Robust Reasoning as a Symmetry-Protected Topological Phase

TL;DR

Robust inference is identified as an effective Symmetry-Protected Topological phase, where logical operations are formally isomorphic to non-Abelian anyon braiding, replacing fragile geometric interpolation with robust topological invariants.

Abstract

Large language models suffer from "hallucinations"-logical inconsistencies induced by semantic noise. We propose that current architectures operate in a "Metric Phase," where causal order is vulnerable to spontaneous symmetry breaking. Here, we identify robust inference as an effective Symmetry-Protected Topological phase, where logical operations are formally isomorphic to non-Abelian anyon braiding, replacing fragile geometric interpolation with robust topological invariants. Empirically, we demonstrate a sharp topological phase transition: while Transformers and RNNs exhibit gapless decay, our Holonomic Network reveals a macroscopic "mass gap," maintaining invariant fidelity below a critical noise threshold. Furthermore, in a variable-binding task on $S_{10}$ ($3.6 \times 10^6$ states) representing symbolic manipulation, we demonstrate holonomic generalization: the topological model maintains perfect fidelity extrapolating $100\times$ beyond training ($L=50 \to 5000$), consistent with a theoretically indefinite causal horizon, whereas Transformers lose logical coherence. Ablation studies indicate this protection emerges strictly from non-Abelian gauge symmetry. This provides strong evidence for a new universality class for logical reasoning, linking causal stability to the topology of the semantic manifold.

Figures (4)

• Figure 1: The Phase Transition of Logical Stability. We compare the logical fidelity of a metric RNN (orange, $N=128$), a standard Transformer (yellow, $N=128$), a normalized RNN control (gray, $N=128$), and our Holonomic Network (blue, $N=32$) on a non-Abelian $S_3$ reasoning task ($L=5$) under increasing semantic noise. Both the Transformer and metric RNN exhibit gapless decay. The Holonomic Network reveals a protected plateau (mass gap), maintaining perfect fidelity up to a critical noise threshold $T_c$. Inset: Principal Component Analysis (PCA) of the hidden state manifold at $T \approx T_c$. The metric RNN (gray fog) exhibits a gapless distribution where logical states overlap. In contrast, the Holonomic Network (red stars) reveals six discrete, topologically separated islands, each corresponding to a unique element of the $S_3$ group. This confirms the emergence of discrete topological sectors protected by an energy barrier.
• Figure 2: Holonomic Generalization and Efficiency. Models were trained on lengths $L \le 50$ (gray region) and tested up to $L=5000$. The Transformer ($\approx 3$M parameters) fails to generalize, collapsing as sequence complexity increases. The Holonomic Network ($\approx 4.6 \times 10^4$ parameters) generalizes perfectly, demonstrating that topological inductive bias is $65\times$ more parameter-efficient than metric scaling for algorithmic tasks.
• Figure 3: Finite-Size Scaling of the Topological Phase. The critical noise threshold $T_c$ is plotted against the logarithm of the gauge rank (network width $N$). The data follows a linear trend $T_c \propto \ln N$, analogous to the topological entanglement entropy $S_{top} \sim \ln \mathcal{D}$ of a non-Abelian anyon model. This logarithmic scaling suggests that the robustness arises from non-local topological order, where the stability barrier grows with the information content of the topological sector.
• Figure 4: Empirical Memory Horizon. We measure the sensitivity of the hidden state $h_t$ to the initial boundary condition $h_0$ via the Jacobian norm $\|\partial h_t / \partial h_0\|_2$. The Metric Phase (red) exhibits exponential decay, indicating a loss of causal history due to contractive dynamics. The Topological Phase (blue) maintains a constant Jacobian of unity ($J(t) \approx 1.0$). This signifies unitary evolution, where the logical state is protected by the $SO(N)$ isometry group, resulting in an infinite correlation length ($\xi \to \infty$).