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. HLOGICDIGITALᵥ1 (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. HLOGICF3ISCAS85ᵥ1 (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. HLOGICEXTRACTION / 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. HLOGICCOQSTDLIBᵥ1 (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.
Vladi Ivanov (Mon,) studied this question.