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

Abstract: This paper is the ninth version (V9) of a prior framework, restating and correcting an explanatory–diagnostic account of the macroscopic behaviour of open, flow-through systems. The central organizing proposition, the Principle of Nonuniformity, is retained 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 most important change in this version is a retraction. The headline result of V8 was the establishment of the extreme-value family as a fifth aggregation channel. This version judges that formulation untenable and demotes it to a second-order readout mounted on the distributional axis. The reason comes from V8’s own argument: the extreme-value type is not determined by itself but by the parent tail, and an object carrying no information independent of the parent cannot stand beside the parent as a channel. The argument is stronger after the demotion—four basic operations yield four first-order structures plus one second-order readout, which is more accurate than “five channels” and no longer needs to answer why the fifth channel never had a ruler of its own. Four further substantive corrections are made. First, V8 had already split the external loss into endogenous renewal and exogenous shock, but the Red-Queen number was not updated and still read as the ratio of the shock term to the relaxation term—inconsistent with the split; this version redefines it as a ratio of two rates, ρRQ ≡ μᵣen/λᵣelax, rewrites the master equation, the minimum maintenance power, and the observable scaling law accordingly, and obtains a new falsifiable prediction: after an equal perturbation, the amount by which the recovery rate exceeds baseline is exactly the renewal rate. Second, the potential structure is refined from one tier into three (quadratic free energy, Landau expansion, large-deviation logarithmic integral), with numerical evidence that the middle tier assigns the ground state incorrectly. Third, the negative result on the independent side is tightened to the derivative of the potential with respect to work vanishes identically, rather than “the potential vanishes”—the latter admits a counterexample. Fourth, three guardrails are added to the extreme-value readout: the Fréchet criterion is regular variation of the tail, not unboundedness of support; the maximum of a lattice parent converges to no non-degenerate limit; and for dependent sequences the type is unchanged while the effective sample size is discounted.