The Triple Lock: A Model of Sequential Bio-Geochemical Filters on the Path to Complex Life

We propose a physically motivated replacement for the speculative factors fl × fi in the Drake Equation. The Triple Lock Index (TLI) evaluates the probability of
complex (eukaryotic) life through three sequential filters: spectral compatibility with oxygenic photosynthesis (Filter 1), a sustained phosphorus cycle via plate tectonics
(Filter 2), and an oceanic habitable zone permitting eukaryogenesis under stellar radiation (Filter 3). Applied to a catalogue of 13 worlds, TLI spans five orders of
magnitude—from ∼5% (Kepler-442b) to < 10−5% (Proxima b)—identifying eight distinct failure patterns. The model correctly predicts Venus’s failure mode (hydro-
sphere loss at Filter 2) in a blind run, generates falsifiable predictions with quan-titative thresholds, and yields N ≈ 56 civilisations in the Galaxy when integrated
into the Drake Equation—consistent with the Fermi Paradox. Monte Carlo analysis (10,000 iterations) reveals that TLI uncertainty for well-characterised worlds is ∼28–
38×, while for M-dwarf planets it reaches 104–107×, driven by unconstrained XUV fluxes—identifying these as priority observational targets for JWST/HWO/LIFE.