The path of the plane is a static curve in 4-dimensional space, yes.
But the plane is not located at any point in the 4-dimensional space, and the position in 4-dimensional space that it doesn't have is not changing over time. Both of those things are required before you can describe the plane as "moving" within the space.
There is no secret backup time that will allow you to track the plane's hypothetical motion along an explicit time dimension. That's not a thing.
I confess. I literally had no idea what you were intending to convey with those two sentences so I restated alternatively what I intended to convey in the hope it might make clear my position (if that was an issue for you) or that I might learn more from your response.
> Where do you think you're contradicting me?
That's not a thought that I thunk.
Therefore I have no response.
> But the plane is not located at any point in the 4-dimensional space
Every point along the 4D path trajectory of the plane in {X,Y,Z,T} is a literal {X,Y,Z} location of that plane at time T.
> Both of those things are required before you can describe the plane as "moving" within the space.
I certainly did not describe the plane as "moving" within R^4.
> There is no secret backup time that will allow you to track the plane's hypothetical motion along an explicit time dimension.
I utterly fail to understand what you intend to convey here.
Although I note that the actual (not hypothetical) velocity of the plane projected onto the time axis is very likely to be on the order of approximately one second per second.
The path of the plane is a static curve in 4-dimensional space, yes.
But the plane is not located at any point in the 4-dimensional space, and the position in 4-dimensional space that it doesn't have is not changing over time. Both of those things are required before you can describe the plane as "moving" within the space.
There is no secret backup time that will allow you to track the plane's hypothetical motion along an explicit time dimension. That's not a thing.