The Principle of Nonuniformity: From Binary Exponentials to Gradient Surplus in Open Systems

This paper restates and extends, as the V7 revision of an earlier framework, an interpretive–diagnostic account of the macroscopic behavior of open, flux-carrying systems. The central organizing statement is renamed from the Law of Nonuniformity to the Principle of Nonuniformity and is recast as a first principle for open systems: the state of an open system deviates from uniformity in both space and time, and this deviation is structural — non-ergodic and accumulable — rather than a mere fluctuation. The dividing criterion between fluctuation and structure is ergodicity breaking (ensemble average ≠ time average). We make the principle’s domain precise by isolating the one regime it does not govern, the ergodic equilibrium floor (the U = 0 baseline), and by explaining why uniformity is the exception at every scale — the universe included — while the principle is stated for open systems. On this footing we retain and develop the quantitative apparatus. Using relative entropy we define two nonequilibrium potentials: 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. Under a mean constraint the exponential law is the maximum-entropy distribution and is therefore the geometric zero of open systems. To offset an ordering quantity against work we introduce an energy-dimensioned ordered free energy 𝒰 ≡ kB·T_*·Uₑxp, and write a dimensionally consistent master equation 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. We clarify the variational status of the dynamics: a single-coordinate relaxation is always an identity rewriting as a gradient flow with no independent physical content; the genuinely falsifiable structural claim is multivariate detailed balance, and the full three-way coupled system generically breaks it, retaining only a weak quasipotential variational structure. 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 in which a single screening-length knob interpolates between exponential and power-law profiles — dissipation compresses the profile to an exponential while vanishing dissipation opens it to a power law; a policy result — uniform additive intervention 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 primal in the zero-constraint limit, while the power-law second-order transition is a secondary form shaped by external constraints. Throughout we separate borrowed mathematical tools from the framework’s own diagnostic conclusions, and give falsifiable test points.