The Expressive Limits of Diagonal SSMs for State-Tracking

Selective state space models (SSMs), such as Mamba, S4D, and LRU, are bounded by transition matrix commutativity (A_t A_t' = A_t' A_t) and solvable affine transformation groups (Aff_D of derived length <= 2). Consequently, stacked multi-layer diagonal networks face severe optimization degradation on non-solvable simple groups such as A_5 due to the exponential circuit emulation depth required to simulate non-abelian commutators. We propose Non-Commutative State Space Models (NC-SSM), their real-orthogonal counterpart SO(3)-SSM, and arbitrary-dimension Cayley-SSM, lifting state transitions to compact Lie groups SU(2), SO(3), and SO(N). Via closed-form Euler-Rodrigues maps and rational Cayley transforms, NC-SSM achieves exact norm-preserving isometry (||U_t|| = 1). We introduce pure Hopf-fibration Bloch projective readouts (S^3/{+-1} =~ S^2 =~ SO(3)) to eliminate sign ambiguity, true quaternion parallel prefix scans (9.06x speedup at T=2048), and Identity-Gated Lie SSMs to eliminate sparse syntax phase drift. Extensive benchmarks across 14 experimental regimes show: (1) NC-SSM achieves 100% tracking on S_3, D_4, Q_8 and simple group A_5, where a 3-layer deep diagonal baseline collapses to 6.60% (p = 8.81e-4); (2) Cayley-SO(5)-SSM breaks Klein's 1884 ceiling on symmetric group S_5 (50.92% vs diagonal 5.25%, p = 0.0015, delivering 7.8x variance reduction over SO(3)); (3) SO(3)-SSM preserves Riemannian manifolds across 300 steps (< 3.12e-6 drift, > 580,000x advantage), achieving 0.04 deg dead-reckoning error and active tangent denoising; (4) NC-SSM achieves 74.36% on Dyck-2 and 30.26% on deep AST scope tracking (p = 0.0081); and (5) ablation confirms strict isometry is mathematically necessary for lossless long-range associative memory.

We study how depth, finite precision, state dimension, and chain-of-thought (CoT) affect the expressive power of multi-layer state-space models (SSMs). For the explicit-table $K$-function-composition problem, a canonical benchmark for sequential information propagation, we prove that any $L$-layer SSM solving $(L+3)$-function composition must satisfy $d^2p=Ω(N/L^3)$, where $d$ is the state dimension and $p$ is the per-scalar precision. Conversely, $K$-function composition is solved exactly by a $(K+1)$-layer generalized SSM with $d=1$ and $p=Θ(\log N)$. This gives a worst-case depth hierarchy for this formal problem family. We then distinguish post-input reasoning, in which all thought tokens are generated after the input, from input-interleaved reasoning, in which thought tokens may be inserted while the input stream is being read. Post-input reasoning does not circumvent our communication-based lower-bound pipeline, whereas input-interleaved reasoning admits bidirectional simulations with general deterministic one-pass streaming algorithms at the granularity of persistent memory. Finally, width and precision are not interchangeable under exact step-preserving simulation in the base affine-state model, but become interchangeable through the streaming-memory characterization once input-interleaved reasoning is allowed.

State Space Models (SSMs) have emerged as a powerful paradigm for efficient long-sequence modeling, offering parallel training and fast linear-time recurrent inference. However, like other recurrent architectures, SSMs must compress an unbounded history into a fixed-size state, which limits context retention and makes precise retrieval over long-range context inherently difficult. To overcome this limitation, we propose Delay State Space Models (DSSMs), a delay differential equation (DDE)-inspired extension of diagonal SSMs that augments discrete SSM recurrences with explicit delayed-state feedback. Making explicit delayed feedback practical requires new stability parameterization, history management, and FFT-training tools. We address these challenges with a practical discretization and parameterization grounded in a simple delay-independent stability condition. To bypass direct time-domain kernel construction, we derive the DSSM transfer function and compute kernels in the frequency domain, using a kernel contour shift to suppress aliasing and recover accurate FFT training. Empirically, DSSMs substantially improve targeted delayed-retrieval tasks while outperforming S4D on most standard sequence metrics and remaining close on the others.