SHiPPO: Recurrent Memory with Transported Polynomial Projections

HiPPO gives recurrent states memory semantics as coefficients of online polynomial projections, but in fixed channel coordinates. Modern selective SSMs, by contrast, rely on token-dependent control and channel interaction. We introduce SHiPPO (Sylvester HiPPO), a transported projection-memory prior that lifts HiPPO coefficient memories into a moving channel frame. For any fixed or realized right-transport path, SHiPPO transports the approximation family and channel metric together; conditional on that path, the state is ordinary HiPPO in a tied moving frame and follows Sylvester coefficient dynamics, preserving the left online-memory operator while adding right-action transport. For selective-SSM execution, we derive a restricted group-local realization with controller-compatible right actions, exponential-adjusted updates, exact block-affine scan, and recurrent decoding. We also give a simultaneous-reducibility criterion identifying when right transports collapse to static mixing plus independent scalar or blockwise banks. Controlled diagnostics show that larger current-token write rank improves ordinary prediction error but cannot recover order-sensitive changes to already-written memory; transported-memory variants recover this signal, which disappears when the transport pathway is removed. A finite-field associative-recall diagnostic with interleaved bindings, operations, and queries provides complementary autoregressive evidence while leaving the preferred right-action realization open. Taken together, these results support SHiPPO as a mechanistically grounded transported-memory prior, with evidence focused on memory mechanisms rather than broad sequence-modeling dominance.