*A glossary of terms that are either new, or used in unfamiliar ways*

### accumulative system

A system in which states are rules and rules update rules. Successive steps in the evolution of such a system are collections of rules that can be applied to each other.

### axiomatic level

The traditional foundational way to represent mathematics using axioms, viewed here as being intermediate between the raw ruliad and human-scale mathematics.

### bisubstitution

The combination of substitution and cosubstitution that corresponds to the complete set of possible transformations to make on expressions containing patterns.

### branchial space

Space corresponding to the limit of a branchial graph that provides a map of common ancestry (or entanglement) in a multiway graph.

### cosubstitution

The dual operation to substitution, in which a pattern expression that is to be transformed is specialized to allow a given rule to match it.

### eme

The smallest element of existence according to our framework. In physics it can be identified as an “atom of space”, but in general it is an entity whose only internal attribute is that it is distinct from others.

### entailment cone

The expanding region of a multiway graph or token-event graph affected by a particular node. The entailment cone is the analog in metamathematical space of a light cone in physical space.

### entailment fabric

A piece of metamathematical space constructed by knitting together many small entailment cones. An entailment fabric is a rough model for what a mathematical observer might effectively perceive.

### entailment graph

A combination of entailment cones starting from a collection of initial nodes.

### expression rewriting

The process of rewriting (tree-structured) symbolic expressions according to rules for symbolic patterns. (Called “operator systems” in *A New Kind of Science*. Combinators are a special case.)

### mathematical observer

An entity sampling the ruliad as a mathematician might effectively do it. Mathematical observers are expected to have certain core human-derived characteristics in common with physical observers.

### metamathematical space

The space in which mathematical expressions or mathematical statements can be considered to lie. The space can potentially acquire a geometry as a limit of its construction through a branchial graph.

### multiway graph

A graph that represents an evolution process in which there are multiple outcomes from a given state at each step. Multiway graphs are central to our Physics Project and to the multicomputational paradigm in general.

### paramathematics

Parallel analogs of mathematics corresponding to different samplings of the ruliad by putative aliens or others.

### pattern expression

A symbolic expression that involves pattern variables (*x*_ etc. in Wolfram Language, or ∀ quantifiers in mathematical logic).

### physicalization of metamathematics

The concept of treating metamathematical constructs like elements of the physical universe.

### proof cone

Another term for the entailment cone.

### proof graph

The subgraph in a token-event graph that leads from axioms to a given statement.

### proof path

The path in a multiway graph that shows equivalence between expressions, or the subgraph in a token-event graph that shows the constructibility of a given statement.

### ruliad

The entangled limit of all possible computational processes, that is posited to be the ultimate foundation of both physics and mathematics.

### rulial space

The limit of rulelike slices taken from a foliation of the ruliad in time. The analog in the rulelike “direction” of branchial space or physical space.

### shredding of observers

The process by which an observer who has aggregated statements in a localized region of metamathematical space is effectively pulled apart by trying to cover consequences of these statements.

### statement

A symbolic expression, often containing a two-way rule, and often derivable from axioms, and thus representing a lemma or theorem.

### substitution event

An update event in which a symbolic expression (which may be a rule) is transformed by substitution according to a given rule.

### token-event graph

A graph indicating the transformation of expressions or statements (“tokens”) through updating events.

### two-way rule

A transformation rule for pattern expressions that can be applied in both directions (indicated with ).

### uniquification

The process of giving different names to variables generated through different events.