The Cylindrical–Septadimensional Chemical Manifold: A Geometric Hypothesis and Applications for Periodic Similarity Representation

This work introduces a cylindrical–septadimensional manifold representation of the periodic system. In this framework, chemical elements are embedded within a curved similarity space defined by electronic configuration, relativistic effects, bonding behavior, and physicochemical descriptors. Periodicity emerges as a cyclic boundary condition along the valence-shell dimension, yielding a cylindrical topology that removes artificial discontinuities present in flat tabular layouts. The septadimensional structure is proposed as the minimal latent dimensionality required to preserve intrinsic variance in elemental properties while maintaining continuous periodic relationships. The model is presented as a geometric and topological representation rather than a new physical theory. Conceptual projections and manifold visualizations are provided to illustrate cylindrical periodicity and higher-dimensional continuity. Potential applications include chemical similarity navigation, materials substitution mapping, descriptor compression for machine learning, periodicity-aware generative chemistry, and exploratory visualization of elemental relationships. This representation offers a unified geometric framework for organizing chemical knowledge in continuous descriptor space.