Closure Formalization
This section formalizes the notion of closure used throughout the Dimensional Emergence Framework (DEF).
Closure is not treated as a metaphysical concept, but as a structural condition on admissible composition.
What closure means in DEF
Section titled “What closure means in DEF”A regime is said to be closed if:
- all admissible compositions of its operative modes
- remain expressible within the same operative set
- under repeated interaction and perturbation.
Closure is therefore a condition on structural completeness, not on isolation.
A closed regime may interact with other regimes,
but it does not require external primitives to maintain internal coherence.
Operative composition
Section titled “Operative composition”Let the operative set be:
{ S, R, D, X̂ }
Closure concerns the admissibility of compositions such as:
- S ∘ R
- R ∘ D
- D ∘ X̂
- and higher-order compositions thereof.
Closure does not require commutativity, associativity, or linearity.
It requires only that admissible compositions do not escape the kernel.
Self-reference and closure
Section titled “Self-reference and closure”Self-referential relations specify how modes can refer to themselves or to each other.
Closure requires that:
- self-referential liftings do not generate new primitive operators,
- recursive application remains bounded,
- and self-reference does not collapse distinctions.
Self-reference enables persistence.
Closure restricts excess.
Constraints as closure conditions
Section titled “Constraints as closure conditions”The five constraints specified in the Core act as closure guards.
Formally, they restrict:
- which compositions are admissible,
- which recursive paths are allowed,
- and which configurations must be rejected.
In particular, the trans-dimensional constraint:
(S · R) ≡ (X̂ · D)
ensures that Existence-side and Happening-side compositions remain mutually consistent.
Finite versus divergent closure
Section titled “Finite versus divergent closure”A key distinction is between:
- finite closure — all admissible compositions remain within a bounded kernel,
- divergent closure — compositions generate an unbounded hierarchy of operators.
DEF requires finite closure at the kernel level.
Divergent behavior may occur within regimes,
but only insofar as it can be re-expressed within kernel constraints.
Closure and regime stability
Section titled “Closure and regime stability”Closure is a necessary condition for regime stability, but not sufficient by itself.
Stability additionally requires:
- bounded self-reference,
- constraint satisfaction,
- and phase ordering.
A regime may temporarily violate closure locally,
but must be able to re-enter closure to remain stable.
Formal freedom
Section titled “Formal freedom”DEF does not prescribe a unique mathematical formalism for closure.
Possible representations include:
- algebraic systems with restricted generators,
- category-theoretic closure conditions,
- operator algebras with bounded recursion,
- or graph-based composition systems.
All are admissible if they respect the Core closure criteria.
This section defines what closure must accomplish,
not how it must be implemented.
Concrete mathematical realizations are explored in subsequent sections.
Closure provides the formal backbone that allows DEF to remain structurally complete
while admitting multiple regime-specific realizations.