Operational Modes

The differentiation into Existence (E) and Happening (H) establishes the minimal condition for dimensionality, but not for stability.

A regime composed solely of Existence and Happening cannot sustain closure under relations.
Additional operative distinctions are required.

---

Operative necessity

For a regime to be stable, it must:

• support persistent distinctions,
• allow relations between distinctions,
• and maintain internal consistency across interactions.

This requires a minimal set of operative modes through which relations can be formed, maintained, and transformed.

---

Definition of operational modes

An operational mode is defined as a fundamental manner in which distinctions can be:

• formed,
• related,
• and preserved within a regime.

Operational modes are not dimensions in a geometric sense.
They are co-equal structural aspects of a regime.

---

The minimal operative set

DEF postulates the following minimal set of operational modes for macroscopic regime stability:

• Structure (S)
  The capacity to sustain distinguishable configurations.

• Space (R)
  The capacity to support relational separation and adjacency.

• Dynamics (D)
  The capacity to support ordered change.

• Exchange (X̂)
  The capacity to support interaction and transfer between distinctions.

Formally, the operative set is denoted as:

$$\mathcal{M} := \{ S, R, D, \hat{X} \}$$

---

Co-equality and irreducibility

No element of $\mathcal{M}$ can be reduced to a combination of the others.

• Structure without Space admits no relational differentiation.
• Space without Structure admits no persistence.
• Dynamics without Structure admits no referent of change.
• Exchange without Dynamics admits no transformation.

Each mode is necessary, and none is sufficient in isolation.

---

Absence of geometric interpretation

The operational modes:

• are not coordinate axes,
• do not presuppose a metric,
• do not define spacetime.

They represent functional capacities, not spatial dimensions.

Any geometric or physical interpretation arises only in later regimes.

---

Relation to dimensionality

The introduction of operational modes constitutes a transition from minimal dimensionality to macroscopic dimensional structure.

Dimensionality, in this sense, refers to the number of independent operative modes required for closure, not to spatial extent.

---

Preparatory formalization

A regime supporting macroscopic stability may be represented as:

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

Where $\mathcal{C}_n$ denotes the closure conditions applicable to the operative set.

The explicit form of closure is introduced in the next section.

---

The identification of operational modes establishes the minimal structural basis required for stable regimes of reality.