A Topological Correspondence Across Four Millennium Problems

Four Millennium Prize Problems -- the Riemann Hypothesis (RH), theYang-Mills mass gap, the Birch-Swinnerton-Dyer conjecture (BSD), and theHodge conjecture -- share a common topological template: a facettedalgebraic body (the regular dodecahedron, rotation group A_5) inscribed ina smooth analytic surface (the Riemann sphere S^2), with a Z_2-symmetryselecting an equator. Each problem asks whether a smooth (analytic) and afacetted (algebraic) description of one object agree. We make thiscorrespondence precise, retaining its verifiable mathematical content:Klein's 1884 integration with the syzygy 1728 * I^5 = E^2 + D^3, the exactspectral gap lambda_1(P^3) = 168 on the Poincare homology sphere, theappearance of Klein's 1728 as the j-invariant normalisation, and the GAGAbridge. We then draw the distinction the earlier version missed: the fourproblems share a structural correspondence but split by logical form. TheRH is equivalent to a Pi-0-1 arithmetic sentence and therefore carries avalid conditional grounding; Yang-Mills, BSD, and Hodge quantify overcontinuous or cohomological objects, sit higher in the arithmetichierarchy, and carry no such grounding. The correspondence is real; itdistinguishes and illuminates the four, but it does not solve them.