Gradientology: Foundations of the Primordial Triad — Treatise XII: The Derivation of the Planetary Engine and the Geological Gradient

This treatise addresses the crucial scalar transition between cosmic physics and biological life by deriving the rocky planet as a necessary phase-separation engine. Following the establishment of universal physical laws in Treatise XI, the Relational Field encounters a ”Bandwidth Crisis”: diffuse cosmic fluxes lack the concentration density required for complex, self-sustaining informational loops. We demonstrate that gravitational collapse of heavy elements into hydrostatic spheres serves as a systematic sorting mechanism for the triadic primitives—Systematization (E), Constraint (C), and Registration (F )—creating the core-mantle-crust architecture as a macro-scale instantiation of the Inversion Principle. The treatise then descends to the micro-scale, deriving alkaline hydrothermal vents as natural geochemical batteries where Iron-Sulfur mineral labyrinths function as the first ”inorganic cells.” We prove that the proton gradient driving early metabolism was a geological given, not a biological invention. Finally, we derive the informational takeover by RNA and the subsequent ”Great Decoupling” through which life internalized the inversion principle via ATP-synthase and DNA-protein architecture, achieving autonomy from its planetary womb. The planetary engine is thus revealed as the necessary intermediate scale that transforms cosmic potential into biological actuality.