The Geometry of Malicious Envy: A Structural Formalization of Constraint Collapse, Adversarial Comparison, and Defensive Stabilization

This manuscript presents a structural, operator-theoretic formalization of malicious envy as a comparison-driven control law, transitioning relational dynamics into the realm of rigorous mathematical physics and control theory. The framework establishes a strict mathematical distinction between benign and malicious envy: benign envy is modeled as self-directed comparison-gap reduction (elevating the self), whereas malicious envy is modeled as target-directed degradation (lowering the target). Crucially, the paper models psychopathic-style escalation not as an emotional state, but as the mathematical collapse of internal (moral/empathic) and external (reputational/consequential) constraint gradients. Using game theory and dynamical systems, the manuscript proves the conditions under which reciprocal cooperation breaks down and target degradation becomes a dominant strategy. To analyze this structurally, the paper lifts the behavioral dynamics into a Hilbert space operator framework. Adversarial collapse is rigorously identified as a Birman-Schwinger spectral rupture, where the adversarial interaction operator reaches the exact threshold of Fredholm failure and its regularized determinant vanishes. Finally, the manuscript formalizes "Defensive Stabilization" via the Unavailability Protocol. It mathematically proves that emotional boundaries are structurally ineffective unless they enforce "access contraction"—physically, financially, or socially shrinking the feasible control set of the adversary. A differentiable JAX numerical simulation is included, demonstrating both the exponential divergence of unconstrained malicious loops and their exact spectral stabilization the moment access contraction clears the Fredholm collapse threshold. This framework is strictly behavioral and non-diagnostic, requiring no mind-reading or inference of hidden motives, operating entirely on observable events, access channels, and structural operator deformations.