Dear Tim:
This is a fantastic essay. And it is very well written.
As we all know, mathematics has been very effective in physics. Its weaknesses to date in modelling physical reality have been twofold:
1. Using open set theory, it does not model that time has an order (whether we interpret this as an order in forward time or an order in backward time);
2. It does not model which directed order (forward order or backward order) corresponds to the observed Arrow of Time.
Your Theory of Linear Structures addresses point 1 - and is therefore important. However, I do not believe it can address the second point. In particular, the initial end point of a line could represent a past instant of time and the final end point of the line could represent a future instant of time - but equally the initial end point of a line could represent a future instant of time and the final end point of the line could represent a past instant of time. The mathematics cannot differentiate between these two interpretations. So, the Theory of Linear Structures, while it can encode an order, will not I believe be able to encode which directed order of a line corresponds to the observed Arrow of Time. Instead, we would have to impose the direction of time from outside the mathematics on the solution - as we do now for open set theory.
What this would mean is that the Theory of Linear Structures - although important - will produce time-symmetric theories, as open set theory does now. We will unfortunately not be able to use it, for example, to have a mathematical theory of evolution. Nevertheless, your work is very good.
If you are interested, in my essay I explain more generally why mathematics cannot, in principle, model the Arrow of Time.
Thank you again for some great research and a clear essay.
Kind regards
Spencer Scoular