Admissible Composition Rules

This section specifies what DEF requires from any notion of composition between operative modes.

DEF does not prescribe a unique algebra, category, or graph calculus.
However, any formalization must implement composition in a way that preserves:

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

This page therefore defines admissible composition as a structural concept.

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1. Composition as a primitive relation

Let the operative modes be:

{ S, R, D, X̂ }

Let ”·” denote a generic composition operator.

In DEF, ”·” does not mean multiplication, convolution, or tensor product by default.
It denotes structured coupling: an admissible way to bind two operative roles into a composite.

A formalism may realize ”·” as:

• operator composition,
• morphism composition,
• graph/hypergraph gluing,
• or typed relation composition.

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2. Partiality of composition

Composition in DEF is generally partial.

Not all pairs of expressions must be composable, and not all compositions must remain admissible.

Formally, one may treat composition as a partial operation:

· : Adm × Adm ⇀ Expr

where Adm is the set of admissible expressions under constraints.

This makes admissibility explicit and prevents implicit closure assumptions.

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3. Non-commutativity and non-associativity (by default)

DEF does not assume commutativity:

A · B ≠ B · A in general.

DEF also does not assume global associativity:

(A · B) · C need not equal A · (B · C).

Associativity may hold within specific regimes or substructures,
but it is not required at the kernel level.

This is necessary to represent:

• ordering effects,
• phase-dependent composition,
• and constraint tension under coupling.

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4. Typing and mode roles

Admissible composition should respect mode roles.

A robust way to enforce this is to type expressions by the modes they involve, e.g.:

type(Expr) ⊆ {S, R, D, X̂}

Typing supports at least three kernel requirements:

• Non-degeneracy: all four modes must remain represented over closure.
• Finite closure: forbids uncontrolled generation of new primitive types.
• Constraint evaluation: enables structured admissibility checks.

Typing may be implemented as:

• static type tags,
• graded components,
• or node/edge labels in graph representations.

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5. Constraint-evaluable composition

A composition rule is admissible only if constraints can evaluate it.

Therefore, each composition step must be checkable by the constraint set C:

A, B admissible
A · B admissible if and only if all relevant constraints are satisfied.

In particular, the trans-dimensional constraint must be representable as an admissibility relation on composites:

(S · R) ≡ (X̂ · D)

A formalism must provide a notion of equivalence or compatibility between such composites.

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6. Self-reference compatibility

Self-reference introduces liftings such as:

S → Sˢ, R → Rʳ, D → Dᵟ, X̂ → X̂ˣ

and the additional self-reference relations:

R → Sˢ, X̂ → Dᵟ, (S · R) → (X̂ · D)

Composition must be defined so that:

• self-referentially lifted expressions remain composable when admissible,
• but repeated lifting does not generate an unbounded operator vocabulary.

Practically, this implies:

• bounded recursion depth,
• restricted rewrite rules,
• or explicit closure filters.

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7. Phase-sensitive admissibility

DEF phases (Entry, Crisis, Resolution) do not impose time dynamics,
but they do impose ordering constraints on admissible compositions.

A formalism is phase-compatible if it can represent:

• a set of admissible compositions in Entry,
• a larger or more coupled admissible set in Crisis,
• and a re-stabilized admissible set in Resolution.

Thus, admissibility may depend on phase labels:

Adm(Entry), Adm(Crisis), Adm(Resolution)

Composition rules may be phase-sensitive without becoming temporal laws.

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8. Minimal admissibility specification

A composition system qualifies as DEF-admissible if it supports:

1. a partial composition operator (not all pairs compose),
2. constraint evaluation on each composition,
3. bounded self-reference under composition,
4. a finite closure criterion,
5. phase-sensitive admissibility sets.

No additional algebraic laws are required at the Core level.

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Scope

This section defines admissible composition structurally, not constructively.

Concrete realizations (algebraic, categorical, graph-based) may now be developed
while remaining aligned with DEF closure, constraints, self-reference, and phase ordering.

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Admissible composition provides the formal entry point for building explicit DEF models
without prematurely committing to a specific mathematics or a specific physical interpretation.