Physics & Math

This section develops the formal and mathematical consequences of the Dimensional Emergence Framework (DEF).

While the Core establishes the structural prerequisites for closure-stable regimes,
the Physics & Math domain addresses how these structures can be formalized, constrained, and related to known physical descriptions.

Scope of this section

The Physics & Math domain answers a different question than the Core:

Given the Core structures, what mathematical forms and physical behaviors are admissible?

Accordingly, this section:

introduces formal representations of closure, constraints, and self-reference,
explores regime-specific realizations of the four-mode kernel,
and connects DEF structures to established physical concepts where appropriate.

It does not introduce new ontological primitives.

From structure to formalization

The Core specifies:

two emergent dimensions (Existence, Happening),
four operative modes (Structure, Space, Dynamics, Exchange),
seven self-references,
five constraints,
and a three-phase ordering.

The task of Physics & Math is to determine:

how these elements can be expressed mathematically,
under which conditions closure is preserved,
and how regime transitions become formally necessary.

Multiple formalizations may be possible; DEF does not assume uniqueness.

Closure and constraints

A central theme of this section is closure under composition.

Topics include:

formal definitions of admissible operator composition,
boundedness conditions for self-referential liftings,
equivalence relations between Existence- and Happening-side couplings,
and criteria for finite versus divergent closure.

These analyses ground the five Core constraints in explicit mathematical terms.

Regime-specific realizations

Different regimes may realize the same Core structure in different ways.

This section therefore explores:

minimal abstract models of the four-mode kernel,
mappings to physical formalisms (e.g. spacetime, interaction, conservation),
and conditions under which familiar physical symmetries emerge.

No single physical theory is privileged; compatibility is evaluated structurally.

Phase structure and ordering

The three-phase structure (Entry, Crisis, Resolution) is treated as an ordering principle.

Here, Physics & Math examines:

how phase separation can be represented formally,
how phase transitions relate to constraint tension,
and how re-stabilization can be characterized without introducing external dynamics.

Phases are not assumed to correspond directly to time evolution.

Relation to established physics

Where relevant, this section discusses correspondences to:

classical and relativistic spacetime descriptions,
quantum and statistical formalisms,
and known invariance or symmetry principles.

Such correspondences are treated as realizations, not derivations of DEF.

DEF does not replace existing theories; it provides a structural frame in which they can be situated.

What this section does not do

To avoid category errors, this section does not:

assert empirical predictions by itself,
fix numerical constants,
or claim experimental validation.

Empirical relevance arises only when a specific formalization is paired with observational interpretation.

How to read this section

Readers may approach this section in two ways:

Top-down: starting from Core constraints and following their formal consequences.
Bottom-up: starting from known physical formalisms and examining how they instantiate DEF structures.

Both approaches are valid and complementary.

This section provides the formal bridge between the structural Core of DEF and its physical and mathematical realizations.