navi-SAD: A New Tool for Probing the Reasoning Mechanisms of Large Language Models from a Dynamical Systems Perspective

navi-SAD is an innovative tool developed by the Project-Navi team to monitor LLM reasoning processes using dynamical systems theory. Its core methods include parallel computation of softmax and linear attention, measurement of their cosine divergence, and reconstruction of internal state attractors using Takens embedding, providing a new framework for understanding the 'black-box' reasoning of LLMs by shifting from static analysis to dynamic process monitoring.

LLM reasoning has long been regarded as a 'black box'—only input and output are known, while the internal process remains unknown. Traditional interpretability methods (attention visualization, neuron activation analysis) struggle to capture the dynamic evolution characteristics of reasoning. In recent years, dynamical systems theory has emerged in neural network analysis; researchers have found that Transformer reasoning can be viewed as a high-dimensional dynamical system, and this insight prompted the development of navi-SAD.

The core innovation of navi-SAD is running two attention mechanisms in parallel: softmax attention (nonlinear, injective) and linear attention (simplified, non-injective), both sharing Q/K/V tensors with Rotary Position Embedding (RoPE). The 'capacity gap' proven by Han et al. (2024) forms its diagnostic foundation; by comparing the output differences of each attention head, the model's dependence on the capacity of nonlinear attention can be captured.

navi-SAD applies the Takens embedding theorem to LLM reasoning analysis:

1. For each (layer, head) pair, compute the cosine distance between the outputs of softmax and linear attention to obtain a scalar time series that changes with generation steps;
2. Treat this sequence as a delayed coordinate observation of the model's residual flow state, and reconstruct the internal dynamic attractor following the Takens theorem;
3. Calculate permutation entropy using Bandt-Pompe ordinal patterns to characterize the complexity of the attractor. Attractor collapse (low permutation entropy) means the internal state loses the complex structure that distinguishes reasoning mechanisms, while a rich attractor (high permutation entropy) retains structural features.

navi-SAD is implemented using the adapter pattern, allowing monitoring code to be injected without modifying model weights. It includes 453 tests (440 CPU +13 GPU) and enforces code standards via CI. Verification based on the Mistral-7B-Instruct-v0.2 model (fp16, eager attention) passes three-level gating:

• Gate0: Non-interference verification—the adapter generates exactly the same tokens and logits as the un-instrumented model under deterministic greedy decoding, verified via layer-wise/step-wise bijection across 32 layers;
• Gate1: Equivalence verification—the recomputed fp32 softmax attention, after native o_proj, matches the output of the native module with cosine similarity ≥0.999996 and relative L2 error ≤0.002759;
• Gate2: Stability verification—after 50 consecutive full generation tests, VRAM usage shows zero growth, CPU RSS growth is only 0.7 MiB, and all records can be serialized/deserialized via gzipped JSONL without memory leaks or drift.

Current Limitations:

1. Cache limitation—measurements are performed under cache-off conditions, and generalization to cache-on (production environment) reasoning has not been verified;
2. Application claims—navi-SAD is a research tool rather than a product, and does not directly claim to detect 'hallucinations' or 'authenticity'. The early TruthfulQA pilot was closed because it failed the permutation null test (p=0.96); Future Directions: Gate3 will be redesigned around synthetic HMM benchmarks, using known fractal dimensions to verify whether permutation entropy can track the fractal dimension of the belief state attractor predicted by Shai et al. (NeurIPS 2024).

navi-SAD engages with recent representational dynamics research (e.g., D2HScore, EigenTrack, Neural Uncertainty Principle, Verbal Uncertainty Mismatch) but has unique features:

• No published method runs two attention mechanisms in parallel on the same frozen weights as a dynamical systems probe;
• Combines known components (linear attention, cosine divergence, delayed coordinate embedding) in a new configuration;
• Uses the Takens framework (treating per-head SAD as attractor reconstruction rather than scalar diagnosis) as a theoretical contribution.