Constraint Network Geometry: Physical Realization, Hierarchical Support and Bottleneck Allocation in EET

What determines the shape of a leaf vein, the branching of a river network, the spiral of a galaxy, or the foam of soap bubbles? Constraint Network Geometry provides a unified answer: all persistent spatial forms in nature are geometric expressions of constraint networks minimizing their total maintenance burden under the inescapable physical logic of Being-as-Resistance, Barrier Asymmetry, and the five operations of The Dynamics of Constraint Networks. Version 2.1 extends the v1.0 framework with five constitutional additions. \textbf{Geometric Dynamics} (Part IX) elevates the geometric burden functional from statics to dynamics, establishing the morphological evolution equation that governs how constraint network geometry changes over time under burden gradients and environmental fluctuations. \textbf{The Encapsulation Spiral} (Part X) provides the positive-feedback growth dynamics that complement the Degeneration Spiral of The Dynamics of Constraint Networks, giving the complete constructive + degenerative lifecycle of constraint networks. \textbf{Morphological Statistical Mechanics} (Part XI) formulates a research program for the statistical mechanics of morphologies, with the embedding partition function rigorously defined in the fixed-topology regime and topological summation posed as an open problem. \textbf{The \(\alpha\)-\(\alpha_{\text{HD}}\) Duality Theorem} (Part XII) establishes the precise mathematical relationship between the macroscopic burden scaling exponent \(\alpha\) and the microscopic encapsulation efficiency \(\alpha_{\text{HD}}\), revealing two critical thresholds that govern when hierarchical organization is beneficial. \textbf{The EET Fixed-Point Theorem} (Part XIII) provides the unified algebraic framework explaining why three independent projection paths---functional, structural, and geometric---converge to the same golden-ratio equation under their respective ideal conditions. Version 2.1 is aligned with EET Core Rules v6.1, adopting the L0 sole meta-axiom architecture, the L1 constitutional presuppositions designation, the Ben-Se terminological convention, and the Creative Asymmetry as an L3 Derived Structural Principle. All v1.0 content is preserved, and all new theorems carry explicit constitutional status labels (CONDITIONAL THEOREM or WORKING HYPOTHESIS) with their premises transparently declared. {Keywords}: Constraint network geometry; geometric burden functional; morphological phase diagram; encapsulation maintenance gain; bottleneck matching; golden ratio; geometric dynamics; encapsulation spiral; morphological statistical mechanics; \(\alpha\)-\(\alpha_{\text{HD}}\) duality; EET fixed-point theorem; Energy-Efficiency Theory