On De-Roughication, De-Hyperization, and De-Nearization: ReductionPrinciples in Advanced Structural Systems

Abstract

This paper formalizes three “de-” processes that convert higher-order uncertainty into crisp structures. First,
we define a cost-sensitive deroughication operator Dα,β that selects boundary elements by rough-membership thresholds; we prove sandwich, extremal, idempotence, and monotonicity properties and show it minimizes a risk. Second,
we specify de-hyperization from (m, n)–SuperHyperStructures to HyperStructures and then to classical operations via
lifting/flattening and a canonical selector, with a recovery result for lifted classical operations. Third, we introduce
de-nearization DeNearε that removes conflicting descriptions to enforce an ε margin; we prove termination, separation,
and maximality. Applications include medical triage, fraud screening, manufacturing, and transit.