The Circle Was Hiding a Helix: Helical Reduced Representations for Exact Vector Symbolic Binding

Vector Symbolic Architectures (VSAs) store symbols as vectors and combine them by binding and bundling. A bundle of $K$ bound pairs in $D$ dimensions holds only about $D/\ln N$ pairs, where $N$ is the codebook size. The cause of this limit lies in the binding operation, not in the vectors. Binding is multiplication in a group $G$, and bundling is addition in the group algebra $\mathbb{C}[G]$. If $G$ is compact, every representation is unitary. Every bound term then has magnitude 1, superposed terms are separable only by statistics, and the capacity bound follows. Bundling cannot escape to the Lie algebra of $G$ either, because distributivity forces any such operation to be a representation. We introduce Helical Reduced Representations (HeRR) bind on the non-compact group $\mathbb{Z}$. Their vectors are Laurent polynomials. Binding is polynomial multiplication, bundling is polynomial addition, and nesting depth is polynomial degree. Circular convolution is this ring modulo $t^D-1$, so existing VSAs are quotients of HeRR. Storing a HeRR in fixed space is a choice of characters of $\mathbb{Z}$, and three independent capacity budgets follow: dimension, number of evaluation points, and numerical precision. Non-unitary characters, absent from compact groups, supply the third. Two exact decoders recover a bundle without observing crosstalk, so the codebook need not be random. At 128 complex numbers of storage, HeRR retrieves 96 role-filler pairs at accuracy 1.00, against 0.13 for FHRR at equal storage. With a correlated codebook, FHRR reaches 0.00 at 8 pairs while HeRR remains exact. The binding capacity limit is therefore a property of compact binding groups, not of vector representations.