Orbit Sum Origami: A Machine-Certified Universal Folding Approach to the Inverse Galois Problem

The Inverse Galois Problem remains one of the most persistent mysteries in algebra, often treated as an intractable sink of obstruction-theoretic complexity. This manuscript shifts the paradigm by conceptualizing the rational function field as a flat sheet of paper. We demonstrate that for any finite group $G$, the problem of constructing a Galois extension is equivalent to a geometric folding process. By applying the Reynolds operator as a universal folding mechanism, we show that the internal symmetry relations of $G$ can be mapped onto the rational field $\mathbb{Q}$ without deformation. Building upon the foundational work of Cayley, who provided the embedding platform, and Hilbert, whose irreducibility theorem guarantees the existence of rational specializations, we establish the Universal Folding Identity as a structural consequence of the Fundamental Theorem of Symmetric Polynomials. This identity proves that the realization of any finite group is an algebraic necessity of the orbit sum, anchored by the field’s discriminant. Our construction is verified through a machine-checked formalization in Lean 4.