GP assumes that the Earth's mass is the Earth's mass (i.e. - an orbital around Earth). I am not aware of any affect the mass of the satellite has on its trajectory, so I'm not sure why you included it.
Which leaves us, in your analysis, three parameters. Vector of position, vector of velocity, and a time scalar. Let's call it a trajectory triplet. This results in 7D trajectory space. Those three are not, however, orthogonal (or even linearly independent).
Just as an isolated example, take a certain satellite triplet. Then take that same satellite's triplet a few seconds later. None of the values of the triplet are the same, and yet it obviously describes the same trajectory.
I am not an astrophysics, so I will not claim absolute knowledge in this field. My limited understanding suggests that all trajectories pass around the equator. Furthermore, for a satellite doing a perfect circle, the speed (scalar) is a direct function of its height. We can, therefor, narrow down the trajectory parameters to:
Height when over the equator
degree of elevation above said height
degree of descent below said height
angle crossing the equator
two phase scalars (one of accounting where above the equator we are talking about, and the other for accounting the possibility of two satellites following each other in the same trajectory).
That's 6 scalars (as opposed to your 2 vectors and two scalars). As far as I can tell (but see disclaimer above), those six are orthogonal. I am not 100% sure the two phases are, indeed, orthogonal, but I am fairly sure you can arbitrarily change any one (or more) of the others and still get a valid and different trajectory.
Still not one dimensional (not sure where that came from), but at least one dimension less than you claimed it was.