CoT-Space: Continuous Reasoning Optimization

• CoT-Space is a continuous reasoning-level semantic space that models chain-of-thought as an optimization process over meaningful reasoning steps.
• It formalizes trade-offs between underfitting and overfitting by tuning CoT length through noise regulation and semantic equivalence, ensuring optimal reasoning trajectories.
• Extensions to multimodal and latent domains demonstrate CoT-Space’s efficiency in prompt design, implicit reasoning tokens, and improved spatial and vision-language tasks.

CoT-Space denotes, in the most explicit recent usage, a continuous, reasoning-level semantic space in which Chain-of-Thought reasoning is analyzed as an optimization process rather than as a discrete token-prediction trajectory. It was introduced to address the mismatch between traditional token-level reinforcement learning and the reasoning-level nature of complex, multi-step thought processes, and it has since become a useful umbrella for related views in which prompts, latent tokens, activation shifts, or multimodal controllers define trajectories through structured reasoning spaces rather than merely longer output strings (Gan et al., 4 Sep 2025, Tong et al., 20 Mar 2026).

1. Definition and conceptual scope

In the formal framework introduced in "CoT-Space: A Theoretical Framework for Internal Slow-Thinking via Reinforcement Learning" (Gan et al., 4 Sep 2025), the central claim is that token-level Markov Decision Process formulations are fundamentally misaligned with Chain-of-Thought. The token-level state space $\mathcal{S}$ is described as exponentially large and discrete, and many distinct token sequences can express the same reasoning content. CoT-Space therefore treats the unit of analysis as semantically meaningful reasoning steps rather than raw token strings.

The framework makes this precise through reasoning step semantic equivalence. For a reasoning step $\xi_l$, the set of semantically equivalent token sequences is defined as

$$\mathcal{T}(\xi_l) = \{ \tau \in \mathcal{A}^* \mid \text{Decode}(\tau) \equiv \xi_l \}.$$

Under the accompanying Expressive Redundancy assumption, the size $|\mathcal{T}(\xi_l, k)|$ grows exponentially with token budget $k$ for each reasoning step. The resulting Continuum Limit Theorem states that, as the number of tokens or reasoning steps increases, the density of semantically meaningful states grows exponentially and the reasoning space converges to a continuous semantic manifold (Gan et al., 4 Sep 2025).

Within this space, a state is written as $s_o = (q, \xi_o)$, comprising the current question $q$ and the sequence of reasoning steps $\xi_o$ completed so far. States where the golden answer $\phi$ is reached are defined as minimums. The objective is then to trace a trajectory toward such a minimum while minimizing a reasoning loss $C(s)$, interpreted as distance to solution (Gan et al., 4 Sep 2025).

A broader theoretical usage appears in work on in-context learning, where CoT is treated as task decomposition over prompts of the form $\xi_l$0, with $\xi_l$1 the input and $\xi_l$2 the final output. This suggests that CoT-Space can also be understood as the space of compositional subtasks made accessible by demonstrations and instructions, rather than only the space of generated thought traces (Tong et al., 20 Mar 2026).

2. Continuous optimization in reasoning-level semantic space

The defining move in CoT-Space is to recast reasoning as continuous optimization. Instead of treating every next token as an action in a vast discrete control problem, the framework models multi-step thought as motion in a reasoning-level semantic space. The dynamics are expressed as a stochastic differential equation,

$\xi_l$3

where $\xi_l$4 is a noise term whose variance decreases with CoT length $\xi_l$5 (Gan et al., 4 Sep 2025).

This formulation yields a direct interpretation of CoT length as an optimization variable. Longer CoTs correspond to smaller optimization noise, with noise scale $\xi_l$6. The analogy drawn in the paper is to learning rate or number of gradient steps: short CoTs imply higher noise and underfitting, whereas excessively long CoTs can reduce noise so much that the policy overfits to quirks or sharp minima (Gan et al., 4 Sep 2025).

The same reasoning-level view also clarifies why token-level descriptions can be inadequate. Distinct surface forms may belong to the same semantic equivalence class $\xi_l$7, so counting tokens or optimizing per-token log-probability does not directly capture the object of interest. CoT-Space therefore provides a semantics-first account of reasoning trajectories, one in which different verbalizations may implement the same step and the same token budget may support very different reasoning paths (Gan et al., 4 Sep 2025).

This continuous-space picture is echoed, in a different form, by work on prompt design. There, prompts are modeled as selectors that extract task-relevant information from the model’s full hidden state during CoT reasoning, and each prompt defines a unique trajectory through the answer space. The prompt-space size is characterized as

$\xi_l$8

where $\xi_l$9 is proportional to the dimension of the internal state and $$\mathcal{T}(\xi_l) = \{ \tau \in \mathcal{A}^* \mid \text{Decode}(\tau) \equiv \xi_l \}.$$0 to the amount of information extracted per step. This suggests a complementary interpretation of CoT-Space as a structured family of reachable reasoning trajectories induced by prompt choice (Zhang et al., 13 Mar 2025).

3. Optimal CoT length, overthinking, and generalization

A central theoretical result of CoT-Space is that convergence to an optimal CoT length is a natural consequence of the trade-off between underfitting and overfitting. On the risk side, the framework gives the information-theoretic upper bound

$$\mathcal{T}(\xi_l) = \{ \tau \in \mathcal{A}^* \mid \text{Decode}(\tau) \equiv \xi_l \}.$$1

On the empirical-loss side, it gives the lower bound

$$\mathcal{T}(\xi_l) = \{ \tau \in \mathcal{A}^* \mid \text{Decode}(\tau) \equiv \xi_l \}.$$2

where $$\mathcal{T}(\xi_l) = \{ \tau \in \mathcal{A}^* \mid \text{Decode}(\tau) \equiv \xi_l \}.$$3 is the minimum CoT depth needed for question $$\mathcal{T}(\xi_l) = \{ \tau \in \mathcal{A}^* \mid \text{Decode}(\tau) \equiv \xi_l \}.$$4. The total error is therefore U-shaped as a function of $$\mathcal{T}(\xi_l) = \{ \tau \in \mathcal{A}^* \mid \text{Decode}(\tau) \equiv \xi_l \}.$$5: small $$\mathcal{T}(\xi_l) = \{ \tau \in \mathcal{A}^* \mid \text{Decode}(\tau) \equiv \xi_l \}.$$6 leads to high bias and underfitting, large $$\mathcal{T}(\xi_l) = \{ \tau \in \mathcal{A}^* \mid \text{Decode}(\tau) \equiv \xi_l \}.$$7 leads to high variance and overfitting, and a unique $$\mathcal{T}(\xi_l) = \{ \tau \in \mathcal{A}^* \mid \text{Decode}(\tau) \equiv \xi_l \}.$$8 minimizes the trade-off (Gan et al., 4 Sep 2025).

The empirical findings reported alongside the theory sharpen this picture. Harder tasks empirically require longer CoTs for optimal performance; larger and more expressive models converge to shorter $$\mathcal{T}(\xi_l) = \{ \tau \in \mathcal{A}^* \mid \text{Decode}(\tau) \equiv \xi_l \}.$$9; different RL algorithms such as PPO, DAPO, and GRPO yield similar converged $|\mathcal{T}(\xi_l, k)|$0 for the same model and task; and more noisy optimization leads to shorter converged CoT length (Gan et al., 4 Sep 2025). The framework uses these observations to explain overthinking as deterioration that occurs when CoT length exceeds the task- and model-appropriate optimum.

Independent empirical work supports the same concern from different angles. In multimodal evaluation, CoT prompting often degrades performance on perception-heavy tasks, suggesting a potentially harmful overthinking behavior, and long reasoning traces do not guarantee better coverage of key reasoning steps (Jiang et al., 13 Feb 2025). In agent settings, CoT training improves prompt actions substantially, while the relative advantage of CoT actions over prompt actions remains similar across checkpoints; later checkpoints are also less likely to revise the action in response to CoT, suggesting greater reliance on the prompt. The same study reports that more than 80% of attention mass and training gradient for predicted action tokens is assigned to prompt tokens across agent environments and models (Liu et al., 25 Jun 2026).

The practical consequence is not simply that CoT should be shorter. Rather, the CoT-Space view implies that length is a regularization variable whose optimal value depends on task difficulty, model capacity, and training noise. This interpretation is consistent with CoT-Valve, which identifies a direction in parameter space that controls the length of generated CoT and compresses reasoning chains with small performance loss: on GSM8K, QwQ-32B-Preview is reduced from 741 to 225 tokens with a minor performance drop from 95.07% to 94.92%, and on AIME from 6827 to 4629 tokens with only one additional incorrect answer (Ma et al., 13 Feb 2025).

4. Prompt-space, demonstrations, and compositional task navigation

A second theoretical strand treats CoT-Space through the lens of in-context learning and prompt design. In "Demonstrations, CoT, and Prompting: A Theoretical Analysis of ICL" (Tong et al., 20 Mar 2026), CoT prompting is formalized as task decomposition, where a complex task is broken into a sequence of $|\mathcal{T}(\xi_l, k)|$1 subtasks and prompts are progressively augmented with intermediate steps. The resulting CoT generalization bound is

$|\mathcal{T}(\xi_l, k)|$2

where $|\mathcal{T}(\xi_l, k)|$3 is the effective Lipschitz parameter for the $|\mathcal{T}(\xi_l, k)|$4-th CoT subtask and $|\mathcal{T}(\xi_l, k)|$5 is the model’s intrinsic ability in that subtask (Tong et al., 20 Mar 2026).

The practical interpretation is explicit: CoT is beneficial when it decomposes the overall task into subtasks that are easy to learn and well identified by demonstrations. If decomposition introduces hard or ambiguous subtasks, CoT can be neutral or harmful. The same framework also proves that, under consistent prompt formats, the influence of the prompt template on the output decays exponentially with the number of demonstrations,

$|\mathcal{T}(\xi_l, k)|$6

where $|\mathcal{T}(\xi_l, k)|$7 is the number of demonstrations (Tong et al., 20 Mar 2026).

The prompt-complexity perspective adds a complementary account. "Why Prompt Design Matters and Works: A Complexity Analysis of Prompt Search Space in LLMs" argues that prompts act as selectors, extracting task-relevant information from hidden state and defining answer-space trajectories via recurrent updates of the form

$|\mathcal{T}(\xi_l, k)|$8

On this view, finding the correct CoT prompt is equivalent to discovering the right algorithm for the task, and naive self-guided prompts such as "think step by step" can severely hinder performance. The paper reports that optimal prompt search can lead to more than a 50% improvement on reasoning tasks (Zhang et al., 13 Mar 2025).

Taken together, these results suggest a prompt-induced version of CoT-Space: demonstrations and instructions do not merely elicit longer explanations, but specify which subtasks become accessible, how stable the corresponding paths are, and how much of answer space is narrowed by the prompt configuration. In this sense, CoT-Space is not only a geometry of generated thoughts, but also a geometry of reachable reasoning programs under prompt conditioning (Tong et al., 20 Mar 2026, Zhang et al., 13 Mar 2025).

5. Latent, implicit, activation-space, and parameter-space realizations

Recent work operationalizes CoT-Space inside model internals rather than in explicit text. In distilled 3D vision-LLMs, Hidden CoT introduces a fixed set of $|\mathcal{T}(\xi_l, k)|$9 learnable, randomly initialized thinking tokens inserted before the question and answer. These tokens are never exposed, verbalized, or decoded; they function as a latent internal scratchpad, receive no direct labels, and are trained only through answer-token loss. In the reported system, the student achieves 8.7x lower inference latency and a 3x reduction in model size while retaining 54–72% of the teacher’s performance, with $k$0 identified as the sweet spot for efficiency vs. accuracy (Asfour et al., 10 May 2026).

In implicit reasoning models, SIM-CoT identifies a latent instability issue: as the number of reasoning tokens increases, training often becomes unstable and collapses because latent representations become homogeneous and lose semantic diversity. The paper measures this through the inter-latent distance

$k$1

and the distance to the vocabulary center

$k$2

SIM-CoT adds an auxiliary decoder during training to align each implicit token with its corresponding explicit reasoning step, then removes the decoder at inference. It boosts Coconut by +8.2% on GPT-2 and CODI by +3.0% on LLaMA-3.1 8B, surpasses the explicit CoT baseline on GPT-2 by 2.1%, and does so with 2.3$k$3 greater token efficiency (Wei et al., 24 Sep 2025).

A related activation-space formulation appears in CoT Vectors. There, a CoT vector is a compact representation encoding task-general, multi-step reasoning knowledge. Extracted CoT Vectors are defined as activation differences between CoT and non-CoT runs, while Learnable CoT Vectors are optimized under a teacher-student objective

$k$4

The paper reports a pronounced U-shaped layer-wise performance curve and interprets it as a three-stage reasoning process: shallow layers for feature extraction and semantic encoding, middle layers for core reasoning with high information density, and deep layers for linguistic expression of the reasoning result (Li et al., 1 Oct 2025).

Parameter-space control provides yet another operationalization. CoT-Valve identifies a direction $k$5 in parameter space, realized by a LoRA branch, that controls CoT length; smaller $k$6 yields longer chains and larger $k$7 yields shorter chains. The authors describe this as a "valve" that controls CoT flow, and the reported interpolation and extrapolation behavior suggests that reasoning length is not wholly entangled with answer correctness (Ma et al., 13 Feb 2025).

These methods are not identical to the RL-based CoT-Space formalism. A plausible implication is that they instantiate different coordinate systems for the same broader problem: continuous reasoning can be represented as latent scratchpads, stable implicit-token manifolds, activation shifts, or controllable parameter directions, each exposing different aspects of internal slow-thinking (Asfour et al., 10 May 2026, Wei et al., 24 Sep 2025, Li et al., 1 Oct 2025, Ma et al., 13 Feb 2025).

6. Multimodal and spatial extensions

CoT-Space has also been extended beyond text-only reasoning. In text-to-image generation, CoT-Diff integrates CoT-style step-by-step spatial reasoning directly into the denoising trajectory of a diffusion model. At each denoising step, an MLLM evaluates the intermediate predicted image $k$8 against the current scene plan $k$9 and prompt $s_o = (q, \xi_o)$0, refines the 3D layout, and injects updated semantic layout masks and depth maps through a condition-aware attention mechanism,

$s_o = (q, \xi_o)$1

The paper states that CoT-Diff outperforms the state-of-the-art method by 34.7% in complex scene spatial accuracy, and gives the concrete example that in “Front Left” relations it achieves 66.4% spatial alignment versus a best previous baseline of 48.4% (Liu et al., 6 Jul 2025).

In medical visual question answering, MC-CoT uses a modular collaborative pipeline in which an LLM decomposes the task into radiology, anatomy, and pathology subtasks, provides guidance for each, and synthesizes the final answer from module outputs. On PATH-VQA, VQA-RAD, and SLAKE, the reported average performance of MC-CoT with LLaVA-1.5-7B and GPT-3.5 is 58.93 recall and 46.07 accuracy, compared with 55.39 and 36.94 for the MLLM-only baseline (Wei et al., 2024).

Process-level benchmarking has made the limitations of multimodal CoT explicit. MME-CoT evaluates six domains—math, science, OCR, logic, space-time, and general scenes—using 1,130 questions with 3,865 annotated key reasoning steps and metrics for quality, robustness, and efficiency. It reports that models with reflection mechanism demonstrate superior CoT quality, with Kimi k1.5 scoring 64.2 on F1 versus 64.0 for GPT-4o, but also that CoT prompting often degrades performance on perception tasks and that only about 60% of reflection steps are valid or productive (Jiang et al., 13 Feb 2025).

These multimodal results broaden the meaning of CoT-Space from an internal semantic manifold for language reasoning to a family of entangled control spaces in which reasoning steps guide perception, geometry, and generation. At the same time, they preserve the core caution of the original framework: more explicit or longer reasoning is not automatically better, and the central problem remains how to align the granularity, stability, and cost of reasoning with the structure of the task (Liu et al., 6 Jul 2025, Wei et al., 2024, Jiang et al., 13 Feb 2025).