Dear Colin,
This is an interesting and important topic. It reminds me of two "somewhat related" ideas I read about recently:
1. The first is from a post-doc of my acquaintance, Se-Jong Kim, who wrote some papers on velocity addition in relativity that relates this to quantum information theory via a group-like algebraic structure called a "loop."
2. The second appears in papers by Piret Kuusk about "geodesic multiplication" in general relativity and possible connections to quantum gravity. The algebraic structure involved is called the "local geodesic loop" and is similar to the structure of the velocity-addition problem.
From reading your paper, I know that neither of these topics is exactly what you are discussing, but there may be interesting analogies. I don't see an email address on your paper, but if you email me, I'd be glad to send you copies of these papers. My email address is on my essay here). Take care,
Ben