Classification and Rigidity of Configuration-Induced Metrics on Finite Sets

This study defines finite metrics obtained by uniformly averaging graph distances over finite families of connected configurations on a fixed vertex set. This construction provides a class of metrics that strictly extends classical graph metrics while remaining entirely combinatorial. We introduce a precise notion of metric equivalence for configuration spaces, develop a canonical reduction theory leading to metric-minimal representatives, and presents a hierarchy of invariants detectable from the induced metric. Explicit constructions show sharp bounds, non-uniqueness phenomena, and intrinsic limits of reconstruction. The results identify configuration-induced metrics as a structured and well-controlled class of finite metric spaces.               

Keywords: 

1. Configuration-induced metrics,
2. Finite metric spaces,
3. Metric equivalence and reduction,
4. Discrete combinatorial geometry