A dimension-independent spectral rounding bound for orthogonal synchronization

Let a finite connected weighted graph carry orthogonal edge labels in O(r), let γ be the spectral gap of its normalized scalar Laplacian, and let Λᵣ be the sum of the lowest r eigenvalues of its normalized connection Laplacian. We prove that nearest-orthogonal rounding of a degree-normalized bottom spectral frame produces a field Q whose normalized squared-Frobenius frustration satisfies Λᵣ/r≤ ν_*≤ ν(Q)≤(3+12/γ)Λᵣ/r ≤18Λᵣ/(rγ). The approximation factor is independent of the matrix dimension. The proof applies a classical Hilbert-Schmidt absolute-value inequality and the graph Poincaré inequality to the positive matrix magnitudes of the spectral blocks. Their global second-moment normalization then controls the rounding error. Every block is rounded by a full singular value decomposition, including singular blocks. No noise hypothesis or separation between the r-th and $(r+1)$-st connection eigenvalues is required.