Spiral Representation of the Periodic Table: A Unified 3D Analysis of Information-Theoretic Functionals for 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 functional in both position and momentum spaces as fundamental descriptors of the atomic densities, the Periodic Table can be represented in our three-dimensional information space as a continuous, highly ordered manifold. The analysis shows that chemical periodicity naturally reveals a geometrical spiral at the coordinates of 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, subshell filling, and electron-configuration anomalies, such as the familiar irregularities seen in chromium and copper. Through the uncertainty principle of complementary analysis in momentum space, more insights are gained by exposing maximal information-theoretic differentiation for lighter atoms and compression among heavy elements. We demonstrate, by considering triadic relationships and complexity properties in relation to the López-Mancini-Ruiz (LMC) and Fisher-Shannon (FS) functionals, 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 spiral representations that continue the characteristic of hydrogen, but also preserving the autonomy of the blocks of elements.