Research collaboration · prototype
An independent collaboration among researchers who have worked on topics around rewrite systems, hypergraphs, causality and emergent spacetime in connection to the Wolfram Physics Project. Its purpose is to document and preserve their contributions, support independent networking, and coordinate possible future independent research in related directions.
Not affiliated with or endorsed by Wolfram Foundation or Wolfram Research Inc. — it does not claim ownership of their names, projects, or materials.
Contributions
A high-level summary of contributions that can be traced to members of the collaboration, independently of the project's branding. Each is backed by the evidence in the detailed timeline.
2018 – 2026
Max PiskunovRichard Assar
The software that made the models computable. Max wrote SetReplace (2018–2021), the engine behind WolframModel, with causal graphs, multiway evaluation and the hypergraph plots used in the project’s publications. Richard wrote HypergraphRewritingEngine (2025–2026), a parallel and GPU multiway engine with exact isomorphism handling.
2020 – 2021
Max Piskunov
Multiway evolution within a single pool of tokens, without global state isomorphism, proposed and first implemented by Max; with it the expressions–events (token–event) graph, a generic multihistory type system, and a bulletin showing that confluence and causal invariance are not equivalent in general.
2020 – 2021
Xerxes Arsiwalla
Multiway rules as type constructors, completions as homotopies, and the rulial limit as an ∞-groupoid: a framing first documented in Xerxes’s Summer School 2020 project and developed with Jonathan Gorard in a series of papers, more than a year before the ruliad essay that cites them.
2023 – 2025
Xerxes Arsiwalla
Homotopy type theory as the building blocks of a theory of pregeometry, and a general theory of observers across physics and computation, in papers led by Xerxes with Hatem Elshatlawy and Dean Rickles.
2023 – 2026
Carlos Zapata-CarrataláUtkarsh BajajPavel HajekNikolay Murzin
Geometry derived from pure hypergraph rewriting. Carlos’s initial vision, from 2023, was a dictionary of differential geometry on discrete combinatorial structures, developed in discussion with Xerxes Arsiwalla and Jonathan Gorard. Stephen Wolfram supported it; he and Carlos came to differ on the approach and its mathematical foundations. Utkarsh worked on discrete difference operators and vector fields on graphs, and later de Rham cohomology of graphs. Nik took part in the live sessions from 2023 and in the 2025 InfraGeometry paclet review. From October 2025, under the Templeton grant Carlos led, Pavel was assigned to infrageometry under Wolfram’s direction, to develop Carlos’s dictionary from the point of view of minimal models of geometric phenomena on graphs. He proposed a pipeline for an emergent spacetime, with spatial and gauge directions, from a single-history branch of the multiway system for given observers, arrived at the notion of renormalizable geometry, and developed research repositories (Infrageometry, InfraAnalysis, InfraCausality, InfraGaugeTheory) as grant deliverables, with contributions from Nik, Dugan Hammock and student collaborators.
2023 live sessions2025 paclet reviewInfrageometry codeCarlos’s statementPavel’s statement
2021 – 2025
Carlos Zapata-CarrataláXerxes ArsiwallaPavel HajekNikolay Murzin
Associativity as a confluence property of hypergraph rewriting, hypermatrix algebra, and an insertion Lie bracket on hypergraphs, suggested by Carlos and Xerxes and generalized by Pavel in a Winter School project (rooted at a hyperedge, related to the commutator of linearized “blow-up” rewriting rules, with graded and higher-arity versions), implemented in the Hypergraph paclet by Nik, Carlos and Pavel.
Heaps of FishLie brackets on hypergraphsHypergraph pacletGraded bracket (commit)Pavel’s statement
2020 – 2025
Nikolay Murzin
Visualizing multispace with 3D multi-perspective layouts (credited to Nik by Stephen Wolfram), and the principal code of the Multicomputation, Quantum Framework, diagrammatic computation and lambda-calculus packages used across the multicomputational work.
2022 – 2024
Utkarsh Bajaj
Compiling automata, Turing and register machines and a functional language into hypergraph rewriting systems, and defining events and causal structure for λ-calculus evaluation through double-pushout hypergraph rewriting.
2024 – 2026
Carlos Zapata-Carratalá
Axiomatic relativity in the manner of Marzke and Wheeler applied to causal graphs from hypergraph rewriting: geometric clocks and signal functions in discrete causal graphs. Also the scientific framing of the Templeton grant “Causality, Observers and Computation”.
People
Current members, with others who worked on the foundations of the project expected to join.
Insertion Lie brackets on hypergraphs; infrageometry: minimal models of geometry on graphs, renormalizable geometry, emergent spacetime from a single-history branch; InfraCausality.
Author of SetReplace, the engine behind WolframModel; local multiway systems.
TimelineStatement forthcoming
Homotopy types and higher categories for multiway systems; pregeometry; observers.
TimelineStatement forthcoming
Multicomputation, quantum and diagrammatic software; multispace visualization.
TimelineStatement forthcoming
Computation as hypergraph rewriting; causal structure of λ-calculus; discrete geometry on graphs.
TimelineStatement forthcoming
Differential geometry and higher-arity algebra; infrageometry; causality and geometric clocks. Led the Templeton grant.
High-performance parallel and GPU multiway hypergraph rewriting.
TimelineStatement forthcoming
Active projects
Concluding
Marzke–Wheeler axiomatic relativity and geometric clocks for causal graphs generated by hypergraph rewriting, and the set-theoretic structure that connects them to the rewriting systems.
Carlos Zapata-Carratalá
Paper in preparation
Lie and Poisson structures on hypergraphs, towards a Hamiltonian formalism for hypergraph rewriting.
Pavel HajekCarlos Zapata-Carratalá
Ongoing
Discrete analogues of differential operators on graphs and hypergraphs, the starting point for several further developments in discrete differential geometry.
Utkarsh BajajCarlos Zapata-Carratalá
Ongoing
Which smooth geometries can arise as infrageometric limits of graphs generated by given rules? We study scaled Gromov–Hausdorff convergence of metric-measure spaces with specified coarse-graining maps, and ask when minimal models of geometric objects and their observables converge to smooth geometric structures and their corresponding observables. This includes weak convergence of the measures associated with vertex and edge densities of multiobjects to measures on limiting geometric objects. How does convergence depend on the rules and observer scale? What is the landscape of possible geometric limits for a given class of rules? Can we explain the emergence of linearity, Riemannian geometry, and symplectic geometry? Under what conditions do discrete constructions whose smooth counterparts agree approach the same result, such as dimension estimates from ball volumes and shell sizes or different constructions of scalar products? At finite observer scales, which smooth geometries best approximate the graph, and how can distance measurements, volume growth, curvature bounds, and topology constrain the possibilities?
Pavel Hajek
Proposed
How does spacetime emerge from hypergraph rewriting models for given observers? We study a single rewriting history through its causal structure, starting with a chosen observer chain and foliation. Nearby observer chains intersect the foliation to represent germs based at the original observer. From these data, we seek spatial slices fibered over ℤ³, with gauge degrees of freedom in the fibers and coarse-graining maps between descriptions at different observer scales. How does spatial connectivity arise from causal relations, and how do objects persist between successive slices? How do the resulting descriptions depend on the observer chain, foliation, and nearby chains? Can different observers obtain compatible descriptions of the same rewriting history? We aim to make the construction explicit and identify the conditions under which spatial, temporal, and gauge structures arise.
Pavel Hajek
Proposed
What are the simplest computational models that exhibit economic phenomena? Following the approach of infrageometry, we explore two directions. In one direction, we ask whether observable economic behavior can emerge from simple systems with resource constraints and local interactions. For example, consider a cellular automaton with an additional rule: whenever the number of black cells exceeds 20, five black cells in the densest region turn white. How does this constraint affect the automaton’s behavior, and how does it recover after these disaster events? Does deleting cells in the densest region have different consequences from deleting them randomly? Which observables have a meaningful economic interpretation, and how do their patterns vary across families of cellular automata? The other infra-direction represents economic states as hypergraphs and economic processes, such as production, exchange, and consumption, as elementary rewriting rules. Can these models recover familiar economic curves, and under what assumptions? Which minimal mechanisms determine stability, shortages, and recovery after disruption?
Pavel Hajek
Ongoing
This site: collecting personal statements from every member and public evidence for each stream of ideas, to be moved to its own domain.
Carlos Zapata-CarrataláMax PiskunovXerxes ArsiwallaNikolay MurzinUtkarsh BajajPavel HajekRichard Assar
Timeline
Main milestones. The detailed timeline lists every entry with its evidence and can be filtered by person.
First commit of SetReplace, Max’s engine for set (hypergraph) substitution systems; a C++ core with proactive match indexing follows in April 2019. It becomes the simulation engine of the Physics Project.
Wolfram Summer School 2019: Max’s project on confluent set substitution systems, built on SetReplace. At this Summer School, Max and Jonathan Gorard urge Stephen Wolfram to restart the physics project.
SetSubstitutionSystem is renamed WolframModel in SetReplace: the function used throughout the project’s publications.
Stephen Wolfram announces the Wolfram Physics Project. The announcement names Jonathan Gorard and Max Piskunov as the two young physicists who, at the 2019 Summer School, urged him to pursue it.
Wolfram Summer School 2020: Nik’s project on multi-perspective 3D graph embeddings and multispace (function MultispacePlot3D). Stephen Wolfram credits the idea of visualizing multispace with a 3D layout to him.
Wolfram Summer School 2020: Xerxes’s project “Homotopic foundations of Wolfram Models” identifies multiway and rulial structures with ∞-groupoids and an ∞-topos, via homotopy type theory (mentor: Jonathan Gorard).
Jonathan Gorard’s bulletin on homotopy type theory and the Wolfram model is “a writeup of work done in collaboration with Xerxes Arsiwalla and Stephen Wolfram”. It credits Xerxes with treating multiway rules as type constructors and with observing that categorical quantum mechanics arises from foliating the rulial multiway system.
“Local Multiway Systems: A New Approach to Wolfram Model Evolution”. Stephen Wolfram writes that Max “has suggested instead the idea of local multiway systems” and did the first implementation.
First public commit of the Wolfram Quantum Framework; the early commit history (quantum bases, states, operators, circuits) is Nik’s. He presents it with Mads Bahrami at the 2022 Wolfram Technology Conference.
“Pregeometric Spaces from Wolfram Model Rewriting Systems as Homotopy Types”, with Jonathan Gorard (Int. J. Theor. Phys., 2024). Six days later, the essay introducing the ruliad cites it.
Nik starts the Multicomputation package (the Multi container and MultiwaySystem); he is its principal code author per the public commit history.
Wolfram Summer School 2022: Utkarsh compiles models of computation (finite automata, Turing and register machines, a small functional language) into hypergraph rewriting systems.
After joining the Wolfram Institute, discussions with Xerxes Arsiwalla and Jonathan Gorard lead Carlos to the idea of infrageometry: a dictionary of conventional differential geometry built on discrete combinatorial data structures, with pure rewrite rules as the computational framework. This dictionary is Carlos’s initial vision for the project, which Stephen Wolfram supports. Carlos leads the conceptual framework; Jonathan contributes applications in physics.
“Infracalculus: Beyond Discrete and Continuous?”, the first of a public series of live sessions on calculus and geometry on hypergraphs with Jonathan Gorard (19 September: “Infrageometry: Beyond Discrete and Continuous?”).
Carlos begins a grant application to the John Templeton Foundation, coordinated with Nicolas Behr, Josh Grochow and Alexander Kolpakov. Its two pillars: metamathematical questions from the Arity Science programme, and infrageometry, deriving geometry-like structures from pure hypergraph rewriting.
“Hypergraph rewriting and Causal structure of λ-calculus”: events and causality for double-pushout hypergraph rewriting, and an algorithm for causal relations in λ-evaluation.
Wolfram Winter School 2025: Pavel’s project “Lie brackets on hypergraphs” generalizes the insertion Lie bracket to hypergraphs and induces Poisson brackets on functions of hypergraphs, asking for a Hamiltonian formalism for hypergraph rewriting. It thanks “Carlos Zapata-Carratalá for setting the direction of the project” and Nik Murzin for pair programming. By Pavel’s account, the bracket was suggested by Carlos and Xerxes; he rooted it at a hyperedge, related it to the commutator of linearized “blow-up” rewriting rules, and explored graded and higher-arity versions and the representability of Lie algebras on subsets of rules.
The John Templeton Foundation awards “Causality, Observers and Computation: A New Approach to Fundamental Physics and Metamathematics” (grant 63655, October 2025 to September 2027), with Carlos Zapata-Carratalá and Stephen Wolfram as project leaders.
First commit of HypergraphRewritingEngine, Richard’s parallel multiway hypergraph rewriting engine with exact isomorphism handling; a CUDA GPU engine and an interactive visualiser follow in December 2025.
The Templeton project starts and Pavel joins the Wolfram Institute as a research fellow on it; he and Carlos hold several sessions ideating and discussing research directions. His first contributions are to the Wolfram Function Repository (among them GromovHausdorffDistance), the DiscreteGeometry paclet and the P vs NP blog project, which thanks him for his help.
Pavel publishes the Infrageometry code base (Infrageometry, SyntheticInfrageometry, InfraAnalysis, InfraCausality, InfraGaugeTheory), supported by Templeton grant 63655. By his account, he was assigned to infrageometry under the direction of Stephen Wolfram, to develop Carlos’s dictionary from the point of view of minimal models of geometric phenomena on graphs. After initial meetings with Wolfram and Nik Murzin, he proposed a concrete pipeline for obtaining an emergent spacetime, with spatial and gauge directions, from a single-history branch of the multiway system for given observers. He implemented observer-scale dependence in minimal models of spatial geometry, arriving at the notion of renormalizable geometry, and a minimal model of fibered graphs, which he began relating to lattice gauge theory with his student collaborator Ioana Milea. He started InfraCausality, a toolkit for discrete special relativity on causal graphs, as a grant deliverable; the repositories also include contributions from Nik Murzin, Dugan Hammock and student collaborators.
“Discrete geometry and de-Rham cohomology of graphs”, Utkarsh’s Oxford MSc dissertation: forms on graphs with coordinate systems and a de Rham theorem (external supervisor: Xerxes Arsiwalla).
Carlos’s position at the Wolfram Foundation ends, and with it his role as project leader of the Templeton grant.
Multi-History Science starts as a research collaboration, with this website and a YouTube channel. For the Arity Science project, whose collaborators are among its members, it is a branching out and a continuation of its research streams.
First Hyperseminar, the seminar series of the Multi-History Science collaboration.
Computational resources
Software written by members of the collaboration for hypergraph rewriting, multiway systems and related mathematics.
Wolfram Language + C++ · 2018–2025
Engine for set and hypergraph substitution systems: WolframModel, causal and token–event graphs, multiway and local multiway evolution, hypergraph plots.
Max Piskunov
C++ / CUDA · 2025–2026
Parallel and GPU multiway hypergraph rewriting with exact canonicalization, deterministic results and an interactive visualiser.
Richard Assar
Wolfram Language paclet · 2023–2026
Hypergraphs, hypergraph rewriting and hypermatrix algebra, including the hypergraph insertion bracket.
Nikolay MurzinCarlos Zapata-CarrataláPavel Hajek
Wolfram Language paclet · 2022–2026
Multiway systems and the Multi container for multicomputational models.
Nikolay Murzin
Wolfram Language paclet · 2021–
Quantum states, operators, circuits and measurements; Nik wrote its early code base, with the Wolfram quantum computation team.
Nikolay Murzin
Wolfram Language paclets · 2020–2025
Compositional string-diagram calculus (with Ian Ford) and lambda-calculus diagrams and trees.
Nikolay Murzin
Wolfram Language · 2026
Experimental code for geometry emerging from large graphs at large observer scales, with companion repositories for causality, analysis and gauge theory.
Pavel Hajek
Python notebook · 2026
A browser notebook for hypergraph rewriting and higher-arity mathematics, from the Arity Science programme.
Carlos Zapata-Carratalá
C++ · 2020–2021
Fast evolvers for Post tag systems and combinators, used in the corresponding studies of computational irreducibility.
Max Piskunov
Organisation
The collaboration's shared GitHub organisation, for its code, notebooks and data.
Open on GitHub →