The Topography of Atomic Stability: Mapping Nodal Stationary Points in Core Density (Z = 1–36)

We investigate the structural topography of atoms for elements Z=1–36 using experimental NIST data. A geometric descriptor, the core density, is defined as ρ = E_core / V, where E_core represents the cumulative ionization energy of the core and V is an effective atomic volume derived from covalent radii. The first ionization energy and core energy exhibit a low Pearson correlation (r = 0.08), consistent with the core and valence occupying distinct structural domains. Using B-spline interpolation and bootstrap resampling (10,000 replicates), we identify statistically significant stationary points — local maxima at Z=2 and Z=18, and a local minimum at Z=10 — that align precisely with shell closures. Bootstrap 95% confidence intervals confirm the significance of these extrema, and a permutation test yields p < 0.0001 for their simultaneous occurrence at the noble gas positions. A sinusoidal fit independently corroborates these locations. The invariance of these nodes across covalent, van der Waals, and ionic radii confirms they are robust features of the electron distribution rather than artifacts of a particular operational definition. The analysis was conducted under a pre-registered blind analysis framework: the descriptor was defined before analysis began, the three radius definitions were applied independently before comparison, and the permutation and bootstrap tests were specified before the data were analysed. The stationary points at Z=2, 10, and 18 emerged from the data's own structure without investigator intervention. This periodic pattern is consistent with the descriptive language of the Dynamic Equilibrium Principle used as a structural descriptor, providing an empirical geometric representation of shell structure. These empirical findings are compatible with that descriptive language but do not rely on it for their validity.