Entropy-Gated Latent Recursion

The evolution of large-scale pre-trained language models has significantly advanced NLP capabilities. Early approaches like beam search and stochastic sampling improved answer diversity but were limited by their reliance on randomness, which often led to redundant trajectories. Recent work introduced recursive reasoning modules (Giannou et al., 2023; Hao et al., 2024), requiring training, to explore latent reasoning paths. Li et al. (2026) proposed entropy-based gating to select uncertain tokens for refinement, but still depended on external modules. The challenge remains to generate diverse, meaningful reasoning paths efficiently without retraining, especially for complex tasks like math reasoning. This paper addresses this gap by proposing a deterministic, structure-based inference axis, leveraging the model’s internal layer span, to produce rich reasoning trajectories without additional training or parameters.

Current inference-time scaling methods primarily depend on stochastic token sampling, which limits the diversity of reasoning paths to noise-driven variations. This stochasticity often results in near-duplicate trajectories, reducing the effectiveness of ensemble and reranking strategies. Furthermore, stochastic methods cannot access certain reasoning pathways that are structurally distinct but deterministic in nature. The core problem is how to systematically generate multiple, diverse, and meaningful reasoning trajectories without retraining or incurring significant computational overhead. Addressing this problem is crucial for improving the robustness, interpretability, and accuracy of large language models in complex reasoning tasks.

The paper introduces three key innovations: 1) The layer span L as a deterministic, structural axis for inference, enabling multiple reasoning paths by re-invoking the top-L layers at high-uncertainty tokens. 2) An entropy gating mechanism that triggers recursive re-application only at uncertain tokens, avoiding unnecessary computation. 3) The formation of an L×T Cartesian sampling space by combining layer span L with temperature T, greatly expanding the candidate pool without retraining. These innovations collectively enable a structured, multi-path reasoning process that is both efficient and effective, surpassing traditional stochastic methods in diversity and performance.

• �� Use a frozen autoregressive transformer with N layers, compute residual states at each layer.
• �� During decoding, calculate token entropy H(pt) to identify uncertain tokens.
• �� When H(pt) exceeds threshold τH, trigger recursive re-application of the top-L layers, up to Kmax iterations.
• �� For each high-uncertainty token, fuse the anchor state with the refined state via a weighted average controlled by α.
• �� Generate multiple reasoning trajectories by varying L (layer span) deterministically.
• �� Combine these with temperature T sampling to form an L×T inference space.
• �� Use self-consistency (EGLR-SC) to aggregate multiple paths via majority voting, improving accuracy.
• �� The entire process requires no training, only inference-time recursion guided by entropy thresholds.

The experiments evaluate 8 instruction-tuned models across six benchmarks, including GSM8K and MATH-500. Hyperparameters like Kmax=3 and τH are auto-calibrated. The study compares greedy, single-axis (L or T), and joint L×T sampling, measuring accuracy, oracle coverage, and path diversity. Ablation studies analyze the impact of L and Kmax. Results show that the joint L×T approach consistently outperforms single-axis methods, with improvements up to +8.2 percentage points on MATH-500. The experiments also compare with FLOP-matched beam search, demonstrating superior performance and diversity, validating the method’s effectiveness across models and tasks.

The joint L×T oracle achieves 91.6% accuracy on MATH-500, surpassing the temperature-only oracle (83.4%) by 8.2 points and the layer-only oracle (81.2%) by 10.4 points. EGLR-SC improves accuracy in 44/48 model-task pairs, outperforming greedy and beam search at matched FLOPs. Different L values produce diverse, deterministic reasoning paths, with no single L dominating, confirming the structural diversity. The combined approach effectively captures reasoning trajectories inaccessible to stochastic sampling alone, demonstrating broad applicability and robustness.

This method can be integrated into systems requiring high reasoning accuracy, such as mathematical problem solving, scientific question answering, and legal reasoning. Its ability to generate multiple, diverse reasoning paths at low cost enhances ensemble, reranking, and reinforcement learning applications. The approach is suitable for deployment in real-time AI assistants, educational tools, and automated reasoning systems, especially where interpretability and robustness are critical. Future extensions could incorporate external knowledge bases and feedback loops for further improvement.

The choice of layer span L is task-dependent and requires tuning, adding complexity. Recursive iterations capped at Kmax may limit exploration of all potential paths, especially in highly complex scenarios. In cases of extremely high uncertainty, the recursive process might not fully capture all relevant trajectories. Computational overhead, though minimal, still exists compared to greedy decoding. Future work should focus on adaptive L selection, external knowledge integration, and scaling to multi-modal reasoning to address these limitations.