arXiv:2607.08773v1 Announce Type: new Abstract: In this work we present a rigorous theoretical framework to a foundational problem of AI safety, namely adversarial robustness. In particular, we show that the adversarial robustness problem can be reduced to a lattice traversal problem. Each element of this lattice corresponds to an interval, i.e., an axis-aligned hyper-rectangle, containing an input point $\mathbf{x}$. Consider a multilayered perceptron classifier (MLP). An interval $I$ constitutes a sound certification if $\mathbf{x} \in I$ and $\mathbf{x}$ can be freely perturbed in $I$ without changing the MLP's prediction. Complementarily, an interval $I$ constitutes a complete certification if $\mathbf{x} \in I$ and when $\mathbf{x}$ moves outside of $I$ the MLP's prediction is guaranteed to change. While the sound certification problem corresponds to the well-studied adversarial robustness, complete certifications have not been examined in the literature. We develop lattice traversal operators, which we apply in a refine & verify iterative scheme. Using formal MLP verifiers, sound maximality and complete minimality are guaranteed. Moreover, we examine objective optimization pr