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A Class of Models with the Potential to Represent Fundamental Physics

The formal, canonical statement of the model. Published in Complex Systems 29(2) (pp. 107–536), mirrored on arXiv as 2004.08210, and presented on the project portal as the Technical Introduction (wolframphysics.org/technical-introduction). This is the reference to cite for precise definitions.

Formal definitions (faithful distillation)

States as hypergraphs (collections of ordered relations). The state is a hypergraph: a finite collection of ordered relations (tuples) among abstract elements. Elements carry no intrinsic properties — only the pattern of relations matters. A hyperedge generalises a graph edge by relating any number of elements. A state is written as a set of tuples, e.g. {{1,2,3},{3,4},{2,4,5}}. Because relations are ordered, the hypergraph is directed.

Rules as rewrites of subhypergraphs. Dynamics are an update rule H₁ → H₂: “a subhypergraph matching pattern H₁ is replaced by a distinct subhypergraph matching pattern H₂” — equivalently a set substitution system. The project’s canonical worked example:

{{x, y}, {y, z}} -> {{w, y}, {y, z}, {z, w}, {x, w}}

Here x, y, z are matched elements and w is newly created. Iterating from a small initial hypergraph grows the structure interpreted as space.

Updating order / event selection. Many matches may exist at each step; each application is an event. A well-defined evolution needs an updating scheme (typically applied to chosen non-overlapping matches). The deeper claim is that for rules of physical interest the choice doesn’t affect observable structure — see causal invariance.

Causal graph. Nodes are events; a directed edge A → B exists “if and only if the input for event B makes use of hyperedges that were produced by the output of event A.” This is the discrete analogue of spacetime causal structure.

Causal invariance. The property “whereby all possible evolution paths yield causal networks that are (eventually) isomorphic as directed acyclic graphs” — the same causal graph regardless of update order. Tied to confluence / the Church–Rosser property. The project presents it as underlying relativistic invariance (“equivalent to Lorentz covariance”) and observer-independence.

Multiway systems & branchial graphs. Instead of committing to one order, follow all permissible rule applications in all orders → a branching graph of states (the multiway system), with a multiway causal graph. Branchial graphs connect states/events that are “branchlike separated” (different branches at the same step) — the substrate for the project’s quantum/Hilbert-space structure. Causal invariance implies branches eventually merge, recovering a consistent objective history.

Key terminology

• Hypergraph — finite collection of ordered relations among featureless elements; the universe’s state.
• Update rule (R) — a rewrite H₁ → H₂; a set/hypergraph substitution system.
• Update event — a single rule application at a specific match.
• Causal graph — events with edge A → B when B consumes output of A.
• Causal invariance — every order yields the same (eventually isomorphic) causal graph; ≈ confluence; linked to Lorentz covariance.
• Multiway system — all possible rule applications in all orders.
• Branchial graph/space — relates branchlike-separated states; substrate for quantum structure.

Sourcing note

The live portal pages block automated retrieval (HTTP 403). The definitions above were cross-checked against the identical text on arXiv (2004.08210) and the Wolfram Institute. Anyone extending this should re-open the live pages to confirm exact wording and the full chapter list.

Grounded facts we can safely state

1. The canonical formal source is “A Class of Models with the Potential to Represent Fundamental Physics” (Wolfram, 2020), in Complex Systems 29(2) and on arXiv (2004.08210).
2. A state is a hypergraph: a finite collection of ordered relations among property-less elements.
3. Dynamics are an update rule H₁ → H₂ rewriting a matching subhypergraph; illustrative rule {{x,y},{y,z}} → {{w,y},{y,z},{z,w},{x,w}}.
4. Each application is an event; the causal graph edges A → B when B uses hyperedges produced by A.
5. Causal invariance = order-independent (eventually isomorphic) causal graph, tied to confluence/Church–Rosser.
6. Multiway systems follow all rule applications in all orders; branchial space is the proposed bridge to quantum mechanics.

Supports concept pages

Hypergraphs & rewriting (formal) · update events & ordering · causal graphs (formal) · causal invariance & confluence · multiway systems · branchial graphs.

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