Do larger models learn increasingly similar representations of the same world? The Platonic Representation Hypothesis (PRH) predicts that they should, but similar can mean native cosine geometry---nearest-neighbour structure induced by cosine similarity before any learned alignment---or linear recoverability: correspondence revealed by a supervised linear map. Using matched Legacy Survey and Hyper Suprime-Cam (HSC) embeddings across five model families, we find that Ridge alignment strengthens the relationship between model size and cross-dataset neighbourhood correspondence relative to native cosine geometry. The effect appears in all five families and persists when both representations are reduced to dimension, so it is not simply that wider embeddings give alignment more dimensions to work with. The learned maps remain far from rotations or uniform scaling, and Sparse Autoencoder (SAE) and Block-Sparse Featuriser (BSF) features give no advantage over the same alignment applied to dense embeddings. Model scale thus improves linear recoverability more clearly than native cosine-neighbourhood correspondence, without establishing convergence toward a shared native metric geometry.
Figures & tables
Figure 1: Held-out mKNN@10 versus log10P ( P : parameter count in millions; Appendix B ). Curves compare native Dense correspondence with Dense+Ridge linear recoverability. (a) Original embedding dimensions: Ridge raises both the level and the size slope ( T=0.00721 ). (b) Independent PCA to 256 dimensions per survey (same feature dimension in both conditions): the slope amplification persists ( T256=0.00864 ).
Figure 2: Filled markers: fraction of Aeff singular directions needed to recover 90% of the Dense+Ridge mKNN lift (median 4.8% ). Open markers: fraction needed for 90% of the map’s anisotropy mass (median 42.7% ). The map is anisotropic across many directions, but most neighbour recovery is concentrated in a much smaller subspace.
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 3: Translation contributes little; the effect is carried by the linear component Aeff . (a) Per-tier alignment gain G over native mKNN, using the full affine map versus the linear component alone with beff removed. Points near the dashed y=x line indicate the translation contributes negligibly (median retained fraction ηA=0.99 ). (b) Family slope amplifications ΔβF under the two conditions; the model-size amplification survives removal of beff ( TA=0.00854 versus Taffine=0.00721 ).
Figure 4: Sparse coding does not overtake the dense baseline at any tested sparsity. (a) Mean residual MSAE+Ridge−MDense+Ridge under inference-time TopK K∈{10,20,40} , with family-bootstrap 95% CIs. Negative values indicate SAE underperforms the matched dense baseline. (b) Equal-family scaling interaction TSAE,K , with per-family slope differences ΔβF,KSAE faint behind it. Both level and size trend stay non-positive across K .