The Law of Nonuniformity: A Universal Principle from Microscopic Binary Exponentials to Macroscopic Gradient Excess

This paper provides a self-contained interpretive–diagnostic framework for the macroscopic behavior of open, flux-carrying systems. Using relative entropy, we define two nonequilibrium potentials: Uᵤnif, referenced to the uniform distribution, applies to closed (state-count-conserving) systems; Uₑxp, referenced to the same-mean exponential distribution, applies to open (flux-conserving) systems. We argue that under a mean constraint the exponential law is the maximum-entropy distribution and is therefore the geometric zero of open systems. So that an ordering quantity can be offset against work, we introduce an energy-dimensioned ordered free energy 𝒰 ≡ kB·T_*·Uₑxp, and on this basis write a master equation (a budget balance) for the evolution of order together with its three-variable coupling to activity and work rate, yielding a maintenance fixed point, a fossil state, a minimum maintenance power, and a critical margin, all dimensionally consistent. We clarify the variational status of the dynamics: a single-coordinate relaxation can always be written as a gradient flow of a complexity free-energy functional, but this is an identity rewriting that carries no independent physical content; the genuinely falsifiable structural claim is multivariate detailed balance, and the full three-way coupled system generically breaks detailed balance, retaining only a weak variational structure in the quasipotential sense. At the stochastic level the dynamics corresponds to an Onsager–Machlup path action, and the fluctuation theorem yields a second law for open systems—whose compact form is a nonpositive free-energy dissipation rate, equivalent to nonnegative internal entropy production. We further give: a dimensionless number distinguishing static from Red-Queen dynamic steady states, with an observable scaling law; a spatial gradient-surplus principle, with a proof that the spatial profile’s “exponential vs. power law” is controlled by a single screening-length knob—dissipation compresses the profile to an exponential, while vanishing dissipation opens it to a power law; a policy result—uniform additive policy cannot prevent multiplicative power-law concentration, and suppressing heavy tails requires acting on the multiplicative growth channel; and an ontological ordering of phase transitions—the exponential first-order transition is the primal transition in the zero-constraint limit, while the power-law second-order transition is a secondary form shaped by external constraints. Throughout, we distinguish rigorously borrowed mathematical tools from the framework’s own diagnostic conclusions, and give falsifiable test points.