Certified Stratified Phase Diagrams for Finite Hopkins Lithography

Abstract

We study an explicitly declared finite partially coherent Hopkins imaging model and develop a proof-carrying framework for topology changes under focus and threshold variation. Three phase diagrams are separated: bifurcations of the spatial critical set, changes in adjacent critical-value ownership, and topology changes of a fixed target contour. The finite Hopkins intensity is a Laurent trigonometric polynomial with exact focus periodicity and reflection symmetry. After sine--cosine polynomialization, the critical and event relations are semialgebraic and therefore admit finite point/arc stratifications over one focus period. At each fixed focus the critical-value set is finite, so strict adjacent lower and upper owners are attained whenever the corresponding side is nonempty. On event-free Morse cells the owning values continue analytically; load-bearing ownership singularities include terminal-fold cliffs, transverse branch-crossing kinks, and reflection-protected pitchforks. Fixed-target contour topology is instead controlled by the target discriminant $J = \tau$, $\nabla_x J = 0$. An explicit level-set transport field proves ambient-isotopy rigidity on every target-event-free focus interval. Newton-polytope and multihomogeneous bounds make the isolated candidate sets explicitly finite, while positive-dimensional event components are handled by semialgebraic event cells. For a frozen 37-source $\text{ArF}$ fixture, under the declared computer-assisted arithmetic contract, we certify a sharp model-relative focus--dose chamber on $|z| \le 10\text{ nm}$, several distinct ownership singularities, transverse target-threshold events, and a target-contour component sequence $1 \to 3 \to 5 \to 4$. All statements concern the declared finite aerial model and are not wafer-process qualification claims.