Geometric Vacuum Selection in the Standard Model: A Two-Anchor Principle from Grassmannian Geometry

We present a geometric framework for understanding the parameter structure of the StandardModel. Starting from the Grassmannian manifold Gr(k,N)—the space of k-dimensional subspaces inan N-dimensional vector space—we demonstrate that two fundamental observables, the weak mixingangle and the gauge-gravity hierarchy, uniquely select the integers (k, n) = (3, 13) with N = k+n =16. This selection is not approximate but exact: no other integer pair satisfies both constraintssimultaneously within experimental tolerances. We provide complete mathematical proofs of globaluniqueness, analyze the robustness of the selection across tolerance variations, and show that theresulting Grassmannian dimension D = k(N −k) = 39 determines the hierarchy between the Planckand electroweak scales. The framework makes over forty parameter-free predictions for StandardModel quantities, with a mean accuracy of 0.1%. We discuss the physical interpretation, connectionsto gauge theory, and implications for the hierarchy problem.