The geometry of collective collapse: A model of regime competition, structural vulnerability, and existential potential

This paper proposes a formal model of collective system dynamics grounded in competition between organizational regimes. A collective system is represented as a set of possible dynamic regimes {R₀, R₁, R₂, ...} that compete to structure the system's observable evolution through replicator dynamics. When a single regime reaches dominance, collective behavior can be approximated as gradient descent on that regime's effective potential — a standard result from nonlinear dynamics applied here to organizational systems. The central contribution is the index Δ of collective vulnerability, which formalizes the distance between the active regime and a baseline viability regime R₀ through three independent and measurable dimensions: functional alignment α, structural redundancy ρ, and deformation history D. Δ predicts the threat threshold required to displace the active regime and therefore the collapse order among comparable systems facing the same perturbation — before the perturbation arrives. A second contribution is the decomposition of the effective potential into a structural invariant V_base and a regime-specific deformation δV_k. V_base is the background potential that surfaces when the active regime can no longer be sustained; collapse is formally the convergence of dynamics onto its gradient. This decomposition resolves the circularity problem in regime identification and connects the model to Terror Management Theory, which provides individual-level microfoundations for why systems systematically build regimes that diverge from V_base. The model generates five falsifiable predictions about collapse order, transition type, reorganization depth, the structural paradox of coherence and fragility, and the asymmetric distribution of α across real systems. Four historical cases illustrate that α, ρ, and D are empirically distinguishable and that Δ produces orderings consistent with observed outcomes.