The Repair Inequality: A General Accounting Law for Persistence Under Selection

Darwinian natural selection is usually formulated in biological terms: heritable variation, differential survival, and reproduction. This paper proposes a more primitive mathematical form. Any persistent structure must withstand state-dependent loss. Let K denote effective selection loss, M denote effective maintenance or repair, and S denote structural reserve. Then persistence is governed by the balance equation dS/dt = M - K. The central result is the repair inequality M >= K. If M > K, reserve grows; if M = K, reserve is conserved; if M < K, finite reserve is depleted and persistence is time-limited. The associated exchange rate R = M/K has a critical boundary at R = 1. Biological evolution is therefore not the definition of selection, but its most visible empirical instance. Organisms persist by metabolism, repair, reproduction, and inheritance because those mechanisms supply maintenance against loss. The same accounting law applies to reliability engineering, survival analysis, open systems, computational search, and resource-bounded reasoning. This paper formalizes the repair inequality, proves elementary lifetime and reserve-buffer theorems, distinguishes effective repair from gross energy input, and shows how replication appears as maintenance-of-maintenance when repair machinery itself degrades.