Topology as Logic: Structural Role Geometry Across Formal, Software, Biological, and Prebiotic Systems

Topology as Logic: Structural Role Geometry Across Formal, Software,Biological, and Prebiotic SystemsIvanov, Vladi DescriptionWe ask whether dependency topology correlates with functional load-bearingorganization as recoverable geometry — not as a metaphor, but as a measurablestructural property detectable by multilayer network analysis. Across sevenindependent substrates, we show that hub persistence and rank divergence underthe Functional Proximity Law recover operational organization that domainexperts describe as logic: axiomatic load-bearing structure in formalmathematics, control and contract structure in legacy software, conserved hubgrammar across ~600 million years of neural evolution, catalytic roleorganization in a published prebiotic autocatalytic network, carry-pathdominance in a 4-bit digital circuit, betweenness persistence in the ISCAS85c432 standard benchmark (n = 196), and a directional formal-systems replicationin the Coq Corelib (n = 17). The central methodological contribution is the betweenness-over-degreedistinction — the "load-bearing logic signal": betweenness centralitypersistence across layers correctly identifies the carry chain in digitalcircuits, the foundational modules in formal mathematics, the control-flowbackbone in legacy COBOL, and the catalytic hub in prebiotic chemistry. Degree-based persistence measures connection density but misidentifies operationallyperipheral nodes (XOR gates, import-dense but logically lightweight modules) asload-bearing. The pre-registered ISCAS85 primary hypothesis was confirmed ondegree (r = 0.426, p = 0.002, Spearman r = 0.551); betweenness r = 0.341(exploratory). The 4-bit ALU betweenness result (r = 0.771, post-hoc) isvalidated against a degree-preserving null (z = +2.68, p < 0.001), confirmingthe signal is not a degree-leakage artifact. The formal-systems claim is supported by two proof-assistant corpora: Lean 4mathlib4 (CONFIRMED, r = 0.777, p = 0.004) and Coq Corelib (PARTIAL, directionconfirmed, r = 0.288, p = 0.287, n = 17 — underpowered). All seven experimentswere pre-registered before analysis. This paper is a companion to the FPL survey paper (arXiv:2604.23639). Itaddresses a focused question: does topology recover the load-bearing logic of asystem, or only its connection density? Pre-registration records:https://github.com/vladi160/preregistrations. --- Version 1 — June 2026 First submission. Seven pre-registered experiments: 1. H_LOGIC_DIGITAL_v1 (4-bit ALU, n = 29): degree h1 CONFIRMED (r = 0.512, p = 0.004); betweenness r = 0.771 (post-hoc; z = +2.68, p < 0.001 vs degree-preserving null). Carry-out nodes load-bearing in both layers; XOR nodes rank gap = 16 (maximum divergence). 2. H_LOGIC_F3_ISCAS85_v1 (ISCAS85 c432, n = 196): h1 CONFIRMED, degree r = 0.426, p = 0.002, Spearman r = 0.551. 54 nodes (28%) show large rank divergence (h3 CONFIRMED). Step betweenness r = 0.341 (exploratory). 3. H_LOGIC_EXTRACTION / F9 (Lean 4 mathlib4, n = 20): CONFIRMED, r = 0.777, Spearman r = 0.733, p = 0.004. Algebra: top declared hub. Analysis: top co-development hub. CategoryTheory / Combinatorics / Order diverge. 4. H_LOGIC_COQ_STDLIB_v1 (Coq Corelib, n = 17): h1 PARTIAL (r = 0.288, p = 0.287); direction positive and confirmed. First cross-proof-assistant directional replication. Step betweenness r = 0.509 (exploratory). 5. F2 COBOL legacy banking (n = 14): CONFIRMED, r(control-flow ↔ copybook- contract) = 0.807, p = 0.002. Topological dormancy signatures: two programs zero-degree in control flow but #2/#3 in data coupling, rank gap = 11. 6. F12c + F13 C. elegans → Drosophila (n = 2,952): Hub grammar cosine ≥ 0.90; F13 Spearman ρ = 0.663, p = 0.002. Cross-species conservation across ~600 Myr of evolution. 7. X9 v2 CatReNet prebiotic chemistry (n = 13): CONFIRMED, r(catalytic ↔ product) = 0.826 vs r(reactant) = 0.496. Molecule 'a' as dominant catalytic hub.