The fifteen properties form coherent clusters reflecting deep structural interdependencies rather than independent design principles.

Entities in the same semantic neighborhood but without a typed relation to this one — candidates for new edges or unrecognized duplicates.

• The fifteen structural properties are not the most fundamental aspect of the theory; they are consequences of the existence of centers and their interdependence.claim0.869
  Alexander's retrospective account of how his theory evolved, demoting the fifteen properties from foundational to derivative status.
• The fifteen properties are the ways centers sustain each other's coherence, thereby contributing to the stability and robustness of natural systems.claim0.843
  Proposed as the reason the properties appear in functionally stable or semistable systems.
• The fifteen properties appear in nature because they are structural complements to the formation of stable and semistable systems, contributing to coherence and stability.hypothesis0.837
• The fifteen properties emerge directly and inevitably from the unfolding of the whole; it is these fifteen properties that guide all differentiations during development.claim0.829
  Alexander's claim that living structure properties are not incidental but are the operative mechanisms of wholeness-preserving transformation
• The fifteen properties are not independent; they overlap, and each property is partly defined in terms of the other fifteen propertiesclaim0.825
  Meta-claim about the logical structure of the properties: the more carefully each is defined, the more it relies on the others, revealing their common origin in the field of centers
• Correspondence analysis reveals four major clusters of fifteen properties with overlapping membership.finding0.824
• Correspondence analysis reveals four clusters of related properties: Cluster 1 (Contrast, Not-Separateness, Roughness, Alternating Repetition, Good Shape), Cluster 2 (Local Symmetries, The Void, Levels of Scale, Good Shape, Positive Space), Cluster 3 (Boundaries, Strong Centers, Deep Interlock and Ambiguity), Cluster 4 (Simplicity and Inner Calm, Echoes, Gradients, Positive Space).finding0.821
  Statistical grouping of properties based on dependency patterns, enabling deeper understanding of their coherence and interaction.
• The fifteen properties appear in nature as a direct result of repeated structure-preserving transformations unfolding the wholeness.claim0.820
  Interpretation that the geometric properties of living structure are not arbitrary but arise inevitably from the smooth unfolding process.