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

Zeyu Jia

arXiv:2610.00329v4cs.LG

TL;DR

Diagonal SSM transitions are efficient but commutative, limiting tracking of non-solvable group words and rigid-body geometry. The paper develops Lie-SSMs, proves algebraic and optimization results, and evaluates cascades on synthetic, odometry, scan, and wearable-sensor tasks. Depth-2 cascades achieve the reported S5 and D16 results and perform across three wearable cohorts.

  • Problem

    Diagonal transitions remain mutually commutative, limiting single-layer SSMs on non-solvable group word problems, while 3D state estimation requires structure-preserving integration on non-commutative Lie groups.

    Source: §1 · p. 1

  • Method

    Lie-SSMs parameterize recurrent transitions on Lie-algebra tangent spaces and map them to Lie groups with the exponential, with depth-2 semidirect and Schreier 2-cocycle cascades for composition.

    Source: §1 · p. 2, §4.4 · p. 12

  • Results

    At T = 128, Lie-SSMs reach 99.90% ± 0.22% / 99.40% ± 0.65% on S5 and 100.00% ± 0.00% on D16, with results also reported across odometry, scans, and three wearable cohorts.

    Source: §1 · p. 3, §7 · p. 26

  • Takeaways & Limitations

    The reported results span synthetic group-word tasks, 6-DoF inertial odometry, parallel scans, and three wearable 3D inertial cohorts using Depth-2 Lie-SO(3) cascades.

    Source: §7 · p. 26

    Theorems 1–2 cover homogeneous compact Lie-group or orthogonal heads with scalar contraction rates; non-commutative prefix scans require d × d precision without non-compact Euclidean dimensions or KV-cache checkpoints.

    Source: §6 · p. 26

Abstract

from arXiv · show

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 $d^*(S_5)=d^*(A_6)=5$ in $SO(5)$. Second, Depth-2 semidirect and Schreier $2$-cocycle cascades resolve $S_5$, $2I\cong SL_2(F_5)$, and $\mathrm{Heis}(F_p)$ in $O(\log T)$ depth, whereas unnormalized contracting coupling collapses exponentially without Directional RMSNorm. Third, we characterize the $O(\varepsilon^{c-1})$ Chen--Magnus--Hall--Witt commutator saddle and $1/|[G,G]|$ coset-centroid dilution, proving $\varepsilon=0$ symmetry breaking and local Polyak--\L{}ojasiewicz 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, $SE_2(3)$ 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.

1. Introduction

The paper introduces Lie-SSMs to address limits of commutative diagonal transitions for non-solvable group words and rigid-body geometry. It develops single-layer obstructions, depth-2 cascades, and gradient analyses, then reports theoretical and empirical results across synthetic, inertial, and recognition tasks.

1. Introduction

  • Lie-SSMs parameterize recurrent transitions on Lie-algebra tangent spaces and map them to the group with the exponential map.The work frames this approach against diagonal and other commutative transitions that cannot track non-solvable group words.Source: §1 · pp. 1–2
  • An o-minimal finite-division criterion yields single-layer obstructions including odd-prime Sylow abelianity and an exponent-30 commutator law excluding S5 from SO(3)^m.The framework also identifies perfect targets and gives an explicit equal-Parikh collision for S5.Source: §1 · pp. 2–3
  • The sharp single-layer threshold for S5 and A6 is d*=5 in SO(5), while SO(4)^m remains obstructed on S5, Heis(Fp), and F20.The paper also states dimension and head-count laws for further group families.Source: §1 · p. 3
  • Depth-2 semidirect and Schreier 2-cocycle cascades resolve S5, Heis(Fp), and binary polyhedral groups; without inter-layer normalization, contracting coupling collapses exponentially.Directional RMSNorm enables continuous Lie-algebra routing in the cascade.Source: §1 · p. 3
  • Small-initialization commutator-separating word pairs have gradient magnitude O(ε^(c−1)), while ε=0 Softmax gradients exhibit 1/|[G,G]| coset-centroid dilution.The paper also proves first-order symmetry breaking at ε=0 and local geodesic-basin Polyak–Łojasiewicz convergence for θmax < π.Source: §1 · p. 3
  • At T = 128, Lie-SSMs reach 99.90% ± 0.22% / 99.40% ± 0.65% on S5, 100.00% ± 0.00% on D16, and strong results across odometry, GPU scans, and three wearable cohorts.The reported cohort results include 95.69% ± 0.14% on UCI HAR, 88.32% ± 0.93% on HAPT-PT 30-subject episodes, and 84.23%±1.00% Mutual-Displacement accuracy on REALDISP.Source: §1 · p. 3

2. Related Work

Related work establishes the computational appeal and algebraic limits of diagonal SSMs, then situates Lie-SSMs among algebraic expressivity, Lie-group recurrence, and reflection-based transitions.

2.1. Structured and Diagonal State-Space Models

  • Diagonal transitions support O(d) per-step updates and O(log T) parallel scans, but their commutativity persists across time steps.The cited background traces diagonal and scalar-identity transitions across S4, S5, LRU, Mamba, and Mamba-2.Source: §2.1 · p. 4

2.2. Algebraic Expressivity, Krohn-Rhodes Theory, and Definable Automata

  • Krohn-Rhodes theory and related results connect diagonal SSM limits to solvable transformation monoids, while Theorem 1 addresses contracting rotation heads without a uniform unnormalized output margin.The cited prior work analyzes bounded-precision and pure state-transition assumptions.Source: §2.2 · p. 4

2.3. Orthogonal, Unitary, and Lie-Group Recurrent Networks

  • Prior orthogonal and Lie-group recurrent networks established norm-preserving transitions, while the paper identifies finite-group expressivity, commutator-saddle, and associative-scan gaps.The cited approaches include Cayley transforms, Lie-algebra exponential maps, and Lie algebra convolutional networks.Source: §2.3 · p. 4

2.4. Householder Reflection Products and Gated Linear Recurrences

  • Householder and planar-rotation approaches motivate alternative transition designs, while the paper proves volume collapse for dissipative rank-1 updates and orientation reversal for single reflections.The paper contrasts these results with its multi-rate criterion, depth-2 cascades, and commutator-saddle analysis.Source: §2.4 · p. 4

3. Lie-Algebraic State-Space Architectures & Associative Parallel Scan

Lie-SSMs parameterize recurrent updates in Lie-algebra tangent spaces and map them to group transitions with the matrix exponential. The section develops parallel scans, depth-2 cascades, cross-head mixing, and structural supervision and readouts.

3.1. Single-Layer Lie-

  • Lie-SSMs map input features to Lie-algebra generators and use matrix exponentials for transitions on SO(d), with corresponding spinor and rigid-body interfaces.For SO(3), Rodrigues’ formula evaluates the exponential; SE(3) uses a coupled exponential for rigid-body twists.Source: §3.1 · p. 5

3.2. Associative Parallel Prefix Scan on Non-Commutative Matrix Lie Groups

  • Associative group and affine-map composition permits O(log T)-depth parallel prefix scans despite non-commutative transitions.The section describes Blelloch scans with O(Td^3) work and Hillis–Steele scans with O(T log T d^3) work.Source: §3.2 · pp. 5–6

3.3. Depth-2 Semidirect and Schreier 2-Cocycle Cascades with Directional RMSNorm

  • Depth-2 cascades track semidirect products and Schreier 2-cocycle extensions by routing through a quotient tracker into a normal-core tracker.Directional RMSNorm normalizes inter-layer states; unnormalized contracting states decay and can collapse downstream routing.Source: §3.3 · p. 6

3.4. Cross-Head Mixing SO(3m) and Differentiable Stiefel Polar Subspace Gating

  • Cross-head SO(3m) mixing expands the generated Lie algebra, while Stiefel Polar projection enforces orthogonality between learned subspace frames.The projection also maintains unit projector norm over full-column-rank frames.Source: §3.4 · p. 7

3.5. Stationary Dense Prefix Supervision and Factored Subdirect Readouts

  • Dense prefix supervision and factored abelianization-plus-core readouts address terminal-gradient decay and coset-centroid variance dilution.The factored loss delivers an undiluted abelianization gradient, activating higher-order commutator brackets after parameters move away from the zero initialization.Source: §3.5 · pp. 7–8

4.1. Topological and Representation Foundations

This section establishes geometric and algebraic foundations for the model’s expressivity arguments. It distinguishes the SU(2) covering of SO(3) from the Hopf fibration and states limits of commutative recurrences and leaky counters.

4.1. Topological and Representation Foundations

  • SU(2) is a two-to-one universal cover of SO(3), whereas the Hopf fibration maps S3 onto S2 with a U(1) fiber.The two constructions have different quotients and base-space dimensions.Source: §4.1 · p. 8
  • Commutative transition operators are invariant to input permutations and therefore cannot distinguish non-commuting generator orders at any hidden dimension.This applies to homogeneous recurrences and affine recurrences whose transition matrices lie in a commutative algebra.Source: §4.1 · p. 8
  • A leaky scalar recurrence approximating the Dyck-1 counter has a finite precision horizon, with the stated bound reaching 310 steps for float32 spacing at λ = 0.95.Using unit roundoff instead gives a stated bound of 324 steps.Source: §4.1 · p. 9
  • Finite subgroups of SO(3) comprise cyclic, dihedral, tetrahedral, octahedral, and icosahedral families; A5 is the only non-solvable one.For SU(2), the listed families include the binary icosahedral group 2I, its only non-solvable finite subgroup.Source: §4.1 · p. 9

4.2. Theorem 1: Unconditional Single-Layer Finite-Division Variety, Sylow/Exponent-30 Obstructions, and Spherical Margin Dichotomy

Theorem 1 characterizes which finite groups homogeneous contracting SO(3) and SU(2) heads can track in a single layer. Its structural laws and collision results yield explicit obstructions for several targets, including S5.

4.2. Theorem 1: Unconditional Single-Layer Finite-Division Variety, Sylow/Exponent-30 Obstructions, and Spherical Margin Dichotomy

  • Exact single-layer tracking with a definable state-only readout is possible if and only if the target group divides a finite product of SO(3) heads, or SU(2) heads, respectively.For fixed parameters, trackability also implies division of the finite supplement generated by surviving heads.Source: §4.2 · p. 9
  • Perfect finite groups are trackable with SO(3) heads exactly when they are subgroups of a finite product of icosahedral groups; the stated SU(2) condition is a central quotient of powers of 2I.With headwise spherical normalization, the central quotients are realized by a one-hidden-layer exact-GELU readout pooling over cosets.Source: §4.2 · p. 9
  • Every odd-prime Sylow subgroup of a finite group dividing SO(3)^m or SU(2)^m is abelian, excluding non-abelian odd-p Heisenberg groups at depth 1.For groups dividing SO(3)^m, SU(2)^m, or SO(4)^m, the odd-p core also satisfies [u^2, x] = e.Source: §4.2 · p. 9
  • Finite groups dividing SO(3)^m obey [u^30, v^30] = e, while those dividing SU(2)^m or SO(4)^m obey a central-involution version of this law.S5 violates the latter law, so it cannot be tracked by these single-layer head families for any finite m.Source: §4.2 · pp. 9–10
  • For S5, equal-Parikh words of length 120 yield an exact unnormalized state collision despite having different group outputs.The stated collision uses words (ct)^30(tc)^30 and (tc)^30(ct)^30 and holds across arbitrary contraction rates.Source: §4.2 · p. 10

Mechanistic Interpretation of Theorem 1.

  • Spherical cap packing bounds reachable-state separation as the number of distinguishable states grows, while non-dividing targets have vanishing uniform separation over horizons.The section also bounds the required readout Lipschitz-to-margin ratio by the inverse separation.Source: §4.2 · p. 10

4.3. Theorem 2: Higher-Rank Orthogonal Dimension Thresholds, Non-Compact Reductions, and Exact Head-Count Laws

Theorem 2 gives dimension and head-count results for finite-group tracking, including sharp single-layer thresholds and limits inherited by low-dimensional rotation groups. It also characterizes selected non-compact reductions and transition constructions.

Mechanistic Interpretation of Theorem 2.

  • SO(4)^m retains single-layer barriers for S5, odd-p Heisenberg groups, and F20, while 2T and 2O divide SU(2) and SO(4) but not any finite product of SO(3).The stated distinction follows from the central exponent-30 obstruction and the contrasting commutator exponents.Source: §4.3 · p. 11

4.3. Theorem 2: Higher-Rank Orthogonal Dimension Thresholds, Non-Compact Reductions, and Exact Head-Count Laws

  • A single SO(5) head tracks both S5 and A6, establishing the stated sharp dimension threshold d* = 5 for these targets.The constructions use signed permutation and deleted permutation representations, respectively.Source: §4.3 · p. 11
  • At every finite cascade depth, SO(3), SU(2), and SO(4) heads track a finite group exactly when all its non-abelian simple composition factors are A5.The characterization excludes An and Sn for n ≥ 6 and PSL2(F7).Source: §4.3 · p. 11
  • For metacyclic Frobenius groups, the SO threshold is k + 1 when k is even and 2k when k is odd.The section gives F21 = C7 ⋊3 C3 as an example with d*SO = 6.Source: §4.3 · p. 11
  • Non-abelian odd-p Heisenberg groups have stated SO thresholds d* = 2p, including d* = 6 for Heis(F3).The representation argument is identified as the Stone–von Neumann Weyl representation.Source: §4.3 · p. 11
  • Under spherical normalization, positive selective scalar gates cancel, and finite subgroups of GL+(d) can be conjugated into SO(d).Finite subgroups of the stated Lorentz group conjugate into SO(d − 1).Source: §4.3 · p. 11

Mechanistic Interpretation of Theorem 2.

  • Anisotropic singular-value splitting collapses normalized trajectories onto a lower-dimensional sphere, whereas chaining 2⌊d/2⌋ Householder reflections generates SO(d).The section contrasts this with dissipative rank-1 updates, which incur volume decay.Source: §4.3 · p. 11

4.3. Theorem 2: Higher-Rank Orthogonal Dimension Thresholds, Non-Compact Reductions, and Exact Head-Count Laws

  • For non-abelian 2-groups with cyclic maximal subgroups, the dihedral group needs one SO(3) head, while the stated semidihedral and modular groups need two.The section also gives an Arf-invariant distinction in SO(3) head counts for certain 2-groups.Source: §4.3 · p. 11

4.4. Theorem 3: Compositional Semidirect and Schreier 2-Cocycle Cascades, Continuous Directional-RMSNorm Routing, and Reset-Monoid / Grassmannian Duality

Theorem 3 uses cascades to compose group trackers and Directional RMSNorm to preserve useful inter-layer routing. It also characterizes finite-precision behavior, cross-head mixing, reset monoids, and multi-subspace projection limits.

4.4. Theorem 3: Compositional Semidirect and Schreier 2-Cocycle Cascades, Continuous Directional-RMSNorm Routing, and Reset-Monoid / Grassmannian Duality

  • Two SO(3) heads track S5, 2I, 2T, 2O, and Q16 at Depth 2, while three track Heis(Fp), with a uniform positive margin.The cascade tracks a group extension through a quotient tracker and a normal-subgroup tracker, composing their states through the action and Schreier cocycle.Source: §4.4 · p. 12, Mechanistic Interpretation of Theorem 3 · p. 13
  • Open-loop spherically normalized recurrences have a guaranteed safe exponential finite-precision horizon, whereas unnormalized contractions can underflow.The theorem also gives a hybrid SSM–attention cache space–precision conservation law; the supplied passages do not specify its full expression.Source: §4.4 · p. 13
  • Interleaved SO(3) heads with adjacent-boundary mixing generate SO(3m), tracking S5, A6, and S6 at m = 2, and PSL2(F7) at m = 3.The result concerns cross-head mixing and its increased representational scope at Depth 1.Source: §4.4 · p. 13
  • Combining isometric group transitions with contractive affine resets tracks the non-commuting reset-group monoid; homogeneous recurrences cannot track the flip-flop monoid U2.The reset-group margin is bounded below by ∆0(F) − 2γ/(1−γ) when this quantity is positive.Source: §4.4 · p. 13

Mechanistic Interpretation of Theorem 3.

  • The cascade represents a group extension by tracking its quotient in Layer 1 and updating its normal subgroup in Layer 2 using the action and Schreier 2-cocycle.The factor set encodes how the chosen transversal section composes, and the resulting recursion twists the normal-subgroup update.Source: §4.4 · p. 12, Mechanistic Interpretation of Theorem 3 · p. 13
  • Without normalization, contracting Layer-1 states collapse the Depth-2 cascade to a state-independent router and the Depth-1 variety, which fails on S5.Directional RMSNorm places the Layer-1 state on the unit sphere, enabling exact affine Lie-algebra routing without discrete intermediate argmax decisions.Source: Mechanistic Interpretation of Theorem 3 · p. 13
  • When c ≥ 3, oblique dual-frame projectors diverge at the simplex singularity, while Stiefel Polar projection eliminates it for d ≥ cdsub.The projection Q(S) = S(S⊤S)^−1/2 has zero maximum coherence and yields an infinite partial-reset burst-survival horizon on the full-column-rank domain.Source: Mechanistic Interpretation of Theorem 3 · p. 13, §4.4 · p. 13

4.5. Theorem 4: Lie-Algebraic Optimization Dynamics, Chen–Magnus–Hall–Witt Commutator Saddles, and Stationary Dense Factored Escape

Theorem 4 analyzes why non-commutative Lie-SSMs can have weak gradients near small initialization and identifies training choices that break this symmetry. It proves local convergence in a geodesic basin and reports empirical validation of the intervening escape trajectory.

4.5. Theorem 4: Lie-Algebraic Optimization Dynamics, Chen–Magnus–Hall–Witt Commutator Saddles, and Stationary Dense Factored Escape

  • Under strict contraction, unnormalized BPTT gradients vanish exponentially, whereas headwise spherical normalization gives a unit suffix tangent Jacobian across arbitrary horizons.The reported unnormalized decay is exponential for Lipschitz linear readouts and double-exponential for homogeneous quadratic Bloch readouts.Source: §4.5 · p. 14
  • The teacher-forced chordal objective satisfies a local Polyak–Łojasiewicz inequality when the geodesic error is bounded by θmax < π.The result applies on the stated chart domain avoiding the first conjugate singularity; principal-branch retraction gives the global chart condition for SO(3).Source: §4.5 · p. 14
  • Word pairs that first differ in the c-th lower-central-series quotient have gradient scaling Θ(ε^(c−1)) under non-degenerate conditions, while lower-degree terms cancel.At ε = 0, first-order word behavior depends only on the abelian Parikh vector, so commutator distinctions emerge only at higher order.Source: §4.5 · p. 14

Mechanistic Interpretation of Theorem 4.

  • Dense per-step prefix supervision and a factored abelianization-plus-core readout yield a non-vanishing expected gradient at ε = 0, breaking first-order abelianization symmetry.For uniform prefix averaging, the normalized gradient is Θ(1/N), rather than the exponentially decaying terminal-only term.Source: §4.5 · p. 15, Mechanistic Interpretation of Theorem 4 · p. 16
  • At ε = 0, terminal-only training at horizon N fails to amplify ε from this saddle: the Θ(1) abelianization gradient is doubly attenuated by coset-centroid variance dilution (1/60 on S5, 1/4 on D16) and Fourier–Markov mixing decay.Source: Mechanistic Interpretation of Theorem 4 · p. 15
  • The non-convex trajectory between origin escape and the local geodesic basin is supported empirically across five seeds on S5 and D16.The local Polyak–Łojasiewicz convergence is a formal guarantee; the intermediate trajectory is reported as empirical validation.Source: §4.5 · p. 14, Mechanistic Interpretation of Theorem 4 · p. 16

5. Comprehensive Multi-Seed Experimental Evaluation

The experiments test Lie-SSM tracking across finite-group benchmarks and show that dense prefix supervision, factored readouts, and Directional RMSNorm resolve distinct S5 and D16 bottlenecks. Results also distinguish non-commutative Lie updates from commutative, planar-rotation, and single-reflection baselines.

5.1. Symbolic and Numerical Verification of the Architectural Hierarchy

  • Table 1 synthesizes architectural thresholds and verification metrics across finite groups, transformation monoids, infinite-state languages, and real-world inertial streams.The evaluation protocol uses five training seeds and disjoint evaluation seeds.Source: §5.1 · p. 16, Table 1 · p. 16, §5 · p. 16

5.2. Non-Abelian Quaternion Group Q8 Benchmark

  • For Q8, the Lie model maintains 100.00% accuracy across horizons, whereas the commutative diagonal model reaches 11.93% at T = 32, near the 12.50% uniform baseline.This benchmark tests tracking of non-commuting quaternion generators.Source: §5.2 · p. 17, Table 2 · p. 17

5.3. Head-to-Head Evaluation on the Non-Solvable Alternating Group A5

  • On A5, Lie-SO(3)-SSM sustains 100.00% ± 0.00% accuracy from T = 16 to T = 256, while the single-reflection DeltaProduct reaches 1.60% ± 0.75% at T = 256.DeltaProduct with two reflections reaches 99.60% ± 0.57% at T = 256.Source: §5.3 · pp. 17–18, Table 3 · p. 17

5.4. Diagnosing and Resolving the S5 Random-Initialization Bottleneck

  • Lie-SO(6)-1L (Dense + Factored) reaches 99.90% ± 0.22% at T = 128 on S5, while Lie-SO(3)-1L (Dense + Factored) reaches 100.00% ± 0.00% across D16 horizons.The S5 cascade with Directional RMSNorm reaches 99.40% ± 0.65% at T = 128.Source: Table 5 · p. 18, §5.4 · p. 19, Table 6 · p. 19
  • On S5, single-layer Lie-SO(3)-1L falls from 85.40% at T = 16 to 2.70% ± 2.56% at T = 128, consistent with its stated variety obstruction.With identical 14,542 parameters, the unnormalized cascade decays to 1.30% at T = 128, while Directional RMSNorm restores accuracy to 99.40% ± 0.65%.Source: §5.4 · pp. 18–19

5.5. Nilpotency-Class-3 Dihedral Group D16 Commutator Saddle-Escape Closure

  • On D16, terminal-only unfactored training reaches 13.60% ± 4.57% at T = 128, while DeltaProduct-SO(3) (nh = 1) saturates near the 25% parity-quotient ceiling.Dense prefix supervision and a factored V4 × C4 readout let the identical Lie-SO(3)-1L model achieve 100.00% ± 0.00% across T = 16, 32, 64, and 128.Source: §5.5 · p. 19, Table 6 · p. 19

5.6. Formal Language Counter Depth (Dyck-2) and De-Confounded 6-DoF Inertial Pose Integration (SE2(3))

The paper compares compact rotations, scalar counters, and extended-pose integration, then benchmarks parallel scans and Lie-SO(3) cascades on inertial recognition. Across these settings, it reports gains for non-compact counters on deeper Dyck-2 nesting, Left-Jacobian coupling in odometry, and cascades on real-world activity and transition tasks.

5.6. Formal Language Counter Depth (Dyck-2) and De-Confounded 6-DoF Inertial Pose Integration (SE2(3))

  • At Dyck-2 depth D = 16, diagonal counter models reach 94.64% ± 0.89%, while pure compact SO(3) rotations reach 76.88% ± 2.16%.The paper attributes the compact-rotation saturation to spherical cap crowding.Source: Table 7 · p. 20, §5.6 · p. 20
  • The Dyck-2 and odometry experiments test a geometric trade-off among compact SO(3) rotations, scalar counters, and non-compact SE2(3)/SE(3) pose manifolds.The odometry evaluation compares integration operators across matched parameter tiers.Source: §5.6 · pp. 19–20, Table 7 · p. 20
  • At 679 parameters, Lie-SE(3)-LeftJac (MLP-679) achieves 2.636 ± 0.167 m ATE, versus 3.165 ± 0.111 m for Euler integration.At 49,575 parameters, Lie-SE(3)-LeftJac (SeqCalib-50.2K, Coupled) achieves 3.124 ± 0.295 m ATE and bounds orthogonality drift to 1.24 × 10^-6.Source: §5.6 · p. 20, Table 7 · p. 20

5.7. Associative Parallel Prefix Scan Hardware Scaling

  • Matched PyTorch CUDA scans reach a (26.03 ± 2.85)× speedup at T = 2048, taking 3.31 ± 0.45 ms versus 85.48 ± 7.70 ms for sequential recurrence.The comparison uses batched matrix multiplications on the same GPU backend.Source: Table 8 · p. 20, §5.7 · p. 20

5.8. Real-World 3D Inertial Human Activity and Postural Transition Pattern Recognition (UCI HAR and UCI HAPT-PT)

  • The inertial-recognition experiments share an SO(3)-equivariant lifting stem and factored readout, isolating the recurrent or sequence operator as the independent variable.The evaluation includes subject-disjoint UCI HAR and HAPT-PT benchmarks, with full-horizon, prefix, and orientation-perturbation protocols.Source: §5.8 · p. 21, Table 9 · p. 22, Table 10 · p. 23
  • Lie-SO(3)-Cascade (Dense, Unfact.) reaches 95.69% ± 0.14% on UCI HAR, and Dir-RMSNorm + Fact. reaches 87.16% ± 0.64% on UCI HAPT-PT.On HAPT-PT, the latter significantly outperforms all 10 non-SO(3) baselines.Source: Table 9 · p. 22, §5.8 · p. 23
  • On UCI HAR and HAPT-PT, multi-horizon prefix supervision improves early-prefix accuracy over terminal-only training by 26.35 and 12.90 percentage points, respectively.At σrot = 45°, Dir-RMSNorm + Fact. retains 82.69% ± 0.75% on HAR and 82.72% ± 2.31% on HAPT-PT.Source: §5.8 · p. 23

5.9. Input-Stem Attribution, Capacity Pareto, 30-Subject Cross-Validation, and Multi-IMU Sensor Displacement (UCI REALDISP)

The section compares input-stem geometry and model capacity across three inertial cohorts, including full-cohort HAPT-PT validation and physical sensor displacement on REALDISP. Results distinguish settings where exact rotation-aware models outperform scalar or raw-coordinate baselines from static postures where global rotation quotienting loses useful gravity information.

Evaluation design

  • The 4-tier evaluation contrasts raw-coordinate, equivariant-concatenation, invariant-scalar, and full-equivariant-frame stems across capacity levels and cohorts.The HAPT-PT cross-validation uses 30 subjects and 358 episodes; REALDISP tests zero-shot on Ideal, Self-Placement, and physical Mutual Sensor Displacement.Source: §5.9 · pp. 23–24, Table 11 · p. 24

UCI HAPT-PT official split

  • 86.23% ± 0.64% on the 9-subject HAPT-PT official split exceeds matched 22K DeepConvGRU (Inv-Scalar) by +5.00 pp and 4× larger 90.5K DeepConvGRU (Inv-Scalar) by +3.58 pp.The comparison evaluates order-sensitive postural transitions under Unif(SO(3)) mounting.Source: §5.9 · p. 24, Figure 4 · p. 25

UCI HAPT-PT 30-subject validation

  • 84.06% ± 1.13% on Win-1074 C3 and 88.32% ± 0.93% on Ep-358 Vote are reported for the 22.3K cascade in 30-subject HAPT-PT cross-validation.The cascade significantly exceeds all six Inv-Scalar backbones after Holm–Bonferroni correction; it also outperforms the unnormalized cascade by +5.64 pp on C3.Source: §5.9 · pp. 24–25

UCI HAR limitation

  • On single-waist UCI HAR static postures, exact global SO(3) quotienting lowers accuracy relative to nominal laboratory-referenced cascades because it removes the gravity-orientation cue distinguishing SITTING from LAYING.Nominal cascades achieve 95.48% ± 0.48% Clean, versus 84.22% ± 2.96% for Cascade Full-Equiv-Frame across gauges.Source: §5.9 · p. 25

UCI REALDISP

  • 84.23% ± 1.00% Mutual accuracy is reported for the REALDISP quarter-turn cascade, while Directional RMSNorm alone raises Mutual accuracy from 79.13% to 83.37%.On REALDISP, Native-Conv1D falls to 17.72%–19.91% under physical Mutual Sensor Displacement; the RMSNorm gain is +4.24 pp (pHolm = 0.0353).Source: §5.9 · p. 25, Figure 4 · p. 25

6. Discussion, Scope, and Limitations

The discussion identifies settings where matching the recurrent transition geometry to the task matters, while bounding the theory and computational scope of Lie-SSMs. It distinguishes scalar-invariant cases from tasks requiring non-commutative frame transport and notes limits on head size and compact-state memory.

Geometric domain of validity

  • Across synthetic word problems and sensor cohorts, matching the recurrent transition manifold to the physical symmetry group consistently outperforms unconstrained or mismatched representations.The discussion contrasts scalar invariants that suffice for stationary cyclic exercises with frame transport needed for order-sensitive transitions and continuous 6-DoF odometry.Source: §6 · p. 26

Task-dependent geometry

  • Pointwise O(3) scalar invariants suffice for stationary cyclic REALDISP exercises, whereas non-commutative SO(3) frame transport is important for time-ordered HAPT-PT transitions.The comparison also identifies a distinct UCI HAR boundary: global SO(3) quotienting removes the static gravity-orientation cue distinguishing SITTING and LAYING.Source: §6 · p. 26

Scope and limitations

  • Theorems 1–2 cover homogeneous compact Lie-group and orthogonal heads, while non-commutative scans require O(d^3) matrix multiplication per reduction node, making compact geometric heads the stated target.The discussion gives d ∈ {3, 4, 5, 6} as the suitable head dimensions.Source: §6 · p. 26
  • Compact rotation manifolds cannot pack exponentially growing free-group trees F2 at fixed precision without non-compact Euclidean dimensions or KV-cache checkpoints.The discussion also identifies custom warp-level CUDA register kernels for SO(3) and SE2(3) scans as a remaining systems direction.Source: §6 · p. 26

7. Conclusion

The conclusion presents Lie-SSMs as a theoretical and empirical framework for non-commutative recurrent modeling. It summarizes single-layer obstructions, depth-2 cascade results, optimization findings, and evaluations across synthetic tasks, odometry, GPU scans, and wearable cohorts.

7. Conclusion

  • The paper establishes a single-layer finite-division criterion for homogeneous contracting SO(3) and SU(2) heads, with variety laws and sharp orthogonal-dimension thresholds among its consequences.The conclusion lists odd-prime Sylow abelianity, odd-p-core squared-automorphism centrality, exponent-30 commutator laws, and minimum head-count laws.Source: §7 · p. 26
  • Depth-2 semidirect and Schreier 2-cocycle cascades resolve S5, 2I, 2T, 2O, Q16, and Heis(Fp) in O(log T) scan depth, while Directional RMSNorm prevents contracting cascade collapse.The conclusion also characterizes commutator saddles and coset-centroid dilution, proving symmetry breaking at ε = 0 and local Polyak–Łojasiewicz convergence for θmax < π.Source: §7 · p. 26
  • The conclusion reports S5 results of 99.90%/99.40% at T = 128 and 100.00% on D16, alongside evaluation on odometry, GPU scans, and three wearable cohorts.Matched-backend GPU scans achieved a reported 26.03× speedup.Source: §7 · p. 26

CRediT authorship contribution statement; Declaration of Competing Interest and Funding

Data and Code Availability

  • The paper states that PyTorch implementations, benchmark protocols, theorem-verification materials, and a reproducibility guide are provided with the submission, and that the three wearable datasets are publicly available.The listed datasets are UCI HAR, UCI HAPT-PT, and UCI REALDISP.Source: Data and Code Availability · p. 27

Appendix A. Proof of Theorem 1 (Single-Layer Finite Division, Variety Laws, and Spherical Margin Dichotomy)

The appendix proves a finite-division criterion by relating normalized state closures to finite-group quotients, then derives algebraic obstructions for single-layer SO(3), SU(2), and SO(4) heads. It also develops the spherical margin consequences of separated or colliding reachable states.

Finite-division proof

  • A multi-rate o-minimal cylindrical flag cell and coordinate-isolated identity padding force state closures for distinct target elements to be disjoint when tracking succeeds.The proof constructs suffixes that approach a common point in the same normalized closure while their contraction coordinates enter the same flag cell, forcing equal decoded outputs.Source: Appendix A · p. 27

Finite division and state separation

  • For any fixed head parameters, a uniform positive normalized-state margin across all word lengths holds exactly when the target-group state closures are pairwise disjoint, implying that the target divides a finite supplement.Conversely, a finite subgroup quotient gives exact tracking with a finite ReLU readout; when the kernel is trivial, a single linear layer suffices.Source: Appendix A · p. 27

Variety laws

  • Finite subgroups of SO(3) and SU(2) have cyclic odd-prime Sylow subgroups, yielding abelian odd-prime Sylow subgroups in their finite direct-product quotients.The proof also derives [u^2, x] = e for x in the odd-p core and exponent-30 commutator restrictions that exclude S5 from the stated single-layer classes.Source: Appendix A · pp. 27–28

Explicit S5 obstruction

  • For c=(12345) and t=(12), η((ct)^30)=(142) differs from η((tc)^30)=(124), although each contains 30 c's and 30 t's; this contrasts with commutation in any finite subgroup of SO(3)^m by Eq. (16).Source: Appendix A · p. 28

Spherical margin dichotomy

  • Spherical cap packing bounds the minimum separation of reachable states, while ReLU decision regions require separate affine regions for Radon-conflicting classes.For F2, the stated bound implies margin decay at fixed compact-head dimension as the number of prefix classes grows.Source: Appendix A · p. 28

Appendix B. Proof of Theorem 2 (Higher-Rank Dimension Thresholds, Weyl–Cartan Reductions, and Head-Count Laws)

The appendix establishes higher-dimensional division thresholds and algebraic obstructions for compact-group heads, alongside representation-dimension and head-count laws. It also reduces certain positive-homogeneous finite-group representations to orthogonal groups.

SO(4) commutator obstruction

  • [u^30, v^30] is central and squares to the identity for groups dividing SO(4)^m, a law violated by S5.The same passage states that odd-prime Sylow subgroups are abelian and [u^2, x] = e on the odd-p core.Source: Appendix B · p. 28

Dimension thresholds and division

  • S5 and A6 first divide SO(5), while 2T and 2O divide SU(2) and SO(4) but not any finite power of SO(3).For cascades of SO(3), SU(2), and SO(4) heads, the supported simple composition factors are restricted to A5.Source: Appendix B · p. 28

Frobenius dimension laws

  • For the stated Frobenius groups, the orthogonal dimension threshold is |H| when |H| is even and 2|H| when |H| is odd.When |H| is even, the SO threshold is |H| if the Sylow 2-subgroup is quaternion and |H| + 1 if it is cyclic.Source: Appendix B · p. 28

Heisenberg dimension threshold

  • Heis(F_p) for p ≥ 3 has minimum faithful representation dimension 2p in SO(d).The passage derives this from a faithful irreducible complex representation of dimension p whose central-element trace is non-real.Source: Appendix B · p. 29

Weyl–Cartan reduction

  • Positive homogeneity cancels positive scalar contractions, and dominated splitting drives the state toward the top subspace before an SO(d) element is expressed by Householder reflections.The passage gives 2⌊d/2⌋ reflections for the planar rotation blocks.Source: Appendix B · p. 29

2-group head-count laws

  • For 21+2r with Arf(q)=1, the group cannot divide SO(3)^r and requires mSO(3)=r+1; the cited passage gives this condition for the minus case.For the group combined with Aodd, the stated head count is r + max(1, Rodd).Source: Appendix B · p. 29

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