Rethinking Non-Isometric Positional Transformations through Rank and Structure

Positional encoding is increasingly formulated through structured transformations of token interactions, where preserving near-isometric geometry is often preferred to avoid contraction or degeneracy. Inspired by state-transition design in linear attention, we revisit whether non-isometric deformation should instead be treated as a controllable source of positional capacity. We introduce positional update rank, $r_{\mathrm{pos}}(P)=\mathrm{rank}(P^{\top }P-I)$, and propose PRISM, which separates expansion order from expansion structure through independent, mixed, and product expansions. Our analysis shows that equal expansion order can yield substantially different operators depending on structure. Experiments across model scales, long-context tasks, and hybrid architectures reveal a consistent moderate-rank regime: expanding beyond rank one is often beneficial, whereas further expansion is not uniformly helpful. These results suggest that non-isometric positional deformation is not merely distortion, but a structured capacity whose utility depends on both its extent and organization.