only X and then trying to calculate Y from that X with simple equation, which leads to two possible solutions of that equation. And without using the multiplication formula, there is no way to know whether the right Y will be even or odd.