Operator Representations of the Kernel

This section outlines mathematical representation families suitable for the DEF kernel.

The goal is not to fix a single formalism, but to specify what any formalism must support in order to represent:

the four operative modes {S, R, D, X̂},
seven self-references,
five constraints (closure guards),
and three-phase ordering (Entry → Crisis → Resolution).

No derivations are given here; this is a representation specification.

Representation requirements

Any representation of the kernel must support:

A finite generator set
The kernel must be generated by a finite set of primitives corresponding to {S, R, D, X̂}.
Admissible composition
There must be a well-defined notion of composition (sequential, relational, or operational) with restricted admissibility.
Self-referential liftings
It must represent the seven self-references as explicit operations without generating unbounded new primitives.
Constraint evaluation
It must encode the five constraints as predicates, invariants, or admissibility filters.
Phase ordering
It must represent ordered traversal of admissible configurations independent of metric time.

Minimal abstract model

Let:

𝕄 = {S, R, D, X̂} be the set of operative modes,
𝕆 be the set of admissible operators generated from 𝕄 under composition,
𝒞 be the set of constraints acting as admissibility conditions.

A regime realization is then characterized by:

a chosen representation space,
a composition rule,
and a constraint filter.

The central closure property is:

admissible compositions remain within 𝕆,
and 𝕆 remains finite under closure.

Algebraic representations

An algebraic representation models modes as generators of an algebraic structure.

Typical options include:

restricted operator algebras, where products are only defined when admissible,
graded algebras, where modes occupy different grades and constraints forbid cross-grade collapses,
non-associative or partially associative algebras, reflecting that kernel composition need not satisfy associativity globally.

Self-references can be represented as lifting operators acting on generators or subspaces.

Constraints appear as:

invariants,
boundedness conditions,
or admissibility predicates on products.

Graph and hypergraph representations

A graph-based representation models:

modes as node types,
compositions as edges or hyperedges,
and admissibility as constraints on allowed motifs.

Self-references correspond to:

reflexive edges,
typed feedback loops,
or controlled recursion motifs.

Constraints are graph-level restrictions:

forbidding certain cycles,
bounding recursion depth,
enforcing equivalences between subgraphs (e.g. SR ↔ X̂D).

Phase ordering may be represented as:

ordered traversal of graph states,
or a transition system over admissible subgraphs.

Category-theoretic representations

A categorical representation treats:

modes as objects or generating morphisms,
compositions as morphism composition,
constraints as commutative conditions or restricted subcategories.

Self-references naturally correspond to:

endomorphisms,
functors mapping structures onto their own representations,
or reflective subcategories.

The constraint (S · R) ≡ (X̂ · D) can be represented as an equivalence of composites (commuting diagrams).

Phase ordering can be modeled as:

a partial order on morphism classes,
or a constrained path structure through a category of admissible configurations.

Dynamical-system representations (structural, not physical)

A dynamical-style representation may model kernel configurations as points in an abstract state space, with transitions defined by admissible compositions.

In DEF terms, this is not physical dynamics, but structural traversal under constraints.

self-references define allowable recurrence,
constraints bound trajectories,
phases define ordered regions: Entry → Crisis → Resolution.

This representation is useful for simulation and regime testing, provided that:

the kernel remains finite under closure,
and transitions do not introduce new primitives.

Representation of the trans-dimensional constraint

The core equivalence:

(S · R) ≡ (X̂ · D)

should be representable in any formalism as one of:

an invariant,
an equivalence relation between composites,
a constraint predicate on admissible compositions,
or a commuting condition in a structured diagram.

This constraint is the primary bridge between Existence-side and Happening-side coupling.

Recommended implementation strategy

For practical modeling, DEF favors a layered strategy:

Start with a graph or operator-algebra representation
(simple, computable, simulatable)
Introduce constraint checking as explicit filters
(admissibility as a predicate)
Represent phases as ordered classes of admissible states
(Entry/Crisis/Resolution as structural traversal)
Only then map to physical interpretation
(metric, invariance, dynamics as regime representations)

This prevents premature identification of structure with spacetime mechanics.

Scope

This section does not select a single formalism.

It specifies the criteria by which a formalism qualifies as a DEF kernel representation:

finite closure,
explicit self-reference without proliferation,
constraint-evaluable composition,
and phase-order expressibility.

Operator representations provide the mathematical substrate required to formalize DEF
while preserving the Core separation between ontology, constraint, and interpretation.