Wolfram Physics Engine Exploration:
Gauge Theory & the Standard Model

Stephen Wolfram launched the Wolfram Physics Project around a radical question: could familiar physics emerge from extremely simple computational rules, rather than being programmed into a simulation as the known equations of relativity, quantum mechanics, or the Standard Model? In the Wolfram model, a possible spatial state is represented by a hypergraph, local rules repeatedly replace small pieces of that graph, and a causal graph records which update events depend on earlier events. Following every possible order of updates produces a multiway graph of alternative histories; slices through that structure produce branchial graphs, which the project relates to quantum states and entanglement.

For suitable rules and under additional assumptions—especially locality, causal invariance, and an appropriate large-scale limit—the project argues that structures resembling continuous space, relativistic causal cones, and aspects of quantum mechanics can emerge. These are proposed mathematical correspondences, not yet a demonstrated model of our universe, and quantum probability is not simply the number of nodes in a hypergraph: Wolfram associates amplitude magnitude with path multiplicity in multiway evolution and phase with position in branchial space. The intuition is a little like Conway's Game of Life, except there is no fixed grid—the network of relationships that may become space is itself continually rewritten. This repository investigates whether gauge structure and field-like dynamics can be built at that microscopic level, with the long-term goal of testing for QED-like behavior and simple bound systems without inserting continuum fields or forces by hand; it does not yet derive QED, particles, physical constants, or molecules.

Stephen Wolfram: Can space and time emerge from simple rules?

Hands-On Introduction to the Wolfram Physics Project

The questions we are trying to answer

This project starts from the interview above. The goal is to take one of Wolfram's unfinished ideas, turn it into a precise computational question, and find an answer—or a specific reason it fails.

Three questions organize the work:

• How does the network become space? Can local rewrites produce a stable large-scale dimension, and what happens when that dimension varies? Wolfram discusses dimension fluctuations at 1:11:17.
• Where would fields and charge come from? Can choices within the rewriting process give rise to internal degrees of freedom and their interactions? He explicitly leaves electric charge unresolved at 1:07:07.
• What could persist as a particle? Can a recognizable structure survive while its underlying network changes, move relative to its surroundings, and interact with another such structure? He identifies the missing particle model at 1:25:38.

Molecules are a later target, not the assumed interpretation of a graph pattern. None of these three questions has been solved here. The calculations below provide starting tools and some restrictions on candidate answers.

1. Start with relationships, not a stage to put particles on

In an ordinary simulation, space is already there. Objects have positions and fields have values at those positions. In the proposal discussed in the interview, the network is what might eventually look like space. There is no separate stage underneath it.

A node is an identifiable abstract element. It is not a voxel, an atom, or a sample of a known field. A hyperedge records a relationship between two or more elements. A rewrite replaces a small pattern of relationships with another, possibly creating new elements.

before = [(0, 1), (0, 2)]
after  = [(0, 2), (0, 3), (1, 3), (2, 3)]

This example replaces two relationships with four, using a fresh node. The integers are IDs, like dictionary keys—not positions or measured values. The tuples list related IDs, not coordinates. Strings could name the same nodes without changing the structure.

It illustrates what a rewrite is; it is not established as a rule that produces matter.

The same rule used by the runnable example below. Node 3 is new. Dot positions help draw the connections; they are not measured positions in space. Even the unchanged endpoint pair (0,2) is a new edge occurrence after this rule consumes and reproduces it.

2. Keep track of what happened—and what else could have happened

A rewrite can use a relationship made by an earlier rewrite. That gives a dependency: the producer must happen before the consumer. Collect those dependencies and you have a causal graph.

There may also be several places where the rule can apply. Following each possibility produces a multiway graph. Histories branch when there are alternatives and merge when they reach equivalent states.

These are different records. A causal edge connects dependent events. A multiway edge connects a state to a possible next state.

A fully specified small example. Above: A and B can happen in either order and reach the same network. Below: C needs a relationship made by each, so its causal graph has two prerequisites. The colors track events, not fields. Node names are retained here; ignoring names can identify the two intermediate graphs as equivalent too. This illustrates the distinction, not quantum interference.

Why retain the alternatives? Wolfram's proposal makes their structure part of the physics question. Selecting one random history may be useful for some measurements, but it does not reproduce the whole multiway system or establish quantum probabilities.

3. Measure whether something like space appears

Stand at a node and count how many other nodes are reachable within one hop, two hops, three hops, and so on. In a region resembling ordinary three-dimensional space, doubling a sufficiently large radius should enclose roughly eight times as many nodes.

That gives a way to ask about dimension without first drawing the graph in 3D. Repeat the measurement at different locations and times: does the same dimension persist, or does the growth pattern change?

This is a direct route into the interview's dimension question. The current probes can make these measurements, but the small rule survey does not establish a three-dimensional regime or physical dimension fluctuations.

flowchart LR
    N["Rewritten network"] --> B["Count nodes within r hops"]
    N --> W["Measure random-walk returns"]
    B --> C["Compare locations, scales, and later states"]
    W --> C
    C --> Q["Does a stable geometric regime appear?"]

d_H(r)=\frac{d\log |B(v,r)|}{d\log r}.

p_t(v,v)\propto t^{-d_s/2}.

4. Ask where internal structure would come from

In the interview, the starting data are relationships and rewrites. There is no instruction to attach a triangle to every node.

The gauge-theory lead is more specific in Wolfram's technical introduction: different local rewrite choices may serve as equivalent descriptions, while a choice made here affects which choices remain possible later. The proposal connects this structure to gauge freedom and field propagation. That connection is something to construct and test.

A fiber is the collection of detailed states or descriptions that lie over one location in a chosen description. To derive one here, we would first need to specify what that description retains and which underlying distinctions it groups together.

An intentionally simple projection: keep the two endpoints and whether they are connected, but hide the route between them. A chain and a loop then look alike to that observer. They can still evolve differently. This explains a fiber as a set of alternatives; it does not derive a physical observer, a graph connection, or gauge equivalence.

InfraGaugeTheory provides a language for graph fibers, projections, connections, and transport. Its stated goals include natural clustering into fibers and obtaining fibered graphs from hypergraph rewriting. Those are directly relevant open construction problems.

What the triangle experiments do—and do not—supply

Our existing finite-gauge studies work in the opposite direction: choose a fiber, calculate its symmetries, then study specified interactions. This is a controlled laboratory for candidate structures. It has not derived the fiber or its dynamics from the bare rewrite system.

Why three vertices? A triangle is the smallest simple graph whose symmetries can act differently when applied in different orders. It is small enough to enumerate exactly. That is a practical reason for a test case, not a reason that nature must use it.

A five-node ring is another possible choice. So is a fifty-node ring. The connections matter as well as the count: ring-preserving maps are rotations and reflections, not arbitrary permutations. Current cycle-reaction results assume odd ring size; an even ring introduces additional algebraic cases. Adding internal vertices is not automatically increasing spatial resolution.

F_x=p^{-1}(x).

G=\mathrm{Aut}(F).

U(v)=sv+a\pmod n,
\qquad s\in\{-1,1\},\quad a\in\mathbb Z_n.

U_{xy}\mapsto g_yU_{xy}g_x^{-1}.

5. Look for something that survives the changing network

Wolfram suggests that particles might be persistent structures in the network, rather than objects placed on it. The useful question is not “does this picture resemble an electron?” It is “what remains the same while the surrounding structure and its constituent nodes change?”

A candidate must survive actual rewrites. Its identity cannot depend on keeping the same node IDs or freezing the region that supports it. Motion must be measured relative to the surrounding network, not the positions assigned by a renderer.

The present fixed-fiber calculations give a useful warning: a prepared defect can support localized graph modes, yet allowed interactions can remove those modes. Localization in one snapshot is not persistence.

These calculations use a supplied two-dimensional lattice, triangle fibers, and prepared link defects. They concern the graph's spectrum. They are not electron orbitals, and no atomic energy scale is assigned.

n_+(L_{\mathrm{bundle}}-12I)
\le\min(Q_\uparrow,Q_\downarrow).

Progress So Far

• Exact finite rewrite histories and event provenance. We can inspect spatial states, dependencies, and alternative histories using the upstream engine. A restricted connection-aware extension preserves these records through edge subdivision. This is a starting point for the interview's causal and multiway questions, not a general evolving gauge field. Construction.

• Intrinsic geometry measurements and a bounded rule survey. Graph-ball and random-walk probes measure structure without taking dimension from a drawing. The present short survey has not found or established emergent 3D spacetime. Scope.

• Exact descriptions of specified finite gauge states. Small-patch results show when local measurements lose information needed to predict an interaction, and how relative alignment restores it. This supplies a concrete test for a future observer-based description; it does not derive that observer or fiber. Patch reconstruction and gluing.

• Restrictions on proposed routes to matter. In studied models, some apparent internal motion is only relabeling; diffuse large fibers reduce to ordinary exchange diffusion; and localized modes can be destroyed by the permitted dynamics. These delimit particular constructions, not all rewriting models. Unary restriction, refinement limit, localization and loss.

Future Work

1. Construct internal alternatives from actual rewrites. On a small, completely explored example, specify a projection or observer and derive its candidate fibers and the maps between them. Determine which choices are equivalent descriptions and which change later observable behavior. A failure to define consistent transport is a useful result; attaching a preferred fiber is not a substitute.

2. Follow the consequences of a local rewrite choice. Compare alternatives with a common prior state and boundary. Track which later matches become possible or impossible, preserving event dependencies. Determine whether any effect survives relabeling, branch merging, and a change of description. This directly tests Wolfram's proposed route from local choices to gauge-like effects.

3. Find and characterize stable geometric regimes. Extend selected rule families far enough to separate growth, finite-size effects, and local dimension variation. Test several seeds and update schedules. Explain any regularity from the rewrite process before interpreting it as space or curvature.

4. Search for persistent, moving structures. Identify candidates by invariant relationships, follow them through replacement of their constituent nodes, and test encounters. Distinguish genuine survival from inactivity, imposed defects, and renderer artifacts.

5. Connect measured dynamics to an effective physical description. If a reproducible geometric or interaction regime appears, derive its scale dependence and observable laws. Quantum amplitudes, interference, physical charge, and mass each need their own construction. Known physics provides comparison targets, not hidden terms in the microscopic updates.

6. Spend compute on the limiting question. Use parallel enumeration for finite rule/transition problems and longer runs for declared persistence or scaling questions. Optimize the limiting operation only when it unlocks an otherwise inaccessible mathematical test. Keep counterexamples, null results, and incomplete runs.

Inspect the experiments

The debugger separates graph state, causal ancestry, branchial slices, and a best-effort spatial projection. It is a way to inspect the calculation—not evidence that the pictured state is three-dimensional.

git clone https://github.com/pirate/wolfram-gauge-physics.git
cd wolfram-gauge-physics
cmake -S . -B build -DCMAKE_BUILD_TYPE=Release
cmake --build build -j
./build/wgphysics_evolve \
  --rule '0,1;0,2->0,2;0,3;1,3;2,3' \
  --init '1,2;1,3' \
  --steps 3 \
  --output out/evolution.json
python3 -m http.server 8765 --bind 127.0.0.1

MIT licensed. This is an independent experimental project, not an official Wolfram Institute or Wolfram Research repository. The attributed Wolfram website screenshot remains the original publisher's material, not part of this repository's MIT license. See figure sources.