New study shows topology reveals hidden logic across math, biology, and software

A groundbreaking preprint by Vladi Ivanov, 'Topology as Logic: Structural Role Geometry Across Formal, Software, Biological, and Prebiotic Systems,' proposes that dependency topology is not just a metaphor but a measurable structural property that correlates with functional organization across radically different domains. The paper pre-registered and tested seven experiments—a rare methodological rigor—using multilayer network analysis on systems ranging from a 4-bit ALU digital circuit and the ISCAS85 c432 benchmark (n=196) to the Lean 4 mathlib4 proof assistant (r=0.777, p=0.004) and Coq Corelib (partial confirmation).

Key findings include that degree-based hub persistence is weak between physical wiring and simulation state-correlation layers (r=0.21 in c432) while betweenness-based persistence is far stronger (r=0.77 in the ALU). The study covered neural connectomics across ~600 million years (C. elegans to Drosophila), a published prebiotic autocatalytic network carrier-path dominance, and legacy COBOL control structures, all showing that topology reveals logical load-bearing organization. This work bridges computer science, neuroscience, and formal mathematics, suggesting that geometry of dependencies is a universal language of structure.