Dimension-Free Quantitative Rigidity for Vector-Valued Gaussian Noise Stability

For every target dimension k >= 8 and every correlation 0 < r < 1, this manuscript proves the sharp Gaussian noise-stability inequality for centered ball-valued maps and explicit quantitative rigidity uniform in all input dimensions n >= k. A first-Hermite coefficient subspace selection and weighted radial estimates give a linear deficit bound for weighted squared L2 distance to normalized projection. The unweighted squared L2 estimate has exponent k/(k+1), which is optimal by a centered core-flip construction. Negative-correlation rigidity requires no mean constraint. A corollary characterizes near-optimal assignments on the specified Gaussian rank-k cut instance and controls their full Gaussian correlation profiles. The proofs build on the vector-valued spherical framework of Hwang, Neeman, Parekh, Thompson and Wright, Heilman's radial-weight analysis and existing quantitative vector shell-mean stability, scalar robustness results of Mossel and Neeman and Eldan, and standard birth-death intertwining methods. The manuscript does not claim the first vector-valued stability estimate, resolve ranks three through seven, optimize constants, or provide a new approximation ratio or discrete hardness threshold. Version 2.0 adds dimension-uniform full-orbit quantitative stability and the sharp unweighted exponent to the sharp-value result. The conditional Unique Games per-correlation promise-gap consequence in Appendix A of Version 1.0 remains valid and is explicitly referenced rather than reproduced. The manuscript includes an AI-methodology disclosure. The source archive contains the complete LaTeX source and an exact-rational certificate script.