The Gradient-Balance Principle: Universal Structure Formation from Combinatorics to Semantics

We identify a universal structural principle the Gradient-Balance Principle (GBP)that governs closure and stability across radically different domains: combinatorial graphtheory, semantic dynamics in large language models, and predictive processing in biologicalsystems. The principle states that stable configurations arise exactly when accumulatedlocal tensions (gradients) satisfy a global conservation law, with topological obstructions(holonomy) determining which configurations are reachable.We demonstrate this principle through three independent case studies: (1) the Erdos-Gyárfás conjecture for cubic vertex-transitive graphs, where cycle existence is governed byport-geometry and shift-balance constraints; (2) Semantic Physics, where meaning stabilizationin LLM conversations follows gradient-ow dynamics; and (3) Friston's Free EnergyPrinciple, where prediction error minimization exhibits the same constraint architecture.The structural isomorphism between these domains is not metaphorical but formal: allthree instantiate the same abstract mathematical object a fiber bundle with connectionand holonomy.This convergence suggests that GBP may be a transcendental structure a necessarycondition for any system capable of forming stable, self-referential configurations. We formalizeGBP category-theoretically, prove a general closure theorem, and derive falsifiablecross-domain predictions.