FIGURES

🔬 NEXAH — Core Figures

This directory contains the canonical figure set
for the NEXAH research layer.

The figures provide a minimal, structured representation of how
dynamical systems are transformed into geometric and structural descriptions.

---

🧭 Organization

main/      → primary figures (paper-ready)
extended/  → supporting figures (deeper analysis)

---

🧠 Conceptual Principle

NEXAH does not assume predefined structure.

Instead, structure is extracted from data:

trajectories → field → structure → transitions → connectivity

---

🔷 Main Figures (Canonical Set)

Figure 1 — Field Structure

Role: Conceptual foundation

Shows how continuous dynamics generate a structured field:

• density organization
• coherent flow regions
• emergence of geometric structure

👉 Interpretation:

structure emerges from dynamics

---

Figure 2 — Transition Geometry

Role: Structural decomposition

Shows how the field organizes into:

• sheets (coherent regions)
• transition zones (gates)
• structured pathways between regions

👉 Interpretation:

transitions are geometric regions, not discrete events

---

Figure 3 — Phase Mismatch & Control

Role: Dynamic interpretation

Illustrates:

• instability as geometric deviation
• phase mismatch as transition precursor
• control via alignment with structure

👉 Interpretation:

stability and control are geometric phenomena

---

🔁 Structural Pipeline (Unified View)

Across all figures:

dynamics → field → structure → transitions → connectivity

---

🔷 Extended Figures (Supporting Layer)

Figure 4–8 — Structural Extensions

Located in extended/

Provide deeper analysis of:

• cross-system comparison (Lorenz, Rössler, Halvorsen)
• regime structure and connectivity
• temporal transition behavior
• full extraction pipeline

⚠️ These figures are:

• supplementary
• not required for core understanding
• intended for extended analysis

---

🔑 Core Insight

Flow generates structure.
Structure organizes motion.
Transitions connect structure.
Connectivity defines global behavior.

---

📄 Suggested Paper Mapping

Role	Figure
Conceptual foundation	Fig. 1
Structural decomposition	Fig. 2
Dynamic interpretation	Fig. 3
Extended analysis	Fig. 4–8

---

⚠️ Scope Clarification

• All results are empirical and data-driven
• No formal topological equivalence is claimed
• Structural elements are observed across systems
• Generality beyond tested systems remains open

---

🧠 Summary

The figure set describes a transformation:

dynamics → structure → transitions → connectivity

NEXAH interprets dynamical systems as:

structured fields with constrained motion

---

🔥 Final Insight

A system is not only a trajectory.

It is a structured space
that constrains how motion can occur.