Trojan Climate Dynamics: Spectral Geometry, Metastability, and the Mathematics of Tipping

Abrupt climate tipping events pose a fundamental challenge to prediction, attribution, and governance. Conventional approaches frame tipping as the crossing of scalar thresholds in control parameters, often associated with local bifurcations, critical slowing down, or variance-based earlywarning signals. In this paper, we develop and rigorously analyze a different mathematical framework. Building on Poisson-Hamiltonian stability theory originally developed to explain the long-lived confinement of Trojan dust clouds near the Lagrange points L4 and L5, and on the Stability-Sustainability framework for open systems, we identify a class of Trojan climate subsystems: forced, dissipative, fast-slow systems whose persistence is governed by quasiinvariants and spectral separation rather than local stability alone. We formulate a canonical stochastic fast-slow climate model, define Trojan invariants and spectral gaps for climate subsystems, and prove a Trojan Tipping Theorem establishing that abrupt regime transitions occur generically through spectral-gap collapse and resonance-mediated transport, even in the absence of deterministic bifurcations. This result formalizes tipping as loss of metastable confinement rather than threshold crossing. We show that this mechanism yields concrete, computable early-warning diagnostics based on the spectral geometry of Markov generators and transfer operators, and we demonstrate their implementation in a reduced AMOC model using operator-theoretic methods directly compatible with CMIP-class diagnostics. We interpret the theory for major climate subsystems-including the Atlantic Meridional Overturning Circulation, monsoon systems, and ice-sheet-ocean coupling-and show why long periods of apparent stability can precede abrupt, irreversible change. Finally, we draw strong conclusions for climate governance: in systems governed by Trojan dynamics, sustainability is mathematically equivalent to the preservation of spectral protection, and delay in reducing forcing constitutes structural risk amplification even in the absence of observable instability.