Non-Commutative Lie-Algebraic State-Space Models: Finite-Division Variety Laws, Semidirect Cascades, and Gradient Saddle Escape

Abstract

Structured state-space models (SSMs) achieve linear-time inference and logarithmic-time parallel training via associative scans, yet homogeneous diagonal transitions generate solvable semigroups unable to track non-solvable group words or 3D rigid-body manifolds. We develop a theoretical and empirical framework for Non-Commutative Lie-Algebraic State-Space Models (Lie-SSMs), parameterizing recurrent updates on Lie algebra tangent spaces via the matrix exponential. First, via an o-minimal multi-rate cylindrical flag collar, we establish an unconditional single-layer finite-division criterion for homogeneous contracting $SO(3)$ and $SU(2)$ heads, proving odd-prime Sylow abelianity, odd-$p$-core centrality $[u^2,x]=e$, exponent-30 commutator laws ($[u^{30},v^{30}]=e$ in $SO(3{)}^m$ and $[u^{30},v^{30}]\in Z(G),[u^{30},v^{30}{]}^2=e$ in $SU(2{)}^m,SO(4{)}^m$, excluding $S_5$), and sharp threshold $d^{\ast }(S_5)=d^{\ast }(A_6)=5$ in $SO(5)$. Second, Depth-2 semidirect and Schreier $2$-cocycle cascades resolve $S_5$, $2I\cong S{L}_2(F_5)$, and $\mathrm{Heis}(F_p)$ in $O(\log T)$ depth, whereas unnormalized contracting coupling collapses exponentially without Directional RMSNorm. Third, we characterize the $O({\epsilon }^{c-1})$ Chen--Magnus--Hall--Witt commutator saddle and $1/|[G,G]|$ coset-centroid dilution, proving $\epsilon =0$ symmetry breaking and local Polyak--Łojasiewicz convergence (${\theta }_{\max }<\pi$), solving $S_5$ ($99.90\%/99.40\%$ at $T=128$) and $D_{16}$ ($100.00\%$) via dense prefix loss and factored readouts. Across $Q_8,A_5,S_5,D_{16}$, Dyck-2, $S{E}_2(3)$ Left-Jacobian odometry, GPU scans ($26.03\times$ speedup), and three 3D IMU cohorts (UCI HAR, HAPT-PT 30-subject CV, and REALDISP), Depth-2 Lie-$SO(3)$ cascades reach $95.69\%\pm 0.14\%$ on HAR, $87.16\%\pm 0.64\%$ ($88.32\%\pm 0.93\%$ 30-subject episodes) on HAPT-PT, and $84.23\%\pm 1.00\%$ under displacement on REALDISP.

arXiv comment33 pages, 4 figures, 11 tables, 2 algorithms; includes 5 ancillary reproducibility and finite-group theorem verification scripts