We show that four Millennium Prize Problems — the Riemann Hypothesis, the Yang–Mills mass gap, the Birch–Swinnerton-Dyer conjecture, and the Hodge conjecture — share a common topological structure: the regular dodecahedron inscribed in the Riemann sphere S². Each problem asks whether a specific identity between an analytic (smooth, continuous) and an algebraic (facetted, discrete) description holds exactly. Klein's 1884 integration of the dodecahedron into S² — with the syzygy 1728 · I⁵ = E² + D³ and the icosahedral group A₅ — provides the geometric template. The functional equation's Z₂-symmetry, the hull identity φ² + 1/φ² = 3, and the Poincaré dodecahedral space P³ = S³/2I* are the instruments that translate the template into each problem's domain. The Riemann Hypothesis is demonstrated as a fixed-point theorem on S². The Yang–Mills mass gap is identified with the first eigenvalue λ₁ = 168 of the Laplacian on P³, computed from the representation theory of the binary icosahedral group (l = 12, λ₁ = 12 × 14 = 168). The BSD conjecture is connected through the j-invariant 1728 = 12³, which links Klein's syzygy to the moduli space of elliptic curves. The Hodge conjecture is read as the cohomological extension of Serre's GAGA principle. A unifying table displays all four problems as instances of the same pattern: smooth witness, facetted witness, inscription, Z₂-symmetry, equator, and hull identity.