Intimate Partner Violence as Coercive Wrong Attractor: Ground State Erasure, Trauma Bonding as Basin Deepener, and the Geometry of Leaving

We apply the Displacement Framework to intimate partner violence (IPV), formalizing the relationship ground state S°_relationship as a five-component equilibrium: physical safety, psychological safety, identity preservation, financial autonomy, and social connection. Coercive control destroys these components incrementally — isolation removes social connection, financial control removes autonomy, gaslighting erodes identity, physical fear removes safety — creating a wrong attractor through systematic S°_relationship erasure.

Nine formal propositions are derived covering: S°_relationship as multi-component safety-autonomy equilibrium, coercive control as incremental component erasure, social isolation as basin-deepening mechanism, trauma bonding through intermittent reinforcement (structurally equivalent to variable-ratio reinforcement in gambling), the exit barrier as the defining feature of the IPV wrong attractor, leaving as the highest-danger moment when D(ξ_exit) peaks, Campbell's Danger Assessment as formal basin-depth measurement, domestic violence shelters as protected alternative basins, and safety planning as exit gradient mapping.

The framework resolves the 'why didn't they leave' question as wrong-attractor blindness: leaving requires traversing D(ξ_exit) that includes financial dependence, housing insecurity, child custody fear, stalking escalation risk, social shame, and legal system failures. Systemic failures are analyzed as landscape deepeners that increase exit barriers rather than reduce them.