Phase Ordering and Constraint Tension

This section examines the relation between phase ordering and constraint tension within the Dimensional Emergence Framework (DEF).

While constraints specify admissible compositions and closure conditions,
phases determine how constraint tension is traversed and resolved.

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Constraint tension

Constraints in DEF are not static prohibitions.
They admit degrees of tension under interaction and perturbation.

Constraint tension arises when:

• multiple constraints are simultaneously engaged,
• self-referential activity increases,
• or cross-dimensional coupling is stressed.

Tension does not imply violation.
It indicates proximity to the boundary of admissibility.

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Phases as ordering relations

The three phases of DEF:

1. Entry
2. Crisis
3. Resolution

do not describe time evolution.
They define an ordering of structural states under increasing and decreasing constraint tension.

Phases regulate how far and in which order tension may develop.

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Entry: low constraint tension

In the Entry phase:

• constraints are weakly engaged,
• self-references are active but loosely coupled,
• compositions remain well within admissible bounds.

Entry establishes a stable initial configuration.
Constraint tension is minimal.

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Crisis: maximal constraint tension

The Crisis phase corresponds to maximal admissible constraint tension.

• self-references are fully engaged,
• cross-dimensional coupling (S · R) ↔ (X̂ · D) is stressed,
• perturbations propagate across multiple modes.

Crisis is the only phase in which:

• reconfiguration is possible,
• regime adaptation can occur,
• or transitions may be triggered.

Crisis does not violate constraints;
it brings them to their limits.

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Resolution: tension release

In the Resolution phase:

• constraint tension decreases,
• admissible configurations are re-established,
• self-references settle into bounded activity.

Resolution restores closure without introducing new operative modes.

Failure to resolve tension leads to:

• divergence,
• regime escape,
• or collapse.

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Ordering necessity

The ordering Entry → Crisis → Resolution is structurally necessary.

• Crisis without Entry lacks admissible grounding.
• Resolution without Crisis yields no adaptation.
• Entry without Resolution prevents persistence.

Skipping a phase results in structural inconsistency.

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Constraint tension and divergence

Divergence occurs when:

• constraint tension exceeds admissible bounds, or
• Resolution cannot be reached from Crisis.

Thus:

• finite closure corresponds to successful Resolution,
• divergent behavior corresponds to unresolved Crisis.

Phase ordering therefore acts as a regulator of divergence risk.

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Regime sensitivity

Different regimes may:

• tolerate different levels of constraint tension,
• exhibit prolonged Crisis phases,
• or require repeated Entry–Crisis–Resolution cycles.

Phase duration is regime-dependent.
Phase ordering is not.

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Scope

Phase ordering does not prescribe dynamics or equations of motion.

It specifies a structural logic of stability, stress, and recovery
that applies across all DEF-compatible regimes.

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Phase ordering provides the mechanism by which DEF regimes
navigate constraint tension without sacrificing closure.