A dimension-independent spectral rounding bound for orthogonal synchronization

Let a finite connected weighted graph carry orthogonal edge labels in $\mathrm{O}(r)$, let $\gamma$ be the spectral gap of its normalized scalar Laplacian, and let $\Lambda_r$ 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 $\Lambda_r/r\le \nu_*\le \nu(Q)\le(3+12/\gamma)\Lambda_r/r \le18\Lambda_r/(r\gamma)$. 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.