Categorical variables arise naturally in many optimization problems but lack the intrinsic order and metric structure required by geometric derivative-free optimization (DFO) algorithms. Existing approaches either impose an artificial ordering, which biases the search, or rely on random neighbor selection, thereby compromising reproducibility. This paper proposes a deterministic treatment of categorical variables based on permutation-induced neighborhood rotation. Categorical values are represented through a fixed canonical encoding, while their neighborhood structure is defined by a permutation that changes periodically according to a deterministic schedule. This mechanism ensures systematic exploration of all categorical alternatives without imposing a fixed artificial order or relying on random number generators. The proposed framework is guided by six design principles and integrates naturally with geometric DFO algorithms operating in normalized search spaces. It preserves full determinism, cache compatibility, and bit-level reproducibility. This preprint documents the core theoretical and algorithmic ideas underlying the proposed DNR framework and establishes priority for the deterministic treatment of categorical variables in geometric DFO.
J. F. A. Madeira (Sun,) studied this question.