Regimes & Closure

The introduction of operational modes enables dimensional structure, but structure alone does not guarantee stability.

For a regime to persist, its internal relations must satisfy specific closure conditions.

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Definition of a regime

A regime is defined as a structured domain in which a fixed set of operational modes supports internally consistent relations.

Formally, a regime is denoted as:

$$\mathcal{R}_n := (\mathcal{M}_n, \mathcal{C}_n)$$

where:

• $\mathcal{M}_n$ is a set of operational modes,
• $\mathcal{C}_n$ is a set of closure conditions defined over those modes.

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Closure as a stability criterion

Closure denotes the ability of a regime to sustain its internal relations under composition.

A regime is considered stable if relations formed within the regime:

• remain expressible within the same operative set,
• do not require external modes for consistency,
• and can be recursively maintained.

Informally, closure ensures that the regime can describe and sustain itself.

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Formal notion of closure

Let $o_i, o_j \in \mathcal{M}_n$ denote operational modes.

Closure requires that relational compositions between modes do not exceed the operative domain:

$$\forall o_i, o_j \in \mathcal{M}_n : \; o_i \circ o_j \in \mathcal{M}_n$$

This expression denotes structural compatibility, not algebraic composition in a strict mathematical sense.

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Instability and breakdown of closure

If closure conditions are violated:

• relations cannot be recursively maintained,
• distinctions degrade or proliferate uncontrollably,
• and the regime loses internal consistency.

Such a regime is structurally unstable.

Instability is not a failure mode but a structural signal that the current operative set is insufficient.

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Regime transition

When a regime cannot satisfy its closure conditions, a regime transition becomes necessary.

Formally:

$$\neg \mathcal{C}_n \;\Rightarrow\; \mathcal{R}_{n+1}$$

A regime transition introduces:

• an expanded or refined operative set,
• new closure conditions,
• and an increased capacity for relational consistency.

The transition is not optional; it is logically necessitated by instability.

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Hierarchical organization of regimes

Regimes form a partially ordered hierarchy:

• lower regimes define minimal distinctions,
• higher regimes resolve instabilities of lower ones,
• no regime invalidates the structure of its predecessors.

Each regime remains locally valid within its domain of applicability.

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Structural, not temporal, progression

Regime transitions do not imply:

• temporal evolution,
• causal progression,
• or physical dynamics.

They represent structural necessity, not historical sequence.

Temporal interpretations arise only when regimes are embedded in later physical or phenomenological contexts.

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Scope limitation

At the Core level:

• closure is defined structurally,
• regime transitions are logical necessities,
• no empirical triggers are assumed.

Concrete realizations of regime transitions are addressed in later sections.

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Regimes and closure conditions establish the criteria by which dimensional structures can persist, transform, or necessitate extension.