Möbius-inspired signed inverse-power representations for pair-potential reconstruction, design, and RDF-driven inversion

Analytic pair potentials provide microscopic models for classical liquids, soft matter, and coarse-grained simulations. We develop a Möbius-inspired signed inverse-power representation for pair-potential reconstruction, potential-shape design, and radial distribution function (RDF)-driven inversion. The strict form uses the Möbius sequence as an arithmetic baseline. The extended form is a general signed integer inverse-power expansion with a continuous overall amplitude. It exactly represents the Lennard-Jones interaction and approximates attractive Yukawa and Morse potentials over specified radial intervals. Potential and force diagnostics establish the limits of short-range and long-range extrapolation. The same representation generates several prescribed interaction shapes and supports discrete inversion searches using shell-averaged force responses with a coordinate-derived finite-box boundary contribution. Six benchmarks span dilute and dense Lennard-Jones states and two temperatures of a repulsive Yukawa system. Independent Monte Carlo simulations reproduce their principal RDF features and selected local structural distributions, while resolving energy and pressure differences. Joint block resampling quantifies competition between nearby analytic forms. A frozen Yukawa approximation also retains close pair structure in one cross-temperature test, with measurable thermodynamic discrepancies. The results support explicit analytic pair models within stated radial and thermodynamic domains and identify the sampling and physical limitations of their structural interpretation.