Quantum Mechanics, Observers, and the Ruliad: An Exposition of the Wolfram Physics Project

The Starting Point: Hypergraph Rewriting

The Wolfram Physics Project begins from a strikingly austere premise: the fundamental structure of the universe is a hypergraph — a generalization of a graph in which edges can connect arbitrary non-empty subsets of vertices — and the dynamics of the universe consist of rewriting rules applied to this hypergraph. A “law of physics,” in this framework, is simply a rule of the form H₁ → H₂, where a subhypergraph matching pattern H₁ is replaced by a distinct subhypergraph matching pattern H₂.

The conceptual parallel to combinatory logic is worth noting: just as in combinator calculus no distinction is drawn between a program and its data (the program executes by applying symbolic transformation rules to its own specification), the Wolfram model draws no distinction between the “background” structure of space and the “foreground” structure of physical processes. Everything — space, time, matter, quantum states — is described entirely in terms of symbolic transformations on hypergraphs.

From this single dynamical primitive, the project claims to recover the major structures of known physics. Space emerges as the large-scale limit of the hypergraph — in much the same way that a continuous fluid emerges from a large collection of discrete molecules. The only information available to define the structure of space is the connectivity of the hypergraph; there is no predefined embedding, no background manifold, no a priori notion of dimension or distance. The dimension of space at a point is determined by counting: one picks a vertex, counts the number of vertices N(r) within combinatorial distance r of it, and reads off the dimension n from the scaling N(r) ~ ar^n. A “flat” region of the hypergraph will exhibit this power-law scaling exactly; a curved region will show deviations. The deviation is precisely captured by the Ollivier-Ricci curvature — a generalization of Riemannian Ricci curvature to discrete metric spaces, defined in terms of optimal transport (Wasserstein distance) between probability measures on neighboring balls. The key formula is that the number of vertices within distance r of a point in a curved spatial hypergraph satisfies N(r) = ar^n(1 - R·r²/6(n+2) + ...), where R is the Ollivier-Ricci scalar curvature. This is the discrete analog of the standard Riemannian result relating ball volumes to curvature.

Time emerges from the causal graph: a directed acyclic graph in which each vertex is a rewriting event and each edge encodes a causal dependency (event B causally depends on event A if B’s input uses hyperedges that were produced by A’s output). The causal graph is the invariant physical content — different choices of updating order correspond to different foliations of the same causal graph, just as different coordinate charts correspond to different descriptions of the same spacetime. Gorard’s central result for general relativity is that, when one extends the geodesic ball analysis from spatial hypergraphs to causal graphs (using discrete analogs of parallel transport and the Riemann curvature tensor), the requirement that the causal graph maintain a fixed limiting dimension is equivalent to the satisfaction of a discrete form of the Einstein field equations. The derivation is closely analogous to the Chapman-Enskog procedure that derives the Navier-Stokes equations of continuum fluid mechanics from discrete molecular dynamics: the large-scale emergent equations are not put in by hand but follow from the microscopic dynamics under an appropriate coarse-graining.

With this picture of how space, time, and gravity emerge from a single deterministic path through the multiway system, we can now turn to the question of what happens when we consider all paths. And quantum mechanics — the subject of this article — emerges from the inherent non-determinism of the rewriting process.

The Source of Quantum Behavior: Non-Determinism and Multiway Systems

Here is the central observation: given a hypergraph and a rewriting rule, the order in which to apply the rule is generically not well-defined. At any given step, there may be multiple matching subhypergraphs to which the rule could be applied, and different choices will produce different successor states. The evolution is therefore non-deterministic.

Rather than choosing a single updating order (which would correspond to fixing a gauge in the general-relativistic interpretation), the Wolfram model parametrizes all possible evolution histories simultaneously using a multiway system: a directed acyclic graph in which each vertex is a hypergraph state and each edge represents a single rewrite step. A single path through the multiway system corresponds to a single deterministic evolution history — a particular choice of updating order, or equivalently, a particular gauge. The multiway system as a whole encodes the totality of possible histories.

The claim, then, is that this multiway system is the quantum mechanical object. More precisely: the branching structure of the multiway system, which parametrizes the non-determinism of hypergraph rewriting, plays the role that superposition and entanglement play in standard quantum mechanics. A single branch is a classical history; the multiway system is the quantum state. This is more than a suggestive analogy — the correspondence with the Feynman path integral, in which the trajectory of a quantum system is described by a functional integral over all possible paths weighted by their amplitudes, is taken seriously and eventually proved to hold in the continuum limit.

Confluence, Causal Invariance, and Their Tension

The relationship between quantum mechanics and general relativity within this framework hinges on a fundamental tension in the theory of abstract rewriting systems: the tension between confluence and non-confluence.

A rewriting system is confluent (equivalently, has the Church-Rosser property) if, whenever two distinct rewrite sequences diverge from a common ancestor, they can always be brought back together to some common descendant. In the Wolfram model, confluence of the hypergraph rewriting rule implies causal invariance — the property that the causal graph (encoding the structure of spacetime) is the same regardless of which path through the multiway system one follows. Causal invariance is the discrete analog of general covariance, and is the condition from which discrete general relativity is derived.

It is worth pausing on what “general covariance” means here, since the analogy is doing real work. In continuum general relativity, general covariance is the principle that the laws of physics take the same form in all coordinate systems — or, equivalently, that the physical content of the theory is invariant under arbitrary smooth diffeomorphisms of the spacetime manifold. The point is that coordinates are a human bookkeeping device, not a physical observable: you can label events in spacetime however you like, and the geometry — encoded in the metric tensor and its curvature — must be the same regardless. Two observers using wildly different coordinate charts will disagree about the numerical values of metric components at a given point, but they will agree on all coordinate-invariant quantities: geodesic lengths, curvature scalars, causal orderings between events. The physics lives in the causal structure, not in the labeling.

In the Wolfram model, the analog of a coordinate choice is an updating order — a decision about which matching subhypergraph to rewrite first at each step. Different updating orders produce different intermediate hypergraph states (just as different coordinate charts produce different component representations of the metric), but if the rewriting rule is causally invariant, the causal graph — the directed acyclic graph encoding which rewriting events were prerequisites for which others — comes out the same regardless. The causal graph is the coordinate-invariant content; the updating order is the gauge. Gorard’s earlier work proved that this discrete causal invariance implies discrete analogs of Lorentz covariance and the Einstein field equations, establishing that the Wolfram model can reproduce general relativity. The quantum story begins precisely where this invariance breaks down.

But quantum mechanics requires non-confluence. The evolution of a quantum system must involve genuine branching — distinct evolution histories that do not reconverge — because it is precisely this branching that encodes superposition. A globally confluent multiway system would correspond to a purely classical universe: one in which, no matter how the evolution proceeds, the same outcome is always reached.

The Wolfram model’s resolution of this tension is both subtle and deeply connected to the theory of measurement. The idea is that the fundamental rewriting rule need not be confluent, but that an observer — by virtue of being a macroscopic structure embedded within the multiway system — necessarily imposes effective confluence through a process analogous to the Knuth-Bendix completion algorithm from universal algebra.

Measurement as Knuth-Bendix Completion

The Knuth-Bendix completion algorithm is a procedure from term rewriting theory that takes a non-confluent rewriting system and attempts to make it confluent by adding new rewrite rules. Specifically, it identifies unresolved “critical pairs” — pairs of states that arise from a common ancestor via different rewrite sequences but that fail to reconverge — and adds rules that force them to converge, thereby imposing an equivalence relation between the two branches.

In the multiway interpretation, Gorard proposes that this completion procedure is formally equivalent to quantum measurement. The key ideas are as follows.

States on distinct branches of the multiway system correspond to orthonormal eigenstates, and the overall multiway evolution corresponds to the unitary evolution of a linear superposition. At the discrete level, each state in a branchlike hypersurface (explained shortly) can be assigned a weight given by the number of distinct evolution paths leading to it through the multiway graph. These path counts are positive integers, and so they cannot directly serve as quantum amplitudes — which must be complex numbers, capable of destructive interference and cancellation. The path counts should be understood as the magnitudes of discrete proto-amplitudes. The full complex-valued amplitude structure emerges only in the continuum limit, once the branchial geometry acquires its complex metric signature (as described in the section on convergence to projective Hilbert space below). Roughly speaking, the phases come from the geometry of the multiway causal graph — specifically, from the imaginary component of the multiway-Minkowski norm, which encodes branchlike separations. The angle quantity e^{iS} that appears in the path integral represents the dispersion angle of geodesic bundles propagating through multiway space, and it is the interaction between these phases across different paths that produces interference. In this sense, interference is not visible at the level of a single branchlike hypersurface; it is a property of the geometry of the space in which those hypersurfaces are embedded.

With this caveat in place, the multiway Born rule gives the probability of “collapsing” to a given eigenstate as the squared magnitude of its amplitude, and the overall framework proceeds as follows.

To state the next idea, we need the concept of a branchlike hypersurface. Recall that a multiway system is a directed acyclic graph whose vertices are hypergraph states and whose edges are rewriting steps, with the graph branching whenever a state has more than one applicable rewrite. This graph has a natural “downward” direction — the direction of time — and so it can be sliced horizontally at any given moment, yielding the set of all states that coexist at that time step. Such a horizontal slice is a branchlike hypersurface. The terminology is chosen by analogy with general relativity: just as a spacelike hypersurface in a Lorentzian manifold is a slice of spacetime at a single moment (a Cauchy surface), a branchlike hypersurface is a slice of the multiway system at a single moment — but now “space” has been replaced by “branch,” and the slice captures not the spatial extent of the universe but the full spread of simultaneously coexisting quantum branches. Two states on the same branchlike hypersurface are separated “in the branch direction” rather than in time; they represent alternative versions of the universe that exist concurrently.

An observer, conceptually, is any persistent structure within the multiway system that perceives a single, definitive evolution history. The observer is not external to the system but is an extended macroscopic entity embedded within it — a subsystem whose internal dynamics must, from its own perspective, appear causally invariant. In order to maintain a coherent internal representation of the world, the observer must have undergone sufficient coarse-graining (branch pair completion) for at least their own local branch pairs to converge.

The mathematical operationalization of this idea is that each observer induces a particular branchlike foliation of the multiway graph: an ordered sequence of non-intersecting branchlike hypersurfaces covering the entire graph, with each hypersurface representing the observer’s notion of “all branches that exist right now.” Different observers — corresponding to different patterns of coarse-graining, or equivalently, different orderings of measurement events — will in general induce different foliations, just as different observers in general relativity induce different foliations of spacetime into spacelike hypersurfaces. The foliation is not the observer; it is the observer’s reference frame.

The critical insight is that if the observer is sufficiently macroscopic — that is, if their internal hypergraph structure constitutes a statistically representative sample of the universe’s hypergraph structure — then the minimal set of completion rules needed to make the observer’s internal representation causally invariant will also make the rest of the universe appear causally invariant from that observer’s perspective. Measurement, in this picture, is the process by which the observer’s coarse-graining forces distinct branches of the multiway system to be identified, collapsing the superposition to a single effective evolution history.

This has the pleasing consequence that the statement of correctness of the Knuth-Bendix algorithm can be interpreted physically: at least in certain cases, sufficient coarse-graining can make the evolution of a quantum-mechanical system appear macroscopically classical. The new rules added by the completion process can also make previously unreachable states of the multiway system accessible — these correspond to quantum interference effects between neighboring branches.

The Uncertainty Principle from Non-Commuting Rewrites

The uncertainty principle emerges naturally from the algebraic structure of rewriting systems. Two rewriting relations are said to commute if, whenever they diverge from a common state, the resulting branches can always be made to reconverge (regardless of the order in which the relations are applied). A rewriting system is confluent if and only if it commutes with itself — this is, in a sense, the content of the Hindley-Rosen commutative union theorem.

In a non-confluent multiway system, there will exist pairs of updating events that do not commute: the outcome of the evolution depends on the order in which these events are applied. Interpreting rewrite relations as linear operators on the (projective) Hilbert space that the multiway system limits to, one obtains the standard commutator structure of quantum mechanics. If two observables correspond to commuting rewrite relations, they can be measured simultaneously with unlimited precision — the time-ordering of the corresponding updates does not matter. If they correspond to non-commuting rewrite relations, the time-ordering affects the outcomes, and measurements of the two observables necessarily take place on distinct branches of multiway evolution. This places a fundamental limit on the precision with which both can be simultaneously prepared, yielding the uncertainty relation.

The unit branchlike distance in the multiway graph is identified with iℏ, a claim that is justified by Sylvester’s law of inertia for quadratic forms (which forces the multiway metric to have complex signature) and by the interpretation of ℏ as the maximum rate of quantum entanglement between global states in the multiway system — an interpretation that is proved to be consistent with the Margolus-Levitin bound on the minimum evolution time between orthogonal quantum states.

Branchial Geometry and the Convergence to Projective Hilbert Space

To make the analogy between multiway systems and quantum mechanics into a theorem, one must prove that the geometry of the multiway system converges to that of complex projective Hilbert space in the continuum limit. Gorard’s proof of this convergence proceeds through several stages.

First, the natural distance on branchial hypersurfaces — the combinatorial metric measuring the branch pair ancestry distance between states — is shown to be equivalent, up to a multiplicative constant, to the information distance from algorithmic complexity theory: the minimum length of a program that transforms one state into the other and vice versa. This is itself a natural extension of Kolmogorov complexity to the bivariate case.

A reasonable objection here is that information distance, being defined in terms of Kolmogorov complexity, is uncomputable — so in what sense can it serve as a metric on a physical space? The answer is that the information distance is not being proposed as a procedure for computing distances in a finite multiway system. At the discrete level, you simply use the combinatorial graph distance on the branchial graph (the shortest path between two vertices), which is trivially computable. The information distance enters the argument only as a characterization of the continuum limit — it identifies what the combinatorial distance converges to as the multiway system grows large, thereby connecting the discrete branchial geometry to the well-studied geometry of statistical manifolds. One might wonder whether restricting to a complexity-bounded variant of information distance (say, polynomial-time computable programs relative to the size of the multiway system) would alter the limiting geometry. For the purposes of the convergence proof, it does not: the asymptotic equivalence between the combinatorial metric and the information distance holds up to an additive logarithmic correction that vanishes in the continuum limit regardless of the complexity bound. But the question of whether the finite-size corrections to the branchial geometry carry computability-theoretic signatures is genuinely open, and potentially interesting for the relationship between computational complexity and quantum mechanics that the Wolfram model claims to illuminate.

Second, in the continuum limit, the discrete information distance converges to the Fisher information metric — the canonical Riemannian metric on statistical manifolds that quantifies the distance between probability distributions.

Third, when one extends from ordinary Riemannian manifolds to complex projective spaces, the Fisher information metric becomes the Fubini-Study metric — the natural Kähler metric on projective Hilbert space, which (when restricted to pure states) reduces to the quantum Bures metric used in quantum information theory.

The remaining step is to explain why the limiting space is projective rather than merely a flat Hilbert space. The answer lies in the structure of the multiway causal graph — the full object that encodes causal relations between rewriting events across all branches of the multiway system, not just within a single evolution history. This graph has three distinct types of separation between events: spacelike, timelike, and branchlike. By Sylvester’s law of inertia (and its generalization by Ikramov to normal matrices), maintaining discrimination between all three separation types across all possible observers requires the metric to have eigenvalues with at least three distinct signs, necessitating the use of complex numbers. The resulting “multiway-Minkowski norm” has the form ‖(t, x, y)‖ = ‖x‖² + i‖y‖² - (1+i)t², where x are spatial coordinates and y are branchial coordinates. The projectivity of the Hilbert space then arises from modding out by the multiplicative group of nonzero complex numbers — a conformal rescaling that leaves the combinatorial structure of the multiway graph invariant.

This lattice-theoretic route to projective geometry — connecting modularity of the multiway lattice (a weakening of distributivity equivalent to the strong diamond property for rewriting systems) with the axioms for continuous geometry in the sense of von Neumann — provides a purely combinatorial characterization of the structure that, in the continuum limit, yields the Hilbert space formalism of quantum mechanics.

Multiway Relativity: Quantum Mechanics as General Relativity in Branchial Space

With the geometry of branchial space established, the central conjecture of the Wolfram Physics Project’s quantum mechanics program can be stated precisely: quantum mechanics is the analog of general relativity for the multiway graph, with branchlike hypersurfaces playing the role of spacelike hypersurfaces, and the Fubini-Study metric taking the place of the spacetime metric tensor.

This principle is called multiway invariance: the requirement that the ordering of timelike-separated measurement events in the multiway graph is preserved under changes of observer (i.e., under different choices of branchlike hypersurface foliation), even though the ordering of branchlike-separated events is not. It is the direct analog of Lorentz invariance, but for “branchtime” rather than spacetime.

This gives rise to a rich set of correspondences. The quantum Zeno effect — the slowing of quantum evolution under rapid repeated measurements — plays the role of gravitational time dilation: as the observer constructs an increasingly steep foliation of the multiway graph (corresponding to increasingly frequent measurements), the perceived evolution of quantum states slows, preventing the maximum entanglement rate from being exceeded. Wave-particle duality arises from the fact that spacelike separation is a special case of branchlike separation in the multiway causal graph: geodesic bundles can appear to correspond to collections of test particles, wave packets, or both, depending on the observer’s choice of multiway causal foliation. The path integral emerges because the multiway evolution is literally a sum over all possible rewriting histories, with path weights determined by the multiway norm.

Elementary particles are interpreted as persistent localized structures in the spatial hypergraph — topological defects preserved by the rewriting rules. The Robertson-Seymour theorem (which states that graph families closed under taking minors are characterized by finite sets of forbidden minors) is interpreted as a combinatorial analog of Noether’s theorem: each forbidden minor corresponds to a conserved quantity, and hence a distinct particle type. The dynamics of these excitations is governed by a discrete Schrödinger equation — a diffusion equation for the path-weight scalar field on the hypergraph, with a discrete Laplacian and an additional potential term, whose Green’s function converges in the continuum limit to the standard non-relativistic propagator.

Bell’s theorem is satisfied through the nonlocality of the multiway causal graph: causal connections exist not only between events on the same evolution branch but also between events on distinct branches, yielding an explicitly nonlocal (but deterministic) theory that violates the CHSH inequality in the same manner as de Broglie-Bohm theory.

The Observer as the Source of Physical Law

A recurring theme in everything described so far is that the observer is not incidental to the physics but constitutive of it. The Second Law of thermodynamics, the continuity of space, the definiteness of measurement outcomes, the uncertainty principle — none of these are properties of the Wolfram model’s raw dynamics. They are properties of what the raw dynamics looks like to an observer with particular characteristics. This is perhaps the most philosophically radical feature of the framework, and it deserves a section of its own.

What are those characteristics? Two features are identified as essential. First, the observer is computationally bounded: they cannot reverse-engineer the computationally irreducible processes unfolding at the level of individual hypergraph rewrites. They are forced to coarse-grain — to treat vast numbers of microscopically distinct states as equivalent — because they lack the computational resources to distinguish them. Second, the observer is persistent in time: they knit together perceptions at successive moments into a single thread of experience, maintaining an identity across the multiway branching rather than fragmenting into separate copies at each bifurcation. These two properties — boundedness and persistence — are what it takes to be an observer in the multiway sense. They are also, as Wolfram has argued, closely related to what we ordinarily mean by consciousness: the subjective sense of being a single locus of experience progressing through time.

Persistence in time is what gives rise to the apparent classicality of the macroscopic world. Because the observer insists on maintaining a single coherent thread of experience, they must continually resolve the branching of the multiway system into a definite history. This resolution is the Knuth-Bendix completion described earlier — but now understood not as an external operation imposed on the system, but as an inevitable consequence of what it means to be a persistent observer embedded within the system. The observer does not choose to collapse the wave function. They cannot help but collapse it, because their own persistence requires it.

This observer-dependence of physical law has an important consequence for the status of quantum mechanics within the framework. The claim is not that the multiway system “generates” quantum mechanics in the way that a differential equation generates its solutions. The claim is that quantum mechanics is what the multiway system looks like to an observer with the two key properties — boundedness and persistence — just as fluid mechanics is what molecular dynamics looks like to an observer who cannot track individual molecules. Different observers, with different computational capabilities or different persistence structures, would in principle extract different effective laws from the same underlying dynamics.

Branchial Graphs and the Categorical Structure

The categorical structure of branchial space has been established rigorously. Gorard and collaborators prove that the multiway system carries the algebraic structure of a dagger-symmetric, compact-closed monoidal category — precisely the categorical structure identified by Abramsky and Coecke as sufficient for formulating quantum mechanics. The tensor product arises from parallel composition on neighboring branches; the dagger from inversion of multiway edges; the compact structure from hypergraph duality. These primitives suffice to define all the standard algebraic operations of finite-dimensional quantum mechanics.

The branchial distance also satisfies the axioms of an entanglement monotone: it is zero for fully-separable states, maximal for maximally-entangled states, and invariant under local unitaries. This is proved using the Cocchiarella et al. geometrical entanglement measure based on generalized Gell-Mann matrices.

Connection to the ZX-Calculus and Quantum Circuits

To test consistency with standard quantum-circuit formalism, the authors develop a detailed embedding of the ZX-calculus — a complete and sound diagrammatic language for reasoning about linear maps between qubits — into the multiway framework. ZX-diagrams are represented as nested operator expressions, and ZX-calculus rewriting rules compile directly as multiway operator system rules. The multiway evolution then represents the space of all possible diagrammatic proofs.

The monoidal structure of the multiway system (given by rulial composition) is shown to be compatible with the monoidal product of ZX-diagrams — first computationally across hundreds of examples, then proved in general via Dixon-Kissinger typed open graphs.

A practical dividend is a novel algorithm for automated diagrammatic reasoning: combining generalized Knuth-Bendix completion (with Bachmair-Ganzinger superposition and causal edge density optimization) with Kerber’s higher-order-to-first-order reduction, the authors obtain a theorem-prover achieving approximately quadratic speedup for simplifying quantum circuits with up to 3000 gates.

Entanglement Entropy in Discrete Spacetime

The deepest test comes from computing entanglement entropy. In the causal set approach, this is done via the Sorkin-Johnston construction: a free scalar field theory on the causal set, with creation/annihilation operators derived from the eigendecomposition of the Pauli-Jordan operator, a distinguished SJ vacuum, and Wightman two-point functions from which entropy is extracted via a generalized eigenvalue problem. The resulting entropy is manifestly covariant and ultraviolet-finite.

In the Wolfram model, an independent entropy comes from the Fubini-Study metric on causal branchial graphs — branchial graphs whose vertices are complete causal histories rather than instantaneous hypergraph states, making the definition manifestly covariant.

The central result — substantiated analytically and numerically across large classes of rules — is that these two definitions are monotonically related. A technical subtlety: algorithmically-generated causal sets need not have integer Hausdorff dimension, requiring analytic continuation of the Green’s functions to non-integer dimension (analogous to dimensional regularization), with uniqueness by Carlson’s theorem.

The Ruliad: Why the Choice of Rule Shouldn’t Matter

There is one further level of structure in the Wolfram model that reframes the entire enterprise, and that bears directly on the question of what it would mean for the project to “succeed.” The constructions described so far — multiway systems, branchial graphs, the convergence to projective Hilbert space — all presuppose a particular hypergraph rewriting rule. But the Wolfram model’s deepest claim is that the large-scale features of physics (general relativity, quantum mechanics, thermodynamics) should not depend on the specific rule chosen. They should be rule-invariant — emergent properties of the rewriting process itself, not of any particular instantiation of it.

The object that makes this claim precise is called the Ruliad. Where a multiway system parametrizes all possible evolution histories for a given rule, the Ruliad parametrizes all possible evolution histories for all possible rules simultaneously. It is the entangled limit of all possible computations: the unique mathematical object obtained by running every conceivable rewriting rule on every conceivable initial condition, and taking the multiway system of the result.

The Ruliad is, by construction, unique — there is only one, since it already contains every possible rule. It is also, at least conjecturally, computationally universal in the strongest possible sense: every computation that can be performed by any formal system is already embedded somewhere within it. Rickles, Elshatlawy, and Arsiwalla, in their philosophical examination of the framework, characterize it as a purely syntactic structure defined independently of any a priori geometric notions — unlike, say, Wheeler’s superspace (whose “points” are already three-dimensional Riemannian geometries), the Ruliad presupposes no spatial, temporal, or geometric structure at all. Everything geometric emerges from the observer’s sampling of it.

This brings us to a point that is philosophically crucial and often misunderstood. The Ruliad, considered “from the outside” — as a totality — has no physics. It contains every possible computation, and therefore, in a sense, every possible law. But a law of physics is not a feature of the Ruliad in itself; it is a feature of what the Ruliad looks like to a particular observer at a particular location within it. The observer’s position is specified along three independent axes: their position in physical space (which part of the hypergraph they occupy), their position in branchial space (which branch of the multiway evolution they are on), and their position in rulial space (which neighborhood of rules they are sampling). Just as our position in physical space determines which galaxies we see, and our position in branchial space determines which quantum measurement outcomes we perceive, our position in rulial space determines which effective laws of physics we experience.

The claim — which functions as a universality conjecture — is that for all observer-positions in rulial space that correspond to entities capable of doing physics (entities with sufficient computational boundedness and persistence to be observers in the sense defined earlier), the large-scale features of the experienced physics are the same: they all see something that looks like quantum mechanics, something that looks like general relativity, and so on. This is analogous to the universality of critical exponents in statistical mechanics, or to the way that the Navier-Stokes equations describe fluid flow regardless of whether the underlying molecules are water, air, or helium — the macroscopic behavior is insensitive to microscopic details, because it arises from generic features of the coarse-graining process.

A natural question arises here: what ensures locality in rulial space? In physical space, locality is enforced by the causal structure — the speed of light limits which events can influence which others, defining light cones and preventing an observer from being affected by arbitrarily distant happenings. Is there an analogous mechanism preventing an observer in the Ruliad from being subject to arbitrary rewrites under rules far from their own? The partial answer seems to be that computational boundedness plays the role that the speed of light plays in physical space. An observer can only sample a finite neighborhood of rulial space — only rules reachable by a small number of rule-transformation steps from their own effective rule — because sampling further would require computational resources they do not have. Just as light cones delimit causal accessibility in spacetime, something like “rulial cones” delimit computational accessibility in rulial space. But there is a circularity worry: the notion of “closeness” in rulial space depends on the rulial metric, which depends on the structure of the rulial multiway system, which in turn depends on the observer’s coarse-graining. The (∞,1)-topos construction discussed below gestures toward a resolution — the Grothendieck topology on the classifying space of rulial multiway systems provides a formal notion of locality that does not presuppose a particular observer — but whether this fully resolves the circularity remains an open question.

Rickles et al. connect this to the tradition of second-order cybernetics: the recognition that no science is possible from a “view from nowhere,” and that the observer must be included in any adequate description of reality. The Ruliad, viewed from nowhere, is an abstract totality. To extract physics from it, one must specify an observer — and the observer, being a part of the Ruliad, is itself a computation within the structure it is observing. This self-referential character (the observer modeling a universe that includes the observer) is a feature, not a bug: it is what makes the framework’s predictions observer-relative without being subjective, since the constraints on what counts as an observer (boundedness, persistence) are themselves objective structural conditions.

If the universality conjecture holds, then the question “which rewriting rule is the correct law of physics?” is, in a precise sense, the wrong question — or at least an observer-dependent one. Different observers at different positions in rulial space may be described by different effective rules, but the large-scale structure is invariant. The open problem, then, is not to find a specific rule that reproduces the Standard Model, but to prove that Standard Model-like structure (non-abelian gauge symmetries, chiral fermions, three generations, and so on) is generic across the relevant region of rulial space — that it follows from the structure of the Ruliad itself, rather than from a fine-tuned choice of rule.

The monoidal structure results discussed earlier — showing that the category of multiway evolution graphs inherits a tensor product compatible with the ZX-calculus — can be understood in this light as a step toward such a universality proof: they show that the quantum-mechanical compositional structure is not an artifact of a particular rule or a particular encoding, but a structural feature of multiway evolution as such.

There is also a deeper mathematical formalization of the Ruliad that deserves mention, since it connects the Wolfram model to some of the most sophisticated machinery in contemporary mathematics. Arsiwalla and Gorard have shown that multiway rewriting systems can be expressed as homotopy types, with rewriting rules playing the role of type constructors and paths through the multiway graph corresponding to proofs of equality between terms. Higher homotopies — homotopies between homotopies, and so on — are constructed by iteratively including higher-order rewriting rules from rulial space. A multiway system equipped with homotopies up to order n is formalized as an n-fold category; the n → ∞ limit, upon inclusion of invertible rewriting relations, yields an ∞-groupoid. By Grothendieck’s homotopy hypothesis (which identifies ∞-groupoids with homotopy spaces), the limiting rulial multiway system thus inherits the structure of a formal homotopy space. The construction extends further: the “classifying space” of all rulial multiway systems — a multiverse of multiway systems — carries the structure of an (∞,1)-topos, the natural home of classical homotopy theory. When equipped with a Grothendieck topology and appropriate adjoint functors, this topos can support cohesive structures in the sense of Schreiber, which synthetically define formal geometric spaces (smooth sets, continuous sets) — providing a purely combinatorial, constructivist route from rewriting rules to the geometric spaces on which physics is formulated. This higher-categorical formalization does not re-derive the emergence of physical or branchial space (which, as discussed at the outset, follows from the Ollivier-Ricci curvature analysis of geodesic balls in hypergraphs and causal graphs for a fixed rule). What it adds is a rigorous justification for why the limiting structures have the right formal properties to support physics: the ∞-groupoid structure ensures that the spaces in question are genuine homotopy spaces, not merely combinatorial graphs that happen to look like them, and the (∞,1)-topos framework provides the categorical environment in which constructions from synthetic geometry — and potentially from higher gauge theory and axiomatic quantum field theory — can be carried out internally.

What This Is, and What It Isn’t

It is worth being clear about the status of these results. What has been established rigorously is:

1. The geometry of the multiway evolution graph converges to that of complex projective Hilbert space in the continuum limit, with the Fubini-Study metric emerging from the discrete information distance on branchial hypersurfaces.
2. The multiway system carries the algebraic structure (dagger-symmetric compact-closed monoidal category) required for categorical quantum mechanics.
3. The ZX-calculus embeds cleanly into the multiway framework, with compatible monoidal structures.
4. The branchial metric satisfies the axioms of an entanglement monotone.
5. The entanglement entropies computed via branchial graphs and via the Sorkin-Johnston construction are monotonically related across large classes of rules.
6. The uncertainty principle, the path integral, the discrete Schrödinger equation, compatibility with Bell’s theorem, and the quantum Zeno effect all follow from the geometry of multiway space and the principle of multiway invariance.

What remains conjectural, or at least incomplete, is the full derivation of quantum mechanics as the unique continuum limit of the multiway formalism — in particular, a complete derivation of decoherence and the Born rule from multiway geometry (rather than by analogy with Knuth-Bendix completion). The multiway analog of the Einstein field equations has been formulated using the methods of Leifer’s “superrelativity,” but its physical consequences remain largely unexplored. And the deepest open question — whether the structure of the Standard Model is a generic feature of the Ruliad, rather than a consequence of a specific rule — remains firmly in the realm of conjecture, though it is arguably the most important question the project faces.

Nevertheless, the conceptual picture is compelling: quantum mechanics arises not as an additional postulate imposed on a classical substrate, but as an inevitable consequence of the non-determinism inherent in any sufficiently general rewriting system, combined with the coarse-graining constraints imposed by macroscopic observers. The branching structure of the multiway system is the quantum state; the branchial graph is the tensor product; the Fubini-Study metric is the geometry of branchial space; causal invariance is general covariance; multiway invariance is the unitarity of quantum evolution; and Knuth-Bendix completion is measurement. If the universality conjecture for the Ruliad holds, then these identifications are not merely features of a cleverly chosen rule, but necessary features of any sufficiently complex discrete rewriting process — making quantum mechanics, in a sense, inevitable. Whether this picture ultimately succeeds as a theory of fundamental physics, it represents a genuinely novel approach to the foundations of quantum mechanics — one rooted not in Hilbert spaces and operators, but in the combinatorics of rewriting and the geometry of the space of all possible computations.

Sources

The following papers and articles form the primary basis for this exposition.