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Emergence of Superposition: Unveiling the Training Dynamics of Chain of Continuous Thought

ICLR 2026 · Large language models › Reasoning and chain of thought · approach: Long CoT dynamics and efficiency

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Analyzes gradient-flow training dynamics of a simplified two-layer transformer on directed graph reachability with continuous CoT, showing the index-matching logit stays bounded (superposition emerges) under the CoT-supervised loss, unlike the unbounded logit under a BFS-style loss.

Problem

Why and how superposition of multiple reasoning traces emerges during gradient-based training of continuous chain-of-thought, given that prior work only showed a construction exists for graph reachability.

Approach

Reparameterizes the second-layer attention with an index-matching logit and shows under single-path CoT supervision (Coconut loss) the logit first grows then stays bounded, balancing exploration and exploitation so the thought becomes a superposition of frontier nodes; the BFS-style loss instead diverges logarithmically.

Key results

• Trained model reaches 96.2% test accuracy on the graph reachability subset with permuted vertex indices.
• Under the CoT (Coconut) loss, the frontier-vs-non-frontier attention logit difference saturates around 60 after roughly 125 epochs in Stage 1, consistent with a bounded index-matching logit.
• Theory: under the BFS loss the index-matching logit diverges at least logarithmically in t, while under the Coconut loss it converges to a finite value when the frontier in-degree is below the maximum in-degree.

Surprising findings

• The BFS-style loss, which explicitly rewards any reachable node, yields an unbounded logit and sharper, less superposed distributions, while the single-path CoT loss stays bounded.

Limitations

• Analysis uses a simplified attention-only two-layer transformer with linear attention, fixed V, and orthonormal embeddings.
• Theory assumes fixed thought distributions in prediction stage; practice shows logit plateaus rather than unbounded growth.
• Candidate-lift mechanism is not always mediated by the <R> token across seeds and settings.

Experimental setup

Datasets: ProsQA subset (graph reachability)

Benchmarks: ProsQA subset (graph reachability)

Metrics: test accuracy attention logit difference (frontier vs non-frontier edges)

Baselines: Coconut-BFS loss

Builds on: Coconut continuous chain of thought expressivity construction (Zhu et al.) ProsQA

Closest papers by meaning

• Reasoning by Superposition: A Theoretical Perspective on Chain of Continuous Thought (NeurIPS 2025 · similarity 0.95)
• Metastable Dynamics of Chain-of-Thought Reasoning: Provable Benefits of Search, RL and Distillation (ICML 2025 · similarity 0.90)
• Continuous Chain of Thought Enables Parallel Exploration and Reasoning (ICLR 2026 · similarity 0.90)
• Let Me Think! A Long Chain of Thought Can Be Worth Exponentially Many Short Ones (NeurIPS 2025 · similarity 0.90)
• Markovian Transformers for Informative Language Modeling (ICLR 2026 · similarity 0.89)
• Compositional Generalization from Learned Skills via CoT Training: A Theoretical and Structural Analysis for Reasoning (ICLR 2026 · similarity 0.89)