So the question must be whether or not that kick can stustain constant numerical value. According to GR the answer is no (...) So the accelerating inertial frame of reference cannot sustain a constant acceleration. This is indesputable high school math surely you cannot possibly disagree with me.
I think our disagreement centres on our definition of time. To an observer in an external, inertial frame, you are absolutely right in that the observed increase in velocity for a given instantaneous force will seem to decrease since otherwise the FE, rocket, whatever, would continue blasting linearly faster all the way through
c, which is impossible. However, to someone
in the accelerating frame they would measure that force (with their own Newton-meter, accelerometer, or whatever) to be the same as it was when they started. If that force was initially measured to be 1g, then there is no reason that the
accelerating observer cannot continue to measure that force to be the same forever (since their perception of time becomes ever more warped compared to that of the external observer, to the point where they experience almost no passage of subjective time).
I appreciate your precision, so I will try to do the same. Remember that to calculate (or measure) velocity (m/s) or acceleration (m/s
2) you need to define what a second is. To take the example you gave of your straight-line function, it is only a straight line if the axes of the graph you plot it on are also linear - it's like plotting y = mx on a logarithmic scale (except in the case of relativity the scale stops at c, with the lines becoming infinitely finely spaced as you approach it) - you see an apparent curve
if and only if you're an inertial observer making measurements of someone else accelerating. As I was saying, it would have to be the whole observable Universe accelerating in one direction uniformally for us not to see the effects of this elsewhere (aberration of light from distant sources being the most obvious effect), so I find this an unpalatable solution which should produce a plethora of evidence against it (and I believe it does).
Does that make sense or do we still not see eye-to-eye on this?