The basic concept of applying our models to physics is to imagine that the complete structure and content of the universe is represented by an evolving hypergraph. There is no intrinsic notion of space; space and its apparent continuum character are merely an emergent large-scale feature of the hypergraph. There is also no intrinsic notion of matter: everything in the universe just corresponds to features of the hypergraph.
There is also no intrinsic notion of time. The rule specifies possible updates in the hypergraph, and the passage of time essentially corresponds to these update events occurring. There are, however, many choices for the sequences in which the events can occur, and the idea is that all possible branches in some sense do occur.
But the concept is then that there is a crucial simplifying feature: the phenomenon of causal invariance. Causal invariance is a property (or perhaps effective property) of certain underlying rules that implies that when it comes to causal relationships between events, all possible branches give the same ultimate results.
As we will discuss, this equivalence seems to yield several core known features of physics, notably Lorentz invariance in special relativity, general covariance in general relativity, as well as local gauge invariance, and the perception of objective reality in quantum mechanics.
Our models ultimately just consist of rules about elements and relations. But we have seen that even with very simple such rules, highly complex structures can be produced. In particular, it is possible for the models to generate hypergraphs that can be considered to approximate flat or curved d-dimensional space. The dimension is not intrinsic to the model; it must emerge from the behavior of the model, and can be variable.
The evolving hypergraphs in our models must represent not just space, but also everything in it. At a bulk level, energy and momentum potentially correspond to certain specific measures of the local density of evolution in the hypergraph. Particles potentially correspond to evolution-stable local features of the hypergraph.
The multiway branching of possible updating events is potentially closely related to quantum mechanics, and much as large-scale limits of our hypergraphs may correspond to physical space, so large-scale limits of relations between branches may correspond to Hilbert spaces of states in quantum mechanics.
In the case of physical space, one can view different choices of updating orders as corresponding to different reference frames—with causal invariance implying equivalence between them. In multiway space, one can view different updating orders as different sequences of applications of quantum operators—with causal invariance implying equivalence between them that lead different observers to experience the same reality.
In attempting to apply our models to fundamental physics, it is notable how many features that are effectively implicitly assumed in the traditional formalism of physics can now potentially be explicitly derived.
It is inevitable that our models will show computational irreducibility, in the sense that irreducible amounts of computational work will in general be needed to determine the outcome of their behavior. But a surprising discovery is that many important features of physics seem to emerge quite generically in our models, and can be analyzed without explicitly running particular models.
It is to be expected, however, that specific aspects of our universe—such as the dimensionality of space and the masses and charges of particles—will require tracing the detailed behavior of models with particular rules.
It is already clear that modern mathematical methods can provide significant insight into certain aspects of the behavior of our models. One complication in the application of these methods is that in attempting to make correspondence between our models and physics, many levels of limits effectively have to be taken, and the mathematical definitions of these limits are likely to be subtle and complex.
In traditional approaches to physics, it is common to study some aspect of the physical world, but ignore or idealize away other parts. In our models, there are inevitably close connections between essentially all aspects of physics, making this kind of factored approach—as well as idealized partial models—much more difficult.
Even if the general structure of our models provides an effective framework for representing our physical universe at the lowest level, there does not seem to be any way to know within a wide margin just how simple or complex the specific rule—or class of equivalent rules—for our particular universe might be. But assuming a certain degree of simplicity, it is likely that fitting even a modest number of details of our universe will completely determine the rule.
The result of this would almost certainly be a large number of specific predictions about the universe that could be made even without irreducibly large amounts of computation. But even absent the determination of a specific rule, it seems increasingly likely that experimentally accessible predictions will be possible just from general features of our models.