Ununiformity Emergence Theory (UET): The Law of Nonuniformity in Open Systems

This paper presents a self-consistent explanatory–diagnostic framework for the macroscopic behaviour of open, flow-through systems. Two nonequilibrium potentials are defined by relative entropy: U_unif, referenced to the uniform distribution, for closed (state-count-conserving) systems; and U_exp, referenced to the same-mean exponential distribution, for open (flux-conserving) systems. Under a constraint on the mean the exponential is the maximum-entropy distribution and hence the geometric zero point for open systems. To let order be offset against work, an energy-dimensioned ordered free energy Ū ≡ k_B·T_*·U_exp is introduced, and a dimensionally consistent master equation for the evolution of order is written on that basis. All three expenditure terms of the master equation are multiplicative—spontaneous loss, endogenous renewal, and exogenous shock all multiply the existing stock of order. This is not a stylistic choice. First, the non-negativity of the KL divergence requires dŪ/dt ≥ 0 at Ū = 0, and an additive expenditure would push Ū into the negative region. Second, the activity equation of the same system already writes the shock as multiplying Φ (one cannot deactivate units that are not yet active), and the order equation must be isomorphic to it (one cannot dismantle structure that does not yet exist). Third, the microscopic reduction gives a loss function that vanishes exactly at the absorbing state, and the multiplicative form is the simplest form satisfying that constraint. The fossil state is consequently an unconditional exact fixed point, and Ū ≥ 0 is preserved automatically. External loss is further split into two ledgers of opposite character: endogenous renewal μ_ren (the system replacing its own components—a means of maintenance) and exogenous shock f_shock (the environment removing them—something the system can only absorb); both are rates. The Red-Queen number is accordingly defined as a ratio of two rates, ρ_RQ ≡ μ_ren/λ_relax, dimensionless and free of both the stock and the shock intensity. Full multiplicativity turns the observable scaling law into a clean linear relation, W/Ū = a + b·μ_ren + c·f_shock, whose three coefficients give three physical quantities and whose model is over-determined and internally checkable. We clarify the variational status of the dynamics (in three tiers) and refine the potential structure into three layers (quadratic free energy, Landau expansion, large-deviation logarithmic integral), with numerical evidence that the middle layer assigns the ground state incorrectly, so that global depths must be read from the logarithmic-integral closed form. The theoretical core is cross-scale critical synchronization: for two levels with critical margins Δ_a, Δ_b and coupling product K, the load-bearing threshold is the vanishing of the dominant slow mode, K_c = Δ_a·Δ_b, with the joint recovery rate vanishing as μ_−(K) = (Δ_a+Δ_b)/2 − √(ε²+K), a curve with no free parameters. Higher-level emergence modifies lower-level margins downward and thereby reshapes K_c itself, yielding an irreducibility argument unavailable to single-level dynamics. Every assertion carries an explicit domain of validity, and one guardrail runs through the whole: shape does not adjudicate mechanism—observing a distributional shape is insufficient to determine the mechanism producing it, and the burden of discrimination lies in perturbation response, not goodness of fit.