Octave-Decade Law: A Unified Scaling Description of Chemical Evolution from the Early Universe to Heat Death

The recognition that many periodic trends recur in groups of eight dates back to New- lands’ 1866 Law of Octaves.Newlands [1866] In the mid-20th century, Robert L. Lehrman introduced the octave–decade mnemonic as a pedagogical device for the observed period- length pattern 8–8–18–18–32. These earlier formulations were qualitative. The present work transforms Lehrman’s terminology into a quantitative, parameter- free Octave–Decade Law1 that reproduces every main-group atomic number exactly from hydrogen to oganesson, derives the +24 insertion from a single relativistic threshold (4f collapse), and extends analytically to superheavy elements, negative-n configurations, and far-future degenerate matter. With four closed-form equations for the main group and a +32 arithmetic progression for transition-metal decade midpoints, the law unifies atomic chemistry, high-pressure physics, and potential composite dark-matter candidates.