155 | Stephen Wolfram on Computation, Hypergraphs, and Fundamental Physics
What this covers
Stephen Wolfram and Sean Carroll discuss a research program that treats fundamental physics as emerging from a computational substrate. The core proposal starts from scratch: space itself is not a background arena but a discrete hypergraph of relational "atoms"—nodes with no intrinsic position, only connections to other nodes—updated according to simple local rules. From this minimal setup, Wolfram argues that both general relativity and quantum mechanics emerge generically as inevitable features of how computationally bounded observers with a single thread of experience perceive the system. The conversation covers what motivates this approach, how the mathematical structures work, and why a computational view of the universe might resolve longstanding puzzles rather than create new ones.
The discussion ranges across several distinct claims. Wolfram defends the principle of computational equivalence—the idea that simple systems can perform computation as sophisticated as anything—and argues that thermodynamic properties like entropy reflect observer limitations rather than objective disorder. He and Wolfram's collaborators have developed branchial space, a mathematical arena representing quantum entanglement through branching histories, and shown that geodesic deflection works identically in both physical spacetime and branchial space, unifying relativity and quantum mechanics under a single framework via causal invariance. The conversation also touches on particles as topological structures in the hypergraph, the nature of measurement as a real thermodynamic process, and prospects for quantum computing—along with more speculative territory like the completion interpretation developed by Jonathan Gorard, which reimagines quantum measurement through the lens of automated theorem proving.
Wolfram argues that fundamental physics can be derived from the bottom up by positing space as a discrete hypergraph of relations updated by simple rules, and that general relativity and quantum mechanics both emerge generically from this single computational substrate as features observed by computationally bounded observers with a single thread of experience.
- Space is not a background but is itself made of discrete relational 'atoms' (nodes in a hypergraph) updated by local rules
- General relativity emerges from geodesic deflection in physical space and quantum mechanics from geodesic deflection in branchial space via the same mathematics
- The laws of physics we observe are inevitable consequences of our being computationally bounded observers who maintain a single thread of experience
Space and time are not fundamental but emerge from discrete atoms of space updated by simple rewrite rules.
- In this framework the only thing in the universe is space; there is no background with things placed in it, so to have an electron you must make it out of features of the pattern of connections between the atoms of space rather than placing a separate object into space.
“If you want an electron, you have to make it out of space, so to speak. You have to make it from features of that pattern of connections between the atoms of space.”
- Unlike most traditional views that treat space and time as the same kind of thing, in these models space is the extent of the spatial hypergraph (following connections) while time is the computational process of updating the hypergraph (the progression of a computation), so the traditional connection between space and time is broken apart and recovering relativity from the model is non-trivial.
“Space is the extent of the spatial hypergraph. Time is the computational process of updating this hypergraph. So time is the progression of a computation.”
- Space is not a pre-existing background in which things are placed; it is itself made of discrete disembodied points (atoms of space) that have no position, only relations (connections) to other points, and continuous-seeming space emerges from these relations only when viewed at large scale, analogous to how continuous fluids emerge from discrete molecules.
“Space is made of something... It's just made of these disembodied discrete points, and each point is just an abstract element, it doesn't have a position”
- The models are fundamentally quantum mechanical; a classical universe could only be obtained as a degenerate case, for example if the universe were built from an ordinary Turing machine where every state has exactly one successor state giving a single thread of time, but generically there are many places where updates can happen, producing branching and hence quantum mechanics.
“I think our models are very fundamentally quantum mechanical.”
General relativity and quantum mechanics are the same mathematics applied to different arenas: physical space versus branchial space.
- The same mathematics that produces the deflection of geodesics by energy-momentum (the Einstein equations) in physical space also produces, in branchial space, the deflection of geodesics that corresponds to the Feynman path integral phase change; thus general relativity and quantum mechanics are basically the same thing in these models, differing only in whether the relevant arena is spacetime or branch time.
“the same mathematics that leads to the Einstein equations in spacetime leads to the Feynman path integral in multiway graphs and branchial space”
- Different orders of applying updates create branching paths of history (a multiway graph), but causal invariance — the property that every branch is eventually followed by a merge when two update orders reach isomorphic hypergraphs — ensures branches that diverge will eventually reconverge, which is why the models yield objectivity in quantum mechanics.
“essentially what causal invariance says is every time there's a branch there will be a subsequent merge”
- In the multiway causal graph, events can be separated in three distinct ways: timelike (one event must occur before another because its output is the input to the other), spacelike (at different positions in physical space), or branchlike (on different branches of quantum history), and the interplay of these three kinds of separation underlies phenomena in black holes and AdS/CFT.
“events can be separated, they can either be timelike... they can be spacelike... or they can also be branchlike, in the sense that they occur on different branches of quantum history”
- Black hole entropy, understood as the number of microscopic states consistent with the macroscopic identification of a black hole, should in principle be computable in these models by counting threads of history in the multiway causal graph, though it is computationally hard due to the exponential number of branches, motivating the local multiway systems optimization being developed.
“given this thing that we say is a black hole, how many possible microscopic states are consistent with this macroscopic observation that it is a black hole. So I would think that we should be able to compute that.”
Observers must maintain single threads of experience and identity through motion to derive the laws of physics.
- Deriving the laws of physics we observe requires that observers maintain a single thread of experience through time and a notion of pure motion (maintaining identity while moving through space) even though the universe is being rewritten enormous numbers of times per second and a moving object is made of different atoms of space at its destination; these two attributes are what is needed to derive general relativity and quantum mechanics.
“this idea of this single thread of experience in time, this idea that it is possible to have pure motion, and it's possible to maintain your identity as you move around in space, those two attributes seem to be what you need to derive generic laws of physics”
- The second law of thermodynamics only holds for computationally bounded observers; a non-computationally-bounded observer who could track every molecular trajectory would not observe the second law at all, because the apparent increase in randomness depends on the observer's inability to compute the detailed microscopic dynamics.
“if we are not computationally-bounded observers and we can figure out what all these trajectories of all these molecules are, we don't get the second law of thermodynamics”
- Many systems are computationally irreducible, meaning the only way to find out what they will do is to run every step, because the observer is itself a computational system that is computationally equivalent to the system being predicted and therefore cannot jump ahead or outsmart it.
“there are lots of systems that are computationally irreducible, in the sense the only way to find out what they'll do is just to run every step”
- The claim that the universe is heading toward a boring high-entropy state with no structure is an artifact of being a computationally bounded observer; a more capable observer might perceive the same gas-in-equilibrium state as amazingly complex microscopic structure, so 'high entropy' reflects the observer's inability to extract structure rather than an objective absence of it.
“when you say it's going to a high entropy state, you're saying, oh, that... All you're saying is, so I as an observer of the kind of observer I am can't make anything out of it, other than to say it's uniform and random.”
- A key prediction of the physics project is that what can be done in our universe is just computation achievable by a Turing machine and not hyper-computation; even if something infinitely more powerful than a Turing machine existed, it would be hidden behind a cosmological event horizon and could never be observed or interacted with.
“what can be done in our universe is just computation and not hyper-computation. That is, it's the kind of computation that can be done by a Turing machine”
- Rather than there being one special simple rule for our universe, one can think of the universe as running all possible rules, with we observers occupying a particular reference frame position in 'rulial space' that samples this universe of all possible universes; because changing the underlying rule also changes the rule for us as embedded observers, a large class of rules all yield the same effective laws of physics.
“you can think of the universe as running all possible rules. And we as observers of the universe are essentially exist in a particular... In a sense-reference frame, in a particular position in rulial space”
- If we believe the physical universe exists, it necessarily follows that mathematics exists in the same way, because both can be described as the set of all possible rules; our doing of mathematics is carving out pieces of the total structure of all mathematics just as we carve out pieces of the universe to observe, motivating a bulk theory of metamathematics analogous to relativity and quantum mechanics in physics.
“if we believe that the universe exists... then it necessarily follows that mathematics exists in the same way”
Large-scale laws are generic features independent of microscopic details, explaining why fundamental theories emerge from discrete models.
- The laws of fluid dynamics (Navier-Stokes) are essentially the same for air and for water despite the underlying molecules and their collision physics being entirely different, illustrating that large-scale laws can be generic and independent of microscopic detail, which is the same reason general relativity and quantum mechanics can be generic in the hypergraph models.
“on a large scale, the laws of fluid dynamics are the same for air and for water. They have some different parameters but they're basically the same laws.”
- When you pick a program at random in the computational universe, even a very simple program can produce extremely complicated behavior; this is the essence of how nature generates complexity, and it contradicts the intuition that simple programs always do simple things.
“if you just sort of pick a program at random, even though the program may be very simple, its behavior may be very complicated”
- From the microscopic atoms of space, by taking many limits at the level of mathematical rigor physicists typically use (and which is no more rigorous than the still-unsuccessful 100-year effort to derive fluid equations from molecular dynamics), the Einstein equations can be derived on large scales, with the advantage that extensive simulations confirm which limits actually work.
“with all of those conditions in physicist-level mathematics, we can derive Einstein's equations”
Particles are not fundamental but locally stable topological deformations in the hypergraph analogous to vortices in fluids.
- In these models particles are not fundamental but are locally stable topological deformations in the hypergraph, analogous to a stable vortex in a fluid; quantum field operators are effective descriptions of these stable structures, and most of the activity of the universe is associated with maintaining the structure of space itself rather than with the particles we observe.
“The particles are locally stable deformations in this hypergraph”
- The infinite sea of virtual particle-antiparticle pairs of quantum field theory, which naively implies an enormous vacuum energy density that should gravitationally curl up the universe, corresponds in these models to the actual rewrite rules of the hypergraph that create the structure of space, making it less mysterious that vacuum fluctuations do not catastrophically curve the universe.
“the sort of limiting case of those virtual particles is these actual rewrite rules in our hypergraph”
- Energy, surprisingly, is very simple in the models: it is the amount of activity in the hypergraph, more formally the flux of causal edges through spacelike hypersurfaces, while momentum is the flux of causal edges through timelike hypersurfaces; this contradicts Wolfram's expectation that energy would require notions of particles to define.
“energy is the flux of causal edges through spacelike hypersurfaces, momentum is the flux of causal edges through timelike hypersurfaces”
- The Higgs mechanism, so far, looks like a hack rather than a fundamental feature, which is disappointing; a goal of the program is to make the Higgs mechanism look less like a hack, but Wolfram knew the Higgs particle would be discovered and was disappointed when it was.
“So far, the Higgs mechanism looks like a hack, and the question is, can we make it look less like a hack”
- Wolfram disputes that the future de Sitter vacuum is a single empty quantum state with no structure; because space itself is made of a teeming collection of atoms of space, the vacuum must have a lot of structure, and even if all identifiable topological particle excitations are pushed beyond the cosmological event horizon, the underlying atoms of space are still active, observable only by a demon-level observer.
“Our vacuum cannot be just the boring, there's nothing there vacuum. Our vacuum has to have a lot of structure in it.”
- Returning to physics after 40 years, Wolfram found that much of fundamental physics had not changed, illustrated by the mass of the lambda particle being essentially the same value he remembered decades earlier.
“The mass of a lambda is 1115 MEV, and I look it up again, it's 1114.962 or something.”
Quantum mechanics is fundamentally about branching multiway graphs where measurement requires real effort to corral branches together.
- It seems you can never win with quantum computing because when computation branches out across many threads in branchial space, the effort required for the observer to corral those threads back together to read the result is at least as great as the gain from the branching; if measurement is honestly accounted for, the apparent speedup may disappear.
“it seems that you can never win with quantum computing, that is that when you branch out in all these different ways, the effort to corral things back together is at least as great as the gain that you get from things branching apart”
- In the completion interpretation (due to Jonathan Gorard), a branching observer perceives a branching universe by corralling branches together — mathematically equivalating two branches to maintain the fiction that definite things happen — and causal invariance guarantees these branches will eventually merge so the observer's forced completions remain consistent; destructive interference occurs when a photon's paths land at opposite ends of branchial space so the observer cannot knit them together into a single photon.
“this is an idea of Jonathan Gorard... what he calls the completion interpretation of quantum mechanics”
- The quantum amplitude should be thought of not as a single complex number but as a magnitude and a phase: the magnitude is determined by path counting in the multiway graph (the number of ways to reach a state), while the phase is determined by the position in branchial space.
“the magnitude of the quantum amplitude is the result of path counting in the multiway graph”
- Measurement in quantum mechanics is not a free instantaneous idealization but takes real effort, as evidenced by the requirement when building a quantum randomness device that a tiny quantum effect must be amplified into a classical number of degrees of freedom via a thermodynamic randomization process for successive measured bits to be uncorrelated.
“that was to me the first sort of sign that there was going to be a cost to measurement, so to speak”
Causal invariance ensures branches diverging in different update orders eventually reconverge, yielding objectivity in quantum mechanics.
- In a discrete graph model Lorentz invariance must be approximate with a maximum boost beyond which the granularity of spacetime becomes apparent; causal invariance in the ordinary causal graph yields ordinary Lorentz invariance, while causal invariance in the multiway causal graph is conjectured to yield the extension to CPT invariance.
“Absolutely, there is a maximum boost, you can't get... In other words, as eventually you start seeing the granularity of spacetime.”
- CPT invariance has a clean formulation in these models: charge conjugation is an inversion (reflection) in branchial space, parity is a reflection of physical space, and time reversal is a reflection in time, so CPT invariance corresponds to the multiway causal graph being invariant under reflection in all three directions, though it has not yet been shown that the models necessarily exhibit CPT invariance.
“C, charge conjugation, which is particle to antiparticle, is effectively like... It is an inversion in branchial space”
- It is looking very likely that local gauge invariance is an inevitable feature of these models, and it is conceivable the actual gauge group (perhaps a subgroup of E8) could be determined by the model even before the reason for three dimensions of space is understood.
“It is looking very likely that it is an inevitable feature of our models that there will be local gauge invariance.”
- In these models the elementary length is much smaller than the Planck length, and the Planck energy is understood not as the elementary energy but as an energy summed over all possible branches in the multiway causal graph, whereas the elementary energy is an individual causal edge in that graph, which clears up the puzzle of why the Planck energy is so large.
“in our models, the elementary length is much smaller than the Planck length”
Three spatial dimensions are likely not generic features and the universe may have cooled from higher dimensions, resolving early-universe problems.
- The three dimensions of space are probably not a generic feature of the models (unlike general relativity and quantum mechanics which seem generic), and the network has no built-in dimension; the universe may have started effectively infinite-dimensional and only gradually cooled down to three dimensions, which would resolve early-universe horizon problems because in high dimensions everything is closely connected.
“the universe started infinite-dimensional and only gradually sort of cooled down to be three-dimensional, and so that then a lot of the problems about these horizon problems in the early universe and so on disappear”
- A core technical difficulty in deriving the early-universe dimension-change equations is that calculus is built on changes with respect to integer numbers of variables; nobody knows how to do calculus with a fractional number of variables (e.g. two-and-a-half variables), so a generalization of calculus to fractional-dimensional space on hypergraphs is needed.
“they never get to do calculus with two-and-a-half variables, that's not a thing, and nobody knows how that works”
- Getting a perfect harmonic oscillator in these models requires closed timelike curves (the history of the universe repeating itself), which is a weird idealization; real physics probably has no perfect harmonic oscillators, illustrating that insisting on standard idealizations stresses the model just as deriving perfect rigid-body mechanics stresses molecular dynamics.
“In order to get that in our models, you have to have closed timelike curves”
- Because the hypergraph network has no built-in dimension and is not microscopically a manifold, it can change topology, tear, or break; in simulations of rapidly rotating black holes near critical angular momentum, a piece of the universe separates off, connected only by a small number of causal edges, potentially producing an observable signature such as shot noise in gravitational radiation.
“for critical, supercritical black holes is a piece of the universe just breaks off, and as it... At the critical angular momentum, what you see is that a piece of the universe is hanging by a thread”
The methodology of finding simplest primitives and observing their behavior differs fundamentally from traditional reverse-engineering physics approaches.
- Cellular automata, despite being fertile minimal models for many phenomena, are inappropriate for fundamental physics because they assume a pre-existing fixed notion of space and time, whereas fundamental physics requires going underneath those notions and building space and time from something more fundamental.
“cellular automata are very minimal models, once you have a fixed notion of space and time... But they assume a pre-existing notion of space and time”
- The technical innovation of representing the model as a hypergraph (where a hyper-edge can connect any number of nodes rather than just two as in an ordinary graph) made the model much easier to deal with and enabled forward progress, even though in the end it does not make much fundamental difference and could have been figured out 25 years earlier.
“In a hypergraph you can have any number of nodes involved in a hyper-edge... it makes the whole model much, much easier to deal with”
- Traditional reductionistic physics works by reverse-engineering from observed phenomena to find an explanation, whereas Wolfram's methodology, learned from 40 years of designing computational languages, is to find the simplest primitives and just see what they do without trying to insert known results like Einstein's equations in advance.
“this idea of find the simplest primitives and then see what they do, this is what I get for having spent 40 years being a computational language designer”
- The multiway graphs can be shown to reproduce categorical quantum mechanics, which is itself known to be equivalent to standard 1920s quantum mechanics; furthermore the models can compile standard quantum circuits into multiway graphs and perform better optimization than standard quantum circuit formalism while necessarily yielding the same answers.
“we do know that our multiway graphs reproduce an approach to quantum mechanics called categorical quantum mechanics, and we can just show that they are equivalent”
- The models keep turning out to be concrete instances of abstract theories such as causal set theory and higher category theory; for example they provide an algorithmic procedure that automatically yields Lorentz invariance (resolving a mystery in causal set theory about why thrown-down events give Lorentz invariance), and the infinity groupoid limit of the models connects to deep questions about why this universe and any universe exists.
“In our models, we have an algorithmic procedure for generating those events, which immediately gives you Lorentz invariance, but yet all the mathematics that people have done in causal set theory... can be applied to our models”
- The mathematical underpinning of the completion interpretation comes from automated theorem proving: applying axioms in all possible ways builds a multiway graph, a theorem's proof is a path through it, and the completions an observer makes to corral quantum branches together are mathematically equivalent to forming lemmas in a proof.
“completions become the formation of lemmas”
- The project has live-streamed its internal working meetings and posts all notes on the web the day after they are made, which has drawn in both professional physicists and computer-oriented people and creates a useful 'generator of reality' where the messy process of being right and wrong in science happens publicly.
“we've live-streamed a lot of our internal working meetings, and we've... All the notes from this project are all posted on the web, basically the day after they're made”
Quantum amplitude comprises magnitude from path counting and phase from branchial position; entanglement has a finite propagation speed.
- In the multiway graphs, bosons are conjectured to be associated with branchings that merge very quickly while fermions are associated with branchings that do not merge quickly and end up at opposite sides of branchial space; at a mathematical level fermions are like the square root of bosons because they branch without merging well, offering a route to deriving the spin statistics theorem.
“we suspect that bosons are associated with, in a sense, branchings that merge very quickly, and fermions are associated with branchings that do not merge quickly”
- Just as a perturbation in physical space expands at the speed of light, a perturbation in branchial space expands at finite speed, implying a maximum entanglement speed in quantum mechanics (a very rough estimate being 10^5 solar masses per second); if correct, this would manifest as limits on very rapid mergers of large black holes and would mean Bell-inequality correlations, though seemingly instantaneous in physical space, are actually bounded propagation in branchial space.
“it also expands at finite speed in our models, and that means there is a maximum entanglement speed in quantum mechanics”
- Black holes may have an entanglement horizon in branchial space outside the physical event horizon, and an observer at the entanglement horizon cannot form a classical thought (cannot collapse a wave function), so a distant observer would conclude such an observer can never determine outcomes — offering a qualitative handle on the black hole information paradox.
“I think what happens is that the observer at the entanglement horizon cannot form a classical thought.”
The computational principle of equivalence implies intelligence and computational sophistication are ubiquitous even in simple systems.
- The principle of computational equivalence holds that once a system passes a very low threshold of complexity, it immediately performs computations as sophisticated as anything can do; there is no gradual increase in sophistication with complexity, so almost any non-trivial program is already maximally computationally sophisticated and universal computation is ubiquitous.
“Once you get above some very low threshold, you're immediately at the max, you're immediately doing computations that are as sophisticated as anything.”
- The anthropic principle is a story of lack of imagination because it assumes life, intelligence and consciousness can only arise in the particular way we have seen; the principle of computational equivalence implies that intelligence is ubiquitous and even very simple systems can be as computationally sophisticated as human brains, so the idea that intelligence requires liquid water is laughable.
“to me, the anthropic principle is sort of a story of lack of imagination, so to speak. Because it's saying the only way that we can have life, intelligence, consciousness, whatever, is the particular way we've seen it.”