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

This section presents the unified theory of macroscopic stationary states for complex systems—the Lawof Nonuniformity. The theory takes the Foundational First Principle (Axiom A0, the binary randomprocess) as its axiomatic substrate, gives all stationary distributions a unified coordinate on the (α, β, γ) three-dimensional manifold, characterizes evolutionary fate by the three control parameters (λ, κ_*, μ_τ), and builds a complete axiomatic system of six axioms (A0–A5) + eight theorems (T1–T8)+ three corollaries (C1–C3). The theory’s central position is that the horizontal-axis position(spectrum coordinate) is the developmental coordinate and does not by itself determine health; thevertical-axis three-state (static / stationary / dynamic expansion) jointly with the systemtype determineshealth. The T7 health criterion is sharpened from a single “residence near the baseline”to a threelayer composite of “vertical-axis matching system type + adaptability + sustained maintenance,”anduses the Red Queen stationary state vs. frozen stationary state refinement to distinguish the intrinsicmechanisms of long-lived systems (intra-cluster vitality + inter-cluster energy / matter / informationexchange + connections under control). The T8 micro phase transition accumulation principle reduces asingle macroscopic phase transition to a cascade of multi-cluster random connections plus mutualreinforcement, corresponding to the literature on Buldyrev (2010) interdependent-network percolation. Axiom A5 (spontaneous evolutionary directionality) forms a precise dual with the Second LawofThermodynamics: isolated systems spontaneously slide toward uniformity, whereas open systems withflux spontaneously slide toward the super–power-law end / runaway / static fossilization. The theoryfamily is cross-domain-isomorphically validated in five domains: astrophysics, life sciences, economics& finance, social networks, and ecology. Core new contributions of this section: the explicitaxiomatization of A0; independent labeling of A1-b (the exponential law baseline); the precise duality of A5 with the Second Law; the T6 head–tail adaptation law; the T7 Red Queen stationary-staterefinement; and the T8 micro phase transition accumulation + engineering duality.