Relational Composition as Parallel Transport: What Per-Entity Frames Force in Attention

Aakash Sarkar, Ananya Mukherjee, Motoshige Sato

A dot product means something only when both vectors share a coordinate system. Content words have one; a fresh entity does not: rotating its coordinates alone changes nothing. We take this per-entity frame freedom as a symmetry and ask what it forces in attention. For orthogonal transports, invariant scores depend only on norms and the transported inner product $\langle x_i,T_{ij}x_j\rangle$; equivariant messages lie in the span of $x_i$ and $T_{ij}x_j$. Endpoint channel count bounds transport rank, so with one vector per entity relations must be supplied; standard attention is the flat sector of a shared frame. Representation theory attaches an integer to each mechanism; the readout decides which. With orthogonal transitions, a convex-cell readout composes a finite group exactly only at or above its smallest faithful real representation dimension. At this minimum, exactness forces a faithful representation. An even readout can decode a cover at smaller width, and the gap is unbounded. Below its threshold, the part of the group a mechanism cannot see sets an accuracy ceiling computable before training. 17/17 fitting seeds at minimal width passed tests through length 1000, versus…