A Rigorous Mathematical Framework for Differentiated Planetesimals: Integrating Mod-9 Invariants, Hexagonal Lattices, and Resonance Anchors in Asteroid 16 Psyche Valuations

Abstract Asteroid 16 Psyche presents a unique paradigm in planetary science as an exposed protoplanetary core. Traditional economic valuations rely on terrestrial market scarcity, yielding unrealistic estimates in the quintillions of dollars. This paper establishes a rigorous mathematical framework that bridges physical compositional constraints—specifically siderophile element distribution within iron-nickel matrices—with a discrete hexagonal lattice model governed by a Mod-9 invariant and a 7-cycle periodic break. Furthermore, we integrate the 3I pulse sequence into the master equation's resonance anchor calculation to evaluate material stability and energy thresholds at the 5184\text{ Hz} frequency limit (72^2).