Notation & Minimal Formalism

This section introduces the minimal formal language used throughout the Core of the Dimensional Emergence Framework (DEF).

The purpose of this notation is structural precision, not mathematical completeness.
All formal expressions introduced here are definitional, not derived.

They serve to:

mark distinctions explicitly,
express structural relations,
and ensure internal consistency across sections.

Formal stance

The Core employs a pre-metric formalism.

Accordingly:

no coordinate systems are assumed,
no metrics are introduced,
no temporal ordering is defined,
no empirical interpretation is implied.

Formal symbols are used strictly to denote relations of distinction and compatibility.

Primitive symbols

The following symbols denote ontological primitives:

Symbol	Designation	Description
B	Being	Undifferentiated ground state
E	Existence	Aspect of persistence or presence
H	Happening	Aspect of change or eventfulness

These symbols do not represent objects or substances.
They denote modes of distinction.

Differentiation operator

Differentiation is denoted by the operator Δ.

$(E, H) := \Delta(B)$

Δ does not denote a temporal process or causal operation.
It denotes the logical emergence of distinction from an undifferentiated state.

Sets and operators

When referring to collections of modes or operators, calligraphic notation is used.

$\mathcal{M} := \{ S, R, D, \hat{X} \}$

Where:

S - Structure
R - Space (Room)
D - Dynamics
X̂ - Exchange

These modes are introduced formally in later Core sections.

Regimes

A regime is denoted as an ordered pair:

$\mathcal{R}_n := (\mathcal{M}_n, \mathcal{C}_n)$

Where:

$\mathcal{M}_n$ is a set of operative modes,
$\mathcal{C}_n$ is a set of closure conditions.

The meaning of closure is defined in the corresponding Core section.

Relational notation

The following conventions apply throughout the Core:

Equality (=) denotes structural equivalence, not identity.
Implication (⇒) denotes logical necessity, not causality.
Composition (∘) denotes relational compatibility, not temporal sequencing.
Negation (¬) denotes logical non-satisfaction of a condition.

Interpretation boundary

Unless explicitly stated otherwise:

no symbol carries metric meaning,
no probabilistic interpretation is assumed,
no empirical claim is made.

All formal expressions in the Core remain structural and regime-relative.