Notation Reference

Every math symbol used in Volume 0, with one-line meaning. Unicode forms are used in Phases 1–2; LaTeX forms (rendered via KaTeX) are used in Phase 3 (chapters 8–11).

Notation legend

Sorts

Symbol	LaTeX	Meaning
$U$	`U`	the universe of entities (kernel)
$O$	`O`	the set of objects
$P$	`P`	the set of properties
$V$	`V`	the set of values
$T$	`T`	the set of transitions
$\mathrm{PROCESS}$	`\mathrm{PROCESS}`	reified transitions, $\rho(T) \subseteq O$

Relations

Symbol	LaTeX	Signature	Meaning
$\mathrm{IS}$	`\mathrm{IS}`	$\mathrm{IS} \subseteq O \times O$	individuation: identity and kind-membership
$\mathrm{HAS}$	`\mathrm{HAS}`	$\mathrm{HAS} : O \to (P \rightharpoonup V)$	attribution: maps object to its property-value partial function
$\mathrm{DOES}$	`\mathrm{DOES}`	$\mathrm{DOES} \subseteq O \times T$	dynamics: relates object to its transitions

Operations

Symbol	LaTeX	Signature	Meaning
$\circ$	`\circ`	$\circ : T \times T \rightharpoonup T$	composition (when interfaces match)
$\rho$	`\rho`	$\rho : T \to O$	reification: a transition becomes an object
$\iota$	`\iota`	$\iota : O \hookrightarrow V$	embedding: an object can be a value

Categorical notation

Symbol	LaTeX	Meaning
$\mathrm{id}_A$	`\mathrm{id}_A`	identity morphism on object $A$
$f : A \to B$	`f : A \to B`	morphism from $A$ to $B$
$g \circ f$	`g \circ f`	composite: $A \xrightarrow{f} B \xrightarrow{g} C$
$\mathcal{K}$	`\mathcal{K}`	the kernel: $\langle U, \tau, \circ \rangle$
$\mathcal{M}$	`\mathcal{M}`	the eight-term algebra
$\mathrm{Hom}_O$	`\mathrm{Hom}_O`	hom-set in category of objects
$\cong$	`\cong`	natural isomorphism
$\dashv$	`\dashv`	adjunction (left adjoint $\dashv$ right adjoint)

Set-theoretic and logical symbols

Symbol	LaTeX	Meaning
$\subseteq$	`\subseteq`	subset or equal
$\subset$	`\subset`	proper subset
$\hookrightarrow$	`\hookrightarrow`	embedding (injective map)
$\rightharpoonup$	`\rightharpoonup`	partial function
$\to$	`\to`	total function or morphism
$\times$	`\times`	Cartesian product
$\in$	`\in`	element of
$\wedge$	`\wedge`	logical and
$\neg$	`\neg`	logical not
$\forall$	`\forall`	for all
$\exists$	`\exists`	there exists
$\exists!$	`\exists!`	there exists a unique
$\equiv$	`\equiv`	definitional equivalence
$\models$	`\models`	semantic entailment / models
$\blacksquare$	`\blacksquare`	Q.E.D. (end of proof)

State and algorithms

Symbol	LaTeX	Meaning
$\sigma$	`\sigma`	state-location function: $\sigma(e) = (p, t_i, I_i, O_i)$
$\kappa$	`\kappa`	the reserved “kind” property (used to desugar IS)
$V_{in}$ , $V_{out}$	`V_{in}`, `V_{out}`	input and output value-interfaces of a transition
$t_i$	`t_i`	the currently-active (or next) transition in an execution
$I_i$ , $O_i$	`I_i`, `O_i`	bound inputs / produced outputs at the current step

Conventions

Math in this volume

Phases 1–2 (chapters 1–7, README): Unicode math symbols (⊆, →, ∘, ⇀, ↪, ×, ∈, ⊂, ι, ρ, σ). Renders in any markdown viewer.
Phase 3 (chapters 8–11): LaTeX math inside $…$ (inline) and $$…$$ (display), rendered by KaTeX. The Unicode symbols continue to render in raw markdown views; KaTeX only kicks in inside math delimiters.

Color palette for diagrams

Use	Color tokens
IS aspect (identity)	green `#16a34a`, dark `#14532d`, light `#f0fdf4`
HAS aspect (attribution)	amber `#d97706`, dark `#78350f`, light `#fffbeb`
DOES aspect (dynamics)	red `#dc2626`, dark `#7f1d1d`, light `#fef2f2`
Subject / brand	indigo `#4f46e5`, dark `#3730a3`, light `#eef2ff`
Structure / connectors	slate `#0f172a`, mid `#475569`, light `#f8fafc`
Kernel / Tier 0 (formal)	teal `#0d9488`, dark `#115e59`, light `#f0fdfa`
Elaboration / desugaring	dashed lines, 1.6px stroke
Reification ( $\rho$ )	double-stroke lines

All SVGs hand-authored, 900×600 viewBox default, system font stack (ui-sans-serif, system-ui, sans-serif).

Status markers

Following the documentation tree convention:

● exists in the running system
◐ partial
○ planned

Most claims in Volume 0 are mathematical, not implementation; markers apply only where a claim depends on a runtime.

Back to Volume 0 README.