The Principle of Nonuniformity: From Binary Exponentials to the Completion of the Extreme-ValueChannel in Open Systems

Abstract:This paper is the eighth version (V8) of a prior framework, restating and extending an explanatory–diagnostic account of the macroscopic behavior of open, flow-through systems. The central organizing proposition, the Principle of Nonuniformity, is stated as a first principle for open systems: an open system departs from uniformity structurally along both a spatial and a temporal axis, a departure continuously supplied by work and paid for by internal entropy production. The full quantitative apparatus is retained and developed. Two nonequilibrium potentials are defined via relative entropy: Uunif, referenced to the uniform distribution, for closed (state-count-conserving) systems; and Uexp, referenced to the same-mean exponential distribution, for open (flux-conserving) systems. The dynamics are governed by an income-minus-expenditure master equation whose external forcing term is now split precisely into an endogenous-renewal part and an exogenous-shock part—a split that gives the Red-Queen number a sharper mechanistic meaning. We further give a dimensionless number distinguishing static from Red-Queen steady states, together with an observable scaling law; and a spatial gradient-surplus principle in which a single screening-length knob interpolates between exponential and power-law profiles, with the screening length now identified microscopically as scaling directly with the diffusion coefficient. The most important new result of this version completes the classification of aggregation limits: whereas the prior framework recognized four canonical steady-state forms (exponential, Gaussian, log-normal, power-law) and explicitly excluded the extreme-value family, this version establishes extreme-value combination as an independent fifth aggregation channel, classified by the Fisher–Tippett–Gnedenko theorem, yielding a complete five-channel classification underwritten jointly by three limit theorems (Lévy–Khintchine, Kesten, Fisher–Tippett–Gnedenko).