Ultrametric Tree Universality: One Geometry, Six Domains

Key Points

• To demonstrate that ultrametric tree geometry underlies hierarchical organization across diverse domains.
• Cross-domain synthesis of existing mathematical structures and geometric properties.
• Examination of applications in quantum error correction, spin glasses, protein folding, cosmology, cognition, and language.
• Development of an interactive explorer to visualize these connections.
• Identified ultrametric tree characteristics as foundational across all six domains.
• Confirmed that cophenetic distance and triadic rigidity are consistently observable in diverse systems.
• Established strong triangle inequality as a unifying principle in hierarchical structures.