The Rhythm of Life and Death: Random Matrix Theory Reveals Level Repulsion in Phanerozoic Mass Extinctions and Radiations

We extend a random matrix theory (RMT) program, previously applied to geological boundaries, faults, mantle plumes, ore deposits, geysers, and glacial outburst floods, to the history of life, testing whether the timing of major macroevolutionary events carries a level-repulsion signature. The motivation is an ecological-niche-shadow hypothesis: Earth's ecological carrying capacity is finite, so a global-scale extinction (which empties niches) or radiation (which saturates them) imposes a recovery or reshuffling interval before the next comparable event, suppressing short recurrence times in the manner of eigenvalue repulsion. We analyze three configurations using a local-unfolding procedure that removes the long-term, non-stationary trend in Phanerozoic event density. (A) Global extinction peaks (n=32 Phanerozoic extinction events including the Big Five): the spacing ratio r-bar = 0.618 lies in the GOE-GUE range, with CV = 0.45, Brody beta = 1.41, and a KS test strongly preferring GOE (D=0.067) over Poisson (D=0.308). (B) Global radiation peaks (n=29 origination events including the Cambrian Explosion and GOBE): r-bar = 0.628, again preferring GOE. Extinction and radiation, the destruction and construction of biodiversity, obey the same repulsion law. (C) Mixed independent clades (forward model, 12 ecologically isolated lineages superposed): the mixture collapses to Poisson (r-bar = 0.359), confirming the spectral superposition theorem. Together with six previously studied physical Earth systems, nine physically unrelated systems now share one principle: GOE-class level repulsion emerges wherever a single long-memory charge-and-release process is isolated, and Poisson emerges wherever many independent sources superpose, from a geyser conduit to the entire biosphere, from minutes to half a billion years. Sixth in a unified RMT program spanning Earth and life.