The Unique Forward-Closed Polyhedral Loop at the Icosahedral Bound: Machine-Verified Proofs in Lean 4

We prove the existence and uniqueness of a closed forward path through regular and semi-regular polyhedra---a cycle using only rectification, dualization, stellation, continuous deformation, and subset extraction, with no operation reversed. Starting from the tetrahedron, forward closure uniquely determines the terminal point: after four stellations, the rhombic hexecontahedron achieves 60-face saturation where faces form a single $I$-orbit with trivial stabilizer, exhausting the rotational degrees of freedom of the icosahedral rotation group (order 60, the maximal polyhedral rotation group in three dimensions). The return path is equally constrained: continuous deformation identifies the convex equilibrium, followed by harmonic-guided extraction at the icosidodecahedron, whose vertices are precisely the saddle points of Poole's degree-6 icosahedral invariant. The cycle exhibits quasicrystalline closure---topological return with intrinsic five-fold orientational ambiguity arising from pentagonal-tetrahedral incommensurability, quantified by the exact algebraic identity $\phi^8 + \phi^{-8} = 47$ and belt width $\omega(B) = 2\pi\phi^{-8}$. Core algebraic and group-theoretic proofs are machine-verified in Lean 4.