Building Latent Spaces out of Sandwiches: Complementary Spectral Factorizations for Graph-Metric Embeddings

For the purpose of defining coherent latent spaces from corpora of text or imageembeddings indexed by a feature graph (as in ArrowSpace \citep{moriondo2026arrowspace}),we study a complementary spectral construction for embedding feature vectors throughtwo slices of a graph Laplacian.Given a symmetric positive-semidefinite Laplacian $L=U\diag(\lambda)U^\top$, a cut$\theta$, and the decomposition $L_{\low}=U\diag(\min\{\lambda_j,\theta\})U^\top$ and$L_{\high}=U\diag(\pospart{\lambda_j-\theta})U^\top$, we embed a row vector as$z_\theta(x)=[xL_{\low}^{1/2}\mid xL_{\high}^{1/2}]$.Although the coordinates depend on the cut, their Gram geometry does not:$\langle z_\theta(x),z_\theta(y)\rangle=xLy^\top$.Consequently, distances, angles, kernel values, nearest-neighbour rankings, andcentroid decisions based only on the concatenated coordinates are cut-invariant.The cut instead controls an interpretable decomposition of each point's fixedLaplacian energy.We characterize this balance as a monotone convex spectral survival curve, provethat every sandwich is a partial-isometric re-expression of the canonical warp$xL^{1/2}$, and extend the result to arbitrary hard or soft multi-slice filter bankswhose squared responses partition a target spectrum.The analysis separates geometry from evidence: complementary sandwiches preserveone graph metric while exposing cut-dependent diagnostics within it.