Computational Closure Model

This page provides a computationally testable formalization of the DEF kernel.

It is not a software engineering guide.
It specifies a minimal executable interpretation of the Core structures in order to:

test finite closure vs divergent composition,
operationalize constraints as admissibility predicates,
represent self-references as explicit rewrite/generation rules,
and encode phase ordering as a state transition structure.

No physical interpretation is assumed.

1. Kernel primitives

Let the operative mode set be:

{ S, R, D, X̂ }

A computational model represents compositions of these modes as finite expressions.

A minimal abstract representation uses:

atomic symbols for modes,
a composition operation (written here as ·),
and explicit self-reference operations (liftings).

No commutativity or associativity is assumed unless imposed by a chosen realization.

2. Expressions and admissibility

Let Expr denote the set of all syntactically constructible expressions, e.g.:

S, R, D, X̂
(S · R), (X̂ · D)
((S · R) · X̂), etc.

Closure is not about Expr being finite (it is not),
but about the existence of a finite admissible subset under constraints.

Define:

Adm ⊆ Expr

where Adm contains exactly those expressions admissible under the constraint set.

3. Self-reference as generation rules

Self-references are represented as explicit rules that generate candidates from admissible expressions.

Minimal self-reference rules include:

S → Sˢ
R → Rʳ
D → Dᵟ
X̂ → X̂ˣ
R → Sˢ
X̂ → Dᵟ
(S · R) → (X̂ · D)

Operationally, self-reference rules are generators:

Gen : Adm → P(Expr)

that propose candidate expressions which must then be filtered by constraints.

4. Constraints as closure guards

Constraints are represented as predicates that decide admissibility.

Let the constraint set be:

C = { C1, C2, C3, C4, C5 }

Each constraint Ci evaluates either:

expressions (local admissibility), or
the current admissible set (global admissibility).

A minimal model treats constraints as filters:

Filter_C : P(Expr) → P(Expr)

that remove candidates which violate closure requirements.

The key cross-kernel condition must be expressible, e.g. as an equivalence predicate:

(S · R) ≡ (X̂ · D)

Other constraints implement boundedness, non-degeneracy, finite closure, and recoverability.

5. Closure iteration

A minimal closure procedure is defined as:

Initialize Adm0 from a finite seed set.
Generate candidates: Cand_k = Gen(Adm_k)
Filter candidates: New_k = Filter_C(Cand_k)
Update: Adm_{k+1} = Adm_k ∪ New_k
Stop if Adm_{k+1} = Adm_k (fixed point), or if divergence is detected.

Finite closure corresponds to reaching a fixed point with finite Adm_k. Divergence corresponds to unbounded growth, failure of constraints, or loss of finiteness criteria.

6. Phase ordering as a transition structure

Phases are represented as labels on admissible states rather than time steps.

Let a kernel state be:

State := (Adm, φ)

where φ ∈ {Entry, Crisis, Resolution}.

Define a tension functional T(State) that measures constraint engagement. Phase transitions are specified structurally:

Entry → Crisis when constraint tension exceeds a regime threshold
Crisis → Resolution when admissibility is restored under decreasing tension
Crisis → Divergence when constraints cannot be satisfied or closure fails
Resolution → Entry under new perturbation (e.g. exchange injection) or re-initialization

This makes phases operational without assuming metric time.

7. Minimal test suite

The model supports regime tests such as:

Finite closure test: does the system reach a fixed point?
Constraint sensitivity: which constraints dominate admissibility loss?
Phase robustness: can Crisis reliably reach Resolution without divergence?
Regime comparison: how do different admissibility rules change stability windows?
Invariance window exploration: when does metric-like coordinatization remain consistent?

These tests do not validate physics.
They validate structural coherence of a proposed formal realization.

Scope

This computational closure model is a formal aid:

it provides an executable interpretation of closure, constraints, self-reference, and phases,
it does not prescribe physical dynamics,
it does not assert empirical predictions,
and it does not commit DEF to a specific mathematical foundation.

Its purpose is to make kernel behavior testable before physical interpretation.