The Periodic Table as an Emergent Helicoidal Manifold: A Unified Information-Theoretic Analysis of the Atomic Elements Z = 1–103

Here we perform a detailed information-theoretic (IT) analysis of atomic electron densities in the periodic table, from hydrogen (Z = 1) to lawrencium (Z = 103). By use of the Shannon entropy, the Fisher information and the disequilibrium functionals in both position and momentum spaces as fundamental descriptors of the atomic densities, the periodic table can be represented in a three-dimensional information space as a continuous, highly ordered manifold. The analysis shows that chemical periodicity naturally emerges as a helicoidal manifold (reminiscent of a helix) at the coordinates of a 3D theoretic-information space (Shannon, Fisher, Disequilibrium), with each period forming one segment within the continuous global trajectory. We find information-theoretic signatures of shell structure, sub-shell filling, and electron-configuration anomalies, such as the familiar irregularities seen in chromium and copper. Therefore, the helicoidal character emerges naturally and is not imposed a priori. Further, through the uncertainty principle of the complementary analysis in momentum space, more insights are gained by exposing maximal information-theoretic differentiation for lighter atoms and compression among heavy elements. Notably, momentum-space analysis reveals that hydrogen occupies a natural intermediate position between helium and lithium based on kinetic energy distribution—contrasting with IT position-space results that emphasize hydrogen’s unique delocalized electron density. Indeed, the 3D IT representation of the elements in position space aligns with the view that H does not belong to either the alkali metals or the halogens, but rather stands as a unique, standalone element. This complementary perspective provides new quantitative support for understanding hydrogen’s dual chemical nature, providing new quantitative insight into ongoing debates about hydrogen’s optimal periodic table position. Furthermore, by considering triadic relationships and complexity properties in relation to the López–Mancini–Ruiz (LMC) and Fisher–Shannon (FS) functionals, we show that atomic complexity increases monotonically along with nuclear charge, and we provide a quantitative measure of how organized atomic electron densities are distributed throughout the periodic system. Based on our IT analyses, the fundamental character of periodicity could be addressed by employing helicoidal representations that highlight the characteristics of hydrogen, while simultaneously preserving the autonomy of the blocks of elements.