Redistribution Capability in Adaptive Liquidity Systems: Mathematical Representations and Structural Properties

Adaptive liquidity systems may possess identical observable compositions and functional organizations while differing in the transformations that remain feasible from their current states. This motivates the introduction of redistribution capability as an additional state property describing the structure of a system's available redistribution possibilities. This paper investigates the mathematical representation and structural properties of redistribution capability. The concept is positioned relative to established notions of admissible controls, reachability, controllability, and viability. Redistribution capability is not identified a priori with any one of these objects. Instead, it is treated as a system-level property whose mathematical realization depends on the information required for a particular analytical task. Five principal representations are examined: scalar, set-valued, graph-based, operator-based, and geometric. Minimal two-, three-, and four-component systems are used to show how these representations preserve different information concerning redistribution magnitude, direction, feasibility, transition paths, transformation mechanisms, and reachable structure. The four-component construction further demonstrates that redistribution capability may depend on functional organization, system architecture, redistribution policy, and environmental conditions rather than on observable composition alone. The paper establishes that these representations are connected by information-preserving or information-reducing mappings and generally form a partially ordered family rather than a universal linear hierarchy. Lower-information representations may be obtained as projections of richer structures, whereas reverse reconstruction is generally non-unique. Consequently, equality under a compact representation does not in general imply structural equivalence of redistribution capabilities. The resulting formulation provides a mathematical basis for treating redistribution capability as a distinct component of adaptive liquidity state descriptions while remaining compatible with established mathematical theories of constrained dynamical systems. It also provides a foundation for subsequent analysis of redistribution dynamics, functional reclassification, and the geometry of adaptive state spaces within Wave Liquidity Redistribution Theory (WLRT).