Irreversible Threat Hysteresis: Non-Equilibrium Asymmetric Potential Well Dynamics and Cusp Catastrophe in Topological Fail-Safe Security Locks

This theoretical monograph establishes the Asymmetric Potential Well Engine (APWE) invariant (Falcon Impossibility F07). We eliminate reversible latching vulnerabilities under persistent adversarial coercion by embedding the physical security state coordinate within a non-equilibrium asymmetric quartic-quintic potential landscape undergoing Thom's Cusp Catastrophe: V(x) = (1/4) x^4 + (1/2) lambda x^2 + mu x + (epsilon/5) x^5. We prove that when coercive threat trajectories cross the bifurcation set Delta = 4 lambda^3 + 27 mu^2 <= 0, the operational state executes an irreversible sub-nanosecond saddle-node transition into a catastrophic attractor basin. In this terminal void basin, the reverse activation barrier diverges to infinity (Delta V_rev -> infinity) via topological collapse, strictly enforcing Landauer's thermodynamic erasure threshold (Delta S_irr >> kB ln 2). We mathematically prove that no physical energy injection, cryptographic key reconstruction, or thermal glitching can restore the cleared state, establishing the ultimate theoretical guarantee for unbreakable non-return security locks.