Bounded Cores for Extremal Kuratowski Orbits in Finite T₀ Alexandrov Spaces

Kuratowski's closure–complement theorem gives a sharp fourteen-set bound for the orbit generated by iterating closure and complement from one subset of an arbitrary topological space. In finite T₀ spaces the situation is more rigid: by work of Herda–Metzler, the fourteen-set bound is never attained, and the largest possible orbit has size ten. We study this finite T₀ phenomenon in the equivalent language of finite posets, with upward-closed open sets and bA = cl(A) = ↓ A. Our first result is an exact point-count refinement: if Kₜ₀(n) denotes the maximum orbit size over all n-point finite T₀ Alexandrov spaces, then Kₜ₀(0) = 1, Kₜ₀(n) = min(2n, 10) (n ≥ 1). We then give a pair-level extremality criterion: a subset has a ten-element orbit exactly when the five canonical sets iA, A, ibA, biA, bA are pairwise distinct. This criterion induces a canonical five-layer decomposition of every extremal pair. Using this decomposition, we prove that deletion-irreducible extremal pairs have at most seven points and classify the resulting bounded cores. Finally, we show that every extremal pair reduces, via layer-preserving deletions, to a layer-terminal core of size at most seven, and conversely arises by reversing layer-preserving one-point extensions. Thus the finite T₀ Kuratowski problem is not merely bounded by ten: its extremal pairs admit a bounded-core structural reduction, while incidence constructions produce arbitrarily large families.