When Reasoning Meets Its Laws

1 INTRODUCTION

• DeepSeek-R1 uses about 300 more reasoning tokens on a squaring question than on the composite “sum, then square” question, alongside a 12.5% accuracy drop.The example illustrates reasoning behavior that differs from human adaptation to problem complexity.Source: §1 · p. 1, Figure 1 · p. 2
• The paper identifies a gap in explicit principles for desirable reasoning, asking how to formalize and evaluate them and whether enforcing them improves general reasoning.It attributes inefficient overthinking or underthinking, which can harm performance, to training data rarely constrained by explicit thinking-budget rules.Source: §1 · pp. 1–2
• LORE formalizes reasoning behavior through a compute law and complementary accuracy law, approximated in practice by monotonicity and compositionality.The paper presents these properties as tractable ways to examine the laws.Source: §1 · p. 2
• LORE-BENCH evaluations find that current models show reasonable monotonicity but lack compositionality, including competitive baselines.LORE-MONO covers diverse domains, while LORE-COMPO is constructed from MATH500 to measure compositionality.Source: §1 · p. 2
• Fine-tuning for compute-law compositionality improves that property and generally improves reasoning capability, with synergistic effects across properties and laws.The introduction reports these findings from validation experiments.Source: §1 · p. 2

2.1 PROBLEM FORMULATION

• LORE represents a model output as a reasoning chain followed by an answer, and defines a composite question by concatenating two questions with a connector prompt.The framework assumes a fixed decoding strategy by default.Source: §2.1 · p. 2
• The paper defines a question’s complexity as the minimum number of unit-cost primitive steps in a valid solution, but computing that minimum is generally intractable.Verifying minimal solution length requires global search over a potentially exponential space.Source: §2.1 · p. 3

2.2 COMPUTE LAW

• The compute law hypothesizes reasoning compute proportional to question complexity, with a model- and decoding-specific linear factor and sublinear overhead.The framework treats introductory and transition tokens as the overhead term.Source: §2.2 · p. 3
• Monotonicity and compositionality are proposed as tractable proxies for the compute law, with the two properties theoretically sufficient to imply it under stated conditions.Monotonicity compares relative complexity; compositionality tests additive compute for independent question pairs.Source: §2.2 · p. 4

2.3 BEYOND COMPUTE: ACCURACY LAW

• The accuracy law hypothesizes exponentially decreasing accuracy with complexity; its proposed properties are non-increasing accuracy with complexity and multiplicative accuracy for independent questions.Under stated conditions, the two accuracy properties imply log accuracy proportional to negative complexity.Source: §2.3 · pp. 4–5

3.1 LORE-MONO

• LORE-MONO orders complexity using 30-step variants of seed questions from math, science, language, and code, generated by scripts and manually checked.The benchmark uses known complexity orderings to assess monotonicity in reasoning compute and accuracy.Source: §3.1 · p. 5

3.2 LORE-COMPO

• LORE-COMPO forms 250 triplets by concatenating distinct-subject MATH500 questions and measures deviations from additive compute or log accuracy using normalized MAD.Lower MAD indicates stronger compositionality, while normalized MAD addresses scale dependence.Source: §3.2 · p. 6

3.3 FINDINGS AND ANALYSIS

• Across 10 evaluated LRMs, compute generally correlates strongly with complexity, while log accuracy mostly declines; DeepSeek-R1-Distill-Qwen-1.5B is weaker.For that model, compute correlation is negative in language (−0.346) and near zero in code (0.151).Source: §3.3 · pp. 6–7, Table 1 · p. 6, Figure 4 · p. 7
• Most evaluated LRMs show strong compute–complexity correlations, but large nMADs and substantial deviations from additive compute indicate weak compositionality, even with length control.Figure 5 compares composite-question compute with the sum for its subquestions.Source: §3.3 · p. 7, Table 2 · p. 7, Figure 5 · p. 7

Proposed Method: SFT-Compo

• Accuracy compositionality is difficult to enforce directly because it does not specify which reasoning path should be selected for supervision.Source: §4 · p. 7
• SFT-Compo pairs questions from distinct categories, concatenates each pair into a composite, and samples K outputs for all three questions from the model or a stronger teacher.Each output includes a reasoning path and its corresponding answer.Source: §4 · p. 8
• It selects combinations whose three reasoning paths yield correct answers and best satisfy the compute-compositionality condition.Each triplet supplies supervised examples for both subquestions and their composite, which are used for SFT.Source: §4 · p. 8

5 EXPERIMENTS

• The experiments ask whether SFT-Compo enforces compositionality and whether it further improves LRM reasoning capacity.Source: §5 · p. 8

5.1 EXPERIMENTAL SETUP

• The evaluation fine-tuned four LRMs on 3.9K question-output pairs for five epochs and assessed compositionality plus general reasoning across six benchmarks.The dataset used sub-question and composite-question triplets; a 14B teacher sampled eight outputs per question.Source: §5.1 · p. 8

5.2 MAIN RESULTS

• 40.5% reduction in reasoning-compute nMAD on the 1.5B model (0.528 to 0.314) and 22.5% on the 8B model (0.423 to 0.328) followed SFT-Compo.Across 1.5B, 7B, and 8B models, SFT-Compo had lower nMADCθ than the base model and aligned more closely with y=x on the 1.5B model.Source: Figure 6 · p. 8, §5.2 · p. 9
• The 8B model gained +5.0 in average Pass@1; SFT-Compo improved all six benchmarks across four model sizes and outperformed SFT in every case.The table reports Pass@1 accuracy averaged across the six benchmarks.Source: §5.2 · p. 9, Table 3 · p. 9

5.3 SYNERGISTIC EFFECT ANALYSIS

• SFT-Compo increased overall reasoning-compute Spearman correlation from 0.875 to 0.977, including a code-domain rise from 0.151 to 0.914.Figure 7a presents the monotonicity comparison.Source: Figure 7 · p. 9, §5.3 · p. 10
• Log-accuracy compositionality nMAD fell 71.1% on the 1.5B model (2.368 to 0.685) and 35.4% on the 7B model (1.170 to 0.756).Figure 7b measures this property with nMADlog Aθ.Source: Figure 7 · p. 9, §5.3 · p. 10

• The framework builds on adaptive reasoning approaches that control a model’s reasoning budget during post-training or at test time.Source: §6 · p. 10
• Post-training work modulates when and how long a model reasons, including variable-length CoT fine-tuning and RL with length penalties.Source: §6 · p. 10
• Inference-time approaches dynamically allocate reasoning budgets, for example through confidence or a secondary controller.Source: §6 · p. 10

• The paper combines theoretical hypotheses with empirical validation.Source: §7 · p. 10
• The authors frame their contribution as a theoretical perspective grounded in human reasoning for improving reasoning in LRMs.Source: §7 · p. 10
• They hope LORE will inspire strategies that guide models toward their optimal paradigms of thinking.Source: §7 · p. 10

ETHICS STATEMENT

• The authors report no known ethical concerns.Source: ETHICS STATEMENT · p. 10

B ADDITIONAL RELATED WORK

• Related work describes LRMs and adaptive reasoning controls at post-training and inference time; prior complexity analysis is noted as limited to a constrained puzzle-solving setting.Source: B ADDITIONAL RELATED WORK · p. 14

C LIMITATIONS

• LORE-MONO has 40 seed questions, independence is proxied by disjoint mathematical concepts, and evaluation focuses on strong open-source LRMs because of budget constraints.Source: C LIMITATIONS · p. 14

D PROOFS AND COROLLARIES

• Under the stated assumptions, compute is exactly proportional to complexity, Cθ(x) = αθκ(x), for finite-complexity questions.The proof reduces compute to a function of complexity and establishes a common compute-per-complexity ratio; the corollary also considers sublinear overhead.Source: D PROOFS AND COROLLARIES · pp. 14–15
• For finite-complexity questions with positive accuracy, the proof derives Aθ(x) = exp(−λθκ(x)); a corollary treats sublinear deviation in the exponent.The argument uses the stated assumptions and a transformed accuracy function that scales linearly with complexity.Source: D PROOFS AND COROLLARIES · pp. 15–16

E.1.1 EXAMPLE SEED QUESTIONS OF LORE-MONO

• LoRe-Mono seed examples span science, language, math, and code, with tasks specifying iterative simulations or computations and requested outputs.Examples include a bioreactor simulation, a letter maze, a recurrence, and code execution.Source: §E.1.1 · pp. 16–17

E.1.2 POTENTIAL SHORTCUT FAILURES

• The authors manually excluded variants with periodic answer patterns because such patterns could let models guess answers without performing the intended computation.The stated screening goal is to ensure benchmark reliability.Source: §E.1.2 · p. 18

E.2.1 ADDITIONAL VISUALIZATION RESULTS

• Visualizations of monotonicity results for five remaining LRMs appear in Figure 8.The supplied caption identifies the figure as visualizations of monotonicity results on remaining LRMs.Source: §E.2.1 · p. 18, Figure 8 · p. 19

E.2.2 "FAIL TO THINK" CASE STUDY

• The weakest model, DeepSeek-R1-Distill-Qwen-1.5B, sometimes generates no reasoning tokens for code questions; one variant-1 output has reasoning length zero.The example output is an empty think block before the answer text.Source: §E.2.2 · p. 20

E.3 ADDITIONAL DETAILS OF LORE-COMPO

• Composite questions use a fixed connector that presents sub-questions in sequential order.The example asks two questions in order, one about a field-walking distance and one about an equation.Source: §E.3 · p. 20

E.4 ADDITIONAL RESULTS OF LORE-COMPO

• Additional visualizations of compositionality results for the remaining LRMs appear in Figure 9.The figure caption describes compositionality visualizations on more LRMs.Source: §E.4 · p. 20

F.1 IMPLEMENTATION DETAILS

• For DeepScaler, questions are categorized with GPT-4.1-mini, then paired across categories to construct sub-question and composite-question triplets.SFT uses a grid search over learning rates {1e-6, 5e-6, 5e-5}, batch size 8, gradient accumulation 2, and warmup ratio 0.Source: §F.1 · p. 20

F.2 ADDITIONAL EXPERIMENTAL RESULTS

• SFT-Compo lowers nMAD and brings reasoning-compute compositionality closer to y = x for DeepSeek-R1-Distill-Qwen-7B and DeepSeek-R1-Distill-Llama-8B.Figure 10 compares each model before and after SFT-Compo.Source: §F.2 · p. 20, Figure 10 · p. 21

G QUALITATIVE EXAMPLES OF LORE-COMPO

• After SFT-Compo, composite-question accuracy rises from 0.25 to 0.875, while reasoning compute rises from 732.13 to 4300.00 for the illustrated DeepSeek-Distill-1.5B example.The two component questions each have accuracy 1.0 before finetuning; their reasoning compute is 2177.25 and 3546.50, respectively.Source: G QUALITATIVE EXAMPLES OF LORE-COMPO · p. 21
• The composite example orders a cookie-selection problem before a polynomial-remainder problem, illustrating the two-question format.The cookie task asks for six indistinguishable cookies from three types; the second asks for a remainder modulo x^2 − 1.Source: G QUALITATIVE EXAMPLES OF LORE-COMPO · pp. 21, 23
• The worked responses give 28 cookie assortments and remainder −13x + 3 for the polynomial example.The cookie response models selections across types, and the polynomial response evaluates at 1 and −1 to determine the remainder.Source: G QUALITATIVE EXAMPLES OF LORE-COMPO · pp. 22, 24