Tension Algebra: A Mathematical Framework for Disequilibrium Dynamics

We develop Tension Algebra, a formal mathematical framework for modeling directed disequilibrium in physical and dynamical systems. A tension element τ = (A → B, Δ) encodes a source, a sink, and a magnitude representing the driving force behind transport phenomena—heat flow, wind, electric current, trade flows. The framework rests on three structural axioms: (1) dissipation—isolated tensions decay toward equilibrium; (2) sustainment—maintaining tension requires external input; (3) cascade—large tensions decompose into smaller ones preserving p-norm structure. We prove 13 theorems establishing the fundamental properties of tension dynamics, including that sustained tension requires input exceeding dissipation (the thermodynamic necessity of energy sources), that parallel tensions combine via a Pythagorean law on squared magnitudes, and that the cascade operation satisfies a triangle inequality. The framework provides a unified language for phenomena ranging from Kolmogorov's −5/3 cascade law in turbulence to the thermodynamics of living systems as self-sustaining tension loops. All results are formalized in the Platonic proof system with 34 verified declarations.

Keywords: tension algebra, disequilibrium dynamics, thermodynamic dissipation, cascade decomposition, formal verification

MSC 2020: 37N99, 70G45, 82C99