Shell formulas for instantons and gauge origami

We introduce the shell formula-a framework that unifies the description of partition functions whose pole structures are classified by Young diagrams of arbitrary dimension. The formalism yields explicit closed-form expressions and recursion relations for a wide range of physical systems, including instanton partition functions of 5d pure super Yang-Mills theory with classical gauge groups, as well as gauge origami configurations such as the magnificent four, tetrahedron instantons, spiked instantons, and Donaldson-Thomas invariants in $\mathbb{C}^3$ and $\mathbb{C}^4$.