One of the problems which has bothered me in the last couple of months is the fact that in a locally finite geometry G, it is improbable to get a notion of geodesics which satisfies the properties: 1. any two points can be joined by a geodesics, 2. there is a global dynamics with local initial conditions. The reason is: since the geometry is locally finite, there are only a fixed amount M of initial conditions. If the number of points in G exceeds M, then there is no geodesics starting from x reaching y unless these finite paths reach every point which is a strong condition. As I was tiling our bathroom last week with hexagonal tiles (the dual graph to a nice flat triangulation of space), why not take the later as an example? Given a triangle in the flat triangulated plane, there are 6 =3! geodesic directions, in which a geodesic can go (using the geodesic dynamical system I have talked about of course, as for shortest paths, there is no local rule giving you the path). You see that in the picture to the right. Obviously, we need to take broken geodesics to connect any two points. To the right there is a picture of our bathroom, which got new tiles (as well as new sink and toilet and before that a new floor). The tiles are the dual to a triangulation of the plane. The 6 geodesics as we defined them go out from every node in this regular graph (every node is a triangle in the triangulation). Note that the dual graph of a regular q-manifold given as a simplicial complex is always (q+1) regular while the graph from the manifold is not regular in general. But you see the problem: if we want a Hopf-Rynov theorem, we can not do that in the finite if we want to have the geodesic given by a deterministic dynamical system and preserve the locally finite nature of space as well as the finitness overall . We could go to the quantum world (using infinity) and look at waves which can be given as expectations of random walks but then we are outside finitism. Any artificial introduction of randomness should be a last resort. (Maybe the architect of our world had to go eventually that way and create things in a quantum fashion just to solve basic connectivity problems. … or piping problems as I would call this now after a few days bathroom renovations).
[By the way, check out my plumber stories ( https://www.youtube.com/watch?v=kY9rToZ2mOs ), from the spring 2019. Stand up is one of the hardest art-forms there are: I solved that task by writing and rewriting the stories (also with the help of members of the Harvard College Stand up Comic Society) and then learning them by heart, also rehearsing with some people. A good stand-up comedian of course can do much impromptu, but that needs much more skills and experience. But I think that it can -like math – be learned. I saw once a documentary about Joan Rivers which gave some insight how she worked, she had a big library of jokes, written down on index cards and learned them. If you have a million jokes, then you can apply them to any situation. But you have to know them by heart. I just wrote again about the insane attitude of math educators who promote that memory is not important. It is all about memory as memory also contains strategies, algorithms, methods. I have sine years claimed that the failure of math education in K-12 in the west is mostly attributed to that idiotic mantra “you shal not learn by heart!” No musician, no climber, no plumber, no commedian, no actor could survive in the profession without a well oiled memory and knowing lots of terminology, procedures, actions, decision making skills. I like to watch a lot of climbing movies. There are many about specific problems like “action directe” which takes also professional climbers several days which includes rehearsing every position of foot or hand or finger, every move has to be known by heart in order to solve the problem. When we were teenager, the German climber Wolfgang Guellich had been our hero and he was the first to solve that problem first. It is somehow like being the first proving discovering a result and then proving the result. By the way, here is one of the latest movies on youtube about this particular boulder problem. ]
So, how do we solve the discrete Hopf-Rynov conundrum without going quantum or abandoning (=betraying) combinatorics? One of the strategies is to use more particles. Instead of introducing randomness ,we can imagine finitely many geodesics moving around. Now, we can, without making space infinite or introduce an infinite concept like a wave (complex valued functions), we can look at this particle gas. Everything is deterministic still. But now, it depends when we start the geodesics. If we release the geodesic now at x we might not reach a particular point y but we might reach it by releasing it a bit later. Determinsm still gives a periodic particle motion of the particle gas, but with many particles, the return times are astronomically large. As space gets bigger, the chance of being able to reach y from x over a time interval is getting larger and larger. It is reasonable to believe that if space has volume V, then we need a time interval of the order log(V) to be able to reach from x any point y simply because making the time interval one larger makes the number of possible geodesics by a constant larger. If we now (as suggested in the movie) think about so many particles that space itself can be considered the particle gas, then the deterministic evolution of the particle gas produces fluctuations of space which renders it possible to reach from x to y just by waiting the right moment to get transported there.
The motion in a complicated deterministic gas is essentially random and if we can get random particle motions, we get expected solutions which look like produced from an “ex machina” random gini. Introducing randomness into models should always be a last resort. It is surrendering to the fact that things have come too complex. One does this all the time in stochastic models. Nobody ever tells you however where the “randomness” comes from. When using a stochastic model, we just admit that we are too dumb to solve the actual problem. So, instead of solving a perfectly deterministic system like a Vlasov dynamical system, one neglect the momentum components of the particles and only look at the particle density which then produces the Boltzman equations, which then of course as we have less information increases entropy. It is a natural thing to do to go from the microcanonical description (which is impractible) to a macrocanonical one where one can work easier. This is always done in statistics. We rarely really know the probability space 
So, once we decide not to go cheap (and use some randomness or quantum evolution) we have to decide how particles moving on geodesics interact. It would be silly to try to simulate some sort of force we know from the continuum. Fundamentally, the forces coming from potentials of Green-functions of Laplacians are all classical meaning that they are non-local. Truly relativistic interactions are very, very difficult. All of GR is essentially looking at a1-body problem using a Schwartzschild type model. All experimental verifications of GR are based on that. Two body problems like BH mergers are models which are a nightmare. They do not solve the true underlying problem but assume that one particle moves in the Schwarzschild metric of the other or that the mass distribution is pre-guessed using classical relativistic intuition. What happens in reality is not so clear. Imagine the two body moving. They produce the energy-momentum tensor T appearing in the Einstein equations. Assume we can solve the Einstein equations, we get a metric in which the particles move on geodesics. If we are lucky, then these solutions are the same than what one has started with. It is a fixed point problem: start with the paths 
So, it is really an interesting question to find a discrete toy model in which particles moving on geodesics interact. There are various things we want to have satisfied:
- Non-interacting particles should move on geodesics
- All particles interact with each other but locally only
- There is a global deterministic and reversible dynamics
- The time evolution can be coxeterized: T=BA with involutions A,B
- The number of particles should be preserved.
- Space itself can be seen as a particle configuration.
- The motion can be described as a motion on divisors.
- The order in which several particles interact at a point does not matter
- The interaction should only involved the orientation of the simplices
Both the requirement to write things as divisors as well as the ability to write the dynamics as a product of two involutions point to wards signed particles. Also the definition of the geodesics hints at that. We used the left rotation but of course there is also right rotation. What I like about the Coxeter description is that the dihedral action is a natural action and that the inverse does not invoke any group extension. The inverse of T=BA is AB.
In the movie, I talk about a possible interaction which satisfies these properties. Whether it does is any good is not clear. I will show some experiments soon. The dynamics has been implemented already in the morning of the talk but no pictures were yet made.
