When Reasoning Meets Its Laws

Abstract: Despite the superior performance of Large Reasoning Models (LRMs), their reasoning behaviors are often counterintuitive, leading to suboptimal reasoning capabilities. To theoretically formalize the desired reasoning behaviors, this paper presents the Laws of Reasoning (LoRe), a unified framework that characterizes intrinsic reasoning patterns in LRMs. We first propose compute law with the hypothesis that the reasoning compute should scale linearly with question complexity. Beyond compute, we extend LoRe with a supplementary accuracy law. Since the question complexity is difficult to quantify in practice, we examine these hypotheses by two properties of the laws, monotonicity and compositionality. We therefore introduce LoRe-Bench, a benchmark that systematically measures these two tractable properties for large reasoning models. Evaluation shows that most reasoning models exhibit reasonable monotonicity but lack compositionality. In response, we develop an effective finetuning approach that enforces compute-law compositionality. Extensive empirical studies demonstrate that better compliance with compute laws yields consistently improved reasoning performance on multiple benchmarks, and uncovers synergistic effects across properties and laws. Project page: https://lore-project.github.io/

• The paper formulates LoRe, establishing compute and accuracy laws that scale with problem complexity.
• It empirically demonstrates that most LRMs follow monotonic trends in compute and accuracy but struggle with compositionality.
• Fine-tuning with compositionality constraints significantly improves both reasoning accuracy and efficiency across benchmarks.

When Reasoning Meets Its Laws: An Analytical Overview

Introduction and Motivation

The paper "When Reasoning Meets Its Laws" (2512.17901) investigates intrinsic patterns underlying reasoning in large reasoning models (LRMs) by developing a formal theoretical framework, LoRe (Laws of Reasoning), designed to characterize the optimal scaling of reasoning behaviors with respect to problem complexity. The motivation is rooted in the observation that state-of-the-art LRMs, even when highly capable, often display computation and accuracy dynamics that are misaligned with human reasoning processes. A canonical anomaly is that LRMs may expend more tokens and achieve lower accuracy on a simple sub-problem than on a more complex composed task, a phenomenon that is systematically visualized in the following benchmarked example:

Figure 1: Anomalous reasoning compute allocation and accuracy drop in DeepSeek-R1 for summation, squaring, and their composition, highlighting deviation from human-like reasoning patterns.

Such irregularities are partially attributed to the lack of explicit, theoretically-motivated guidelines for how models should allocate compute or target accuracy as functions of problem complexity. This paper formalizes and empirically probes these desiderata through novel “laws of reasoning” and provides actionable evaluation protocols.

Theoretical Framework: The Laws of Reasoning

The LoRe framework introduces two core laws:

• Compute Law: The expected reasoning compute $C_\theta(x)$ should scale linearly with the intrinsic complexity $\kappa(x)$ of question $x$, i.e., $$C_\theta(x) = \alpha_\theta \kappa(x) + o(\kappa(x))$$.
• Accuracy Law: The probability of correct solution $A_\theta(x)$ should decay exponentially with complexity, $A_\theta(x) = \exp(-\lambda_\theta \kappa(x))$.

Since direct computation of $\kappa(x)$ is generally intractable, the framework proposes assessment via two empirically tractable properties:

• Monotonicity: For any two instances $x_1$, $x_2$ with $\kappa(x_1) \leq \kappa(x_2)$, compute must satisfy $C_\theta(x_1) \leq C_\theta(x_2)$ and accuracy must satisfy $A_\theta(x_1) \geq A_\theta(x_2)$.
• Compositionality: For independent $x_1$ and $x_2$, $$C_\theta(x_1 \oplus x_2) \approx C_\theta(x_1)+C_\theta(x_2)$$ and $$A_\theta(x_1 \oplus x_2) \approx A_\theta(x_1)A_\theta(x_2)$$.

This framework encapsulates the desired scaling characteristics analogous to classic learning scaling laws, but applies them specifically to the reasoning trace and compositional accuracy structures:

Figure 2: The LoRe framework connects laws of compute and accuracy with their empirical manifestations via monotonicity and compositionality.

LoRe-Bench: Empirical Probing of Reasoning Laws

Monotonicity Benchmark (LoRe-Mono)

To measure monotonicity, the authors introduce LoRe-Mono, a synthetic dataset spanning math, science, language, and code domains, where each seed question generates 30 variants with controlled, strictly increasing complexity:

Figure 3: Question variant generation protocol, incrementing complexity by increasing the number of primitive steps applied to each seed.

Compositionality Benchmark (LoRe-Compo)

For compositionality, LoRe-Compo is constructed from pairs of independent questions sampled from different pre-defined domains (e.g., algebra and geometry) and their conjunction, directly enabling rigorous compositionality metrics using the normalized mean absolute deviation (nMAD).

Empirical Evaluation and Findings

Monotonicity Results

Across ten recent LRMs, including DeepSeek-R1 (multiple capacities), Phi-4-mini-reasoning, and models featuring reasoning length control, the following key observations emerge:

• Most models show high Spearman correlation between constructed complexity and reasoning compute, and negative correlation with accuracy—indicating monotonicity is largely satisfied for major model families.
• Notably, certain models (e.g., DeepSeek-R1-1.5B) fail monotonicity in more complex or under-trained domains (e.g., language), and significant deviations are visible in variance across tasks.

  Figure 4: Visualization of reasoning compute and log accuracy versus complexity variant index, showing near-linear and monotonic scaling in most domains.

Compositionality Results

Empirical compositionality is substantially absent in all evaluated models, even those with explicit reasoning length control:

Figure 5: Empirical compositionality assessment for reasoning compute, with broad deviations from the ideal $y=x$ line, confirming lack of additivity across independent sub-tasks.

Figure 6: nMAD metrics quantifying departure from compositionality in compute and accuracy.

Improving Compositionality via Supervised Fine-Tuning

Given this deficiency, the authors devise a fine-tuning (SFT-Compo) method: For each triple $(x_1, x_2, x_{12})$, they sample multiple reasoning outputs, select optimal paths that best satisfy compositionality (matching sum of token counts), and use these as supervised targets for further model tuning.

This procedure leads to:

• Strong improvements in compositionality as measured by reduced nMAD in compute (e.g., $0.528 \to 0.314$ for DeepSeek-R1-1.5B).
• Consistently improved Pass@1 accuracy across general reasoning benchmarks (GSM8K, MATH500, AIME, AMC, OlympiadBench), with substantial gains such as $+7.4$ or $+5.0$ on individual benchmarks.
• Crucially, gains persist when compared to a baseline SFT that only uses correct outputs without compositionality optimization, implying that enforcing the law—rather than additional teacher supervision per se—is responsible for the improvement.

Figure 7: Monotonicity restoration in code reasoning for 1.5B models after compositionality enforcement.

Synergistic Effects

• Enforcing compositionality not only enhances that property but also improves monotonicity and accuracy compositionality, suggesting coupling and mutual reinforcement among the proposed laws.
• These results support the hypothesis that models systematically aligned with optimal reasoning laws generalize more robustly and allocate cognitive compute more effectively.

Figure 8: Compositionality improvements—alignment with the $y=x$ ideal—across additional LRM architectures.

Implications and Future Directions

This work provides foundational theoretical and empirical tools to analyze and optimize the "thinking patterns" of LRMs. By moving beyond heuristic prompts or reward shaping and toward explicit adherence to computational and statistical laws, model designers are equipped to align reasoning models with optimal performance regimes observable in humans.

Practical consequences include:

• Principled pre- and post-training regimes: Incorporating compositional compute-based SFT as an explicit training objective.
• Benchmarking and failure diagnosis: Detection of anomalous scaling, overthinking, and underthinking in model development.
• Scalable generalization: Facilitating compositional generalization and calibration in complex domains (math, code, science), which are critical for alignment and interpretability.

Theoretically, this framework may be extended to additional laws (e.g., error propagation, variance scaling) and inspire connections with meta-reasoning, uncertainty calibration, and model-based planning in LLMs.

Conclusion

"When Reasoning Meets Its Laws" (2512.17901) delivers an articulation of optimal reasoning scaling principles, supported by new benchmarks and actionable fine-tuning strategies. The coupling between empirical properties (monotonicity and compositionality) and latent model optimality offers a precise criterion for advancing LRM reasoning quality, with direct implications for both cognitive alignment and robust downstream task performance.