Universal Limit Equation (ULE): A General Boundary Operator for Multidomain Constraint and Stability Analysis

The Universal Limit Equation (ULE) is proposed as a dimensionless boundary operator for representing the governing limit state of systems subject to multiple active constraints. The canonical formulation is ULE[x(·), u(·)] = supₜ maxᵢ ρᵢ(x(t), u(t), t), with the universal admissibility condition ULE≤1, where each normalized limit-state function ρi is defined such that ρi≤1 if and only if the corresponding constraint is satisfied. For static demand-capacity systems, the formulation reduces to ULE = maxᵢ(Dᵢ / Cᵢ) The framework provides a common mathematical representation for heterogeneous constraints while preserving domain-specific physical or operational models. The paper develops the canonical formulation, admissibility interpretation, bottleneck and non-compensation properties, static and demand-capacity reductions, stability reserve, dynamic extensions, optimization-gate interpretation, worked examples, and falsifiability criteria. This Version 1.0 is presented as a theoretical preprint and establishes the formal mathematical framework; broader empirical universality remains a subject for subsequent validation. Author: Mohamed Haseeb C M © 2026 Mohamed Haseeb C M. Licensed under CC BY 4.0.