Scale relativity and fractal space-time may give you the answer you are looking for. Basically distances and the smoothness of space depends upon what scale you are looking at, specifically in a fractal-like space.

https://arxiv.org/abs/0812.3857

If I understand clearly, this is about how to take a manifold and make it discrete such that Lorentz invariance is conserved.

This is not what I was thinking though. Two problems:

• Time and space are fundamentally separated in the Wolfram model, and Lorentz invariance is guaranteed in "most" graphs automatically. The referenced text is interesting but maybe not that relevant.

• If I understand it correctly, the idea to make a manifold discrete is to pick points on a manifold discontinuously (an intuitive approach indeed). The metric between these points is obtained by the metric from the manifold. This would, for example, be equivalent to me requiring the diagonal distances in the grid to be $\sqrt{2}$.

Especially the second problem was one of the main points of my post. Even when you can embed a graph, you simply cannot generally choose a metric like that. There is no way to know which manifold your graph would "fit into" (and they are certainly not unique).

The entire idea is to build a graph from "bottom up" so to say, rather than already having a manifold and build a graph onto it. Also it should be mentioned (not only for this comment, but some of the others too), that even if you have a set of points you don't have a graph. You still need to define the edges as well.

Having a graph with a manifold-induced metric is also kinda weird, since graphs should be thought of as abstract concepts which can (if desired) be drawn/embedded in an infinite different possible ways. Having a graph on a manifold "requires" you to embed it on that manifold in some sense.

This paper might be of relevance: https://arxiv.org/pdf/gr-qc/0311055.pdf

How do we create a grid which obeys the Pythagorean Theorem on large scales?

That's also called Euclidean distance

Can we create a large scale isotropic graph?

Both of these questions were resolved in this recent post of mine: https://community.wolfram.com/groups/-/m/t/1945186

I'm working on something similar, actually, I have some problem with the floating-point approximation in the really particular rule I'm using. Theoretically, the rule will converge for a portion of the euclidean space, but that isn't actually happening for approximation errors, so I'm rewriting all with a geometrical substitution rule (a little more complex to implement)

The method I'm using (the circle mapping of the euclidean space) is intended to preserve some fundamental rotational symmetry. For the distance, I'm using the number of jumps, even if in the paper the part about Minkowskian distance is REALLY incomplete and inaccurate about the mapping of the coordinates on the spatial, causal and brachial space.

However, these (very beautiful) methods are not based on replacement rules. What I was experimenting with was a particular mapping of the Euclidean space through a specific substitution rule (which I am still studying) to preserve all (too much) of the Euclidean properties, even at an intermediate scale. For experimental working, I'm using a lot of rotation. Theoretically, with the right bounding condition the algorithm will need to converge, but the floating-point approximation breaks it all.

Furthermore, your methods are for the spacetime causal graph (if I understand decently, but I still have many doubts about the various mappings of the Wolfram model), while here we were trying to build a purely Euclidean space, therefore working on the space hypergraph.

I would also like to add that from the 4D Minkowsian method it is not evident, exactly as in the paper, how the mapping of the spacetime points on the whole lattice on which the Minkowskian distance is then defined, leaving room for an arbitrariness that leads to the complete breakage of any metric model based directly on a norm dependent on the coordinates and not on the topological characteristics of the graph.

In the end, you can't have a metric if you don't first specify how to interpret the topology, including the coordinates of the space.

But I may have understood cabbage for potatoes.

16-cell honeycomb that provides you with coordinates

How?

If there is an arbitrariness to obtain the coordinates, and it is not possible to obtain it directly from the topology, it is a huge problem.

I believe that the use of lattice coordinates to define the Minkowskian norm is a huge conceptual error because it requires as a fact present here on page 13, that there is a valid graph embedding on a Euclidean space, and does not clearly demonstrate that it is guaranteed a priori.

The solution can come free from the removal of the concept of microscopic coordinate and be based entirely on the topology and the metric of the graph, defining the metric using the typical distance function of a graph, i.e. the minimum jump number to connect two points. In this way there is a very important criterion for mapping coordinates, that is that they must respect the metric just introduced. In this way, an effective, various and universal coordinate system is obtained, because there is the certainty that it preserves the most important characteristic of the graph, it's metric, to which everything is conceptually traced.

Questions for both your approaches (Leuenberger and Pasqua):

Can you specify more clearly what edges and vertices your eventual graphs consist of? When you intertwine two grids, Leuenberger, what edges do you create? Or, Pasqua, what are the edges and vertices of your graph? I assume the vertices are the crossings, while the edges are the lines "doing the crossing"? It seems to me to become extremely complicated to determine distances between points, so can you elaborate on that?

And mayby I am simply a bit slow, but how do you obtain Euclidean space in the limit? Remember that you cannot simply embed a graph in the plane and expect to "see" the correct distances.

I assume we all use the basic graph metric, otherwise, can you specify what other metric you use and justify why?

I think I've made a graph capable of approximate on large scale the euclidean space. The method to make it is literally "cheating" because is produced iterating a simple algorithm on the euclidean 2D space to produce a suitable mapping of the space dependant by a single parameter (an angle that is obtainable dividing by an integer $2\pi$, so in some sense a single positive integer) In the image (I apologize for GeoGebra, but I used the first thing I had available on the computer), I've used an angle of $15°$.

Here you can find the project (I apologize again for having done the constructions manually, I'm thinking to write a real program that maybe compute some macroscopic distance and angle). If you change the partition angle (with $1°$ is visible) the partition cover finer and finer way the space. The use of a $\frac{2\pi}{n}$ angle is needed to obtain a FINITE graph from the computation. This with the fact that our universe is actually finite, is sufficient to say that at the end of the algorithm we have a finite number of nodes and edges. I'm going right now to make some suitable test at various scale.

An important needed step of the algorithm is to remove the double-counting of some points, and in the case, the node already exists, connect the node for which the algorithm is being applied with the existing node instead of introducing a new one.

To make sure the mapping preserves the circular symmetry from every point, the algorithm needs to be run for every node in the graph. The graph can be seen as completed when a successive application of the algorithm on any node, preserve the graph. I suppose this is possible from the finite space to map and from the $\frac{2\pi}{n}$ angle used for the partition.

PS: I apologize if I have not explained myself clearly, I prepare some programs and as soon as possible I publish them to make the concept clearer and finally have numbers in hand.