Independently pretrained vision (DINOv2) and text (MiniLM) encoders develop graph-Laplacian eigenvalue spectra of near-identical intrinsic complexity, yet the functional map between their spectral eigenbases is far from diagonal or orthogonal -- a decoupling of 'how much structure' from 'how it is oriented' that bounds spectral cross-modal alignment methods and underperforms ambient-space alignment on retrieval

Sarkar (2026) constructs k-nearest-neighbor graphs over 1,000 Flickr30k image-caption pairs separately in DINOv2 (vision) and all-MiniLM-L6-v2 (text) embedding spaces, computes each graph's normalized Laplacian eigenbasis (a manifold-respecting analogue of PCA), and fits a functional map C between the two truncated spectral bases via regularized least squares with a Laplacian-commutativity penalty. Three diagnostics decompose cross-modal geometric compatibility: the normalized eigenvalue-spectrum distance between the two encoders is just 0.043 (both follow the same concave growth profile from ~0.03 to ~0.66), indicating the two independently trained manifolds capture a comparable amount of structure at each scale -- consistent with the Platonic Representation Hypothesis. However, the functional map C exhibits near-zero mean diagonal dominance (<0.05, vs. >0.7 typical of near-isometric shape correspondence) and a large orthogonality deviation (70.15, vs. <0.1 typical), meaning the i-th eigenvector of one manifold does not correspond to any single eigenvector of the other -- energy scatters into off-diagonal bands (rows ~15, ~30) rather than concentrating on the diagonal. This decoupling, termed the "spectral complexity-orientation gap," persists across anchor budgets (|S|=5 to 500), spectral truncations (k_s=10 to 100), and a second text encoder (mpnet), ruling out estimation artifacts. Practically, the functional map substantially underperforms ambient-space baselines on cross-modal retrieval: at |S|=500 anchors, Procrustes achieves 55.5% i2t Recall@1 versus only 4.3% for the functional map (12.9x gap, which widens rather than narrows with more anchor supervision); a composability test (composing a DINOv2-to-MiniLM map with a MiniLM-to-mpnet map) shows the composed map is 4.3x worse than a direct map, consistent with multiplicative error compounding of two already-weak individual maps rather than a failure of the composability mechanism itself. The paper argues this is a genuine structural property of independently-trained neural representations rather than a limitation of the functional-map computation, since varying anchor budget, spectral truncation, or text encoder choice does not close the gap.