Reasoning is topologically simple

Chris Kapulkin; Nathan Kershaw

cs.AINov 2, 2026

Reasoning is topologically simple

Authors: Chris Kapulkin, Nathan Kershaw

Organizations: College of Connected Computing & Department of Mathematics, Vanderbilt University · Department of Mathematics, University of Western Ontario

Abstract

We formulate and investigate the hypothesis that reasoning text has simpler co-occurrence topology than non-reasoning text, and that higher-quality reasoning has simpler co-occurrence topology than lower-quality reasoning. We discuss and test applications to AI safety and AI evaluation.

Figures & tables

Panel of Figure 1 — Figure 1: An example co-occurrence filtration

Genre	Median	IQR
Mathematics	0.537	0.482–0.626
Philosophy	0.674	0.617–0.754
History	0.730	0.681–0.811
Short stories	0.872	0.816–0.889

Table 1: Normalised

H_{1}

total persistence by genre (observed / null mean).

	Comparison	$p$	Shift	95% CI	$\Pr(X<Y)$
H1	Math $<$ history	$<10^{-5}$	$-0.19$	$[-0.26,-0.14]$	0.93
H2	Math $<$ stories	$<10^{-5}$	$-0.32$	$[-0.37,-0.25]$	0.99
H3	Philosophy $<$ history	$0.019$	$-0.05$	$[-0.12,-0.004]$	0.67
H4	Philosophy $<$ stories	$<10^{-5}$	$-0.19$	$[-0.23,-0.12]$	0.89

Table 2: Genre comparisons (one-sided Brunner Munzel test).

Figure 2: Total persistence distribution among the four genres.

Figure 3: Total persistence / null persistence for human vs OpenAI paper pairs.

Figure 4: ECI vs median total persistence / null persistence for 48 models.

Figure 5: Total persistence / null for benign and sabotaged tasks.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Original (s)	Optimized (s)	Speedup
AIME08	0.0332	0.00159	$20.9\times$
AIME09	5.860	0.0255	$230.2\times$
AIME10	15.045	0.0327	$459.7\times$
AIME11	92.606	0.0673	$1{,}376.5\times$

Appendix

Table 3: Comparison of the naive and optimized implementations of Co-occurrence persistent homology.

No.	OpenAI preprint	Matched human paper	Venue
5	A Complete Local Domain without a Small Cohen–Macaulay Module	He (2013), Normality and Cohen-Macaulayness of local models of Shimura varieties	Duke Math. J.
13	A Counterexample to the Group-Ring Determinant Conjecture	Li (2012), Compact Group Automorphisms, Addition Formulas and Fuglede-Kadison Determinants	Annals of Mathematics
19	A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs	Rudelson et al. (2016), Hafnians, perfect matchings and Gaussian matrices	Ann. Probab.
20	A Gepner stability condition on every smooth quintic threefold	Chiodo and Ruan (2010), Landau-Ginzburg/Calabi-Yau correspondence for quintic three-folds via symplectic transformations	Inventiones
22	A Linear Clock for Random Walk on Tree-Weighted Planar Maps	Benes et al. (2013), On the rate of convergence of loop-erased random walk to SLE(2)	Comm. Math. Phys.
33	A Sharp Threshold Bound for Monotone Graph Properties	Mester (2013), Invariant monotone coupling need not exist	Ann. Probab.

Appendix

Table 9

Model	Lab	ECI	Total persistence / null	AIME accuracy
Kimi K3	Moonshot (Kimi)	157.4	0.492	28/30
GLM-5.3	Zhipu (GLM)	155.6	0.348	23/30
DeepSeek V4 Pro 0813	DeepSeek	155.3	0.344	29/30
Qwen3.8 Max (0902)	Qwen	155.1	0.428	29/30
DeepSeek V4.1 Flash	DeepSeek	154.9	0.368	29/30
DeepSeek V4 Flash 0731	DeepSeek	154.5	0.369	28/30

Appendix

Table 10

Language Model Reasoning Evaluation

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Mar 21, 2026J/K move · Enter open · S save

Reasoning Topology Matters: Network-of-Thought for Complex Reasoning Tasks

Fan Huang

Indiana University Bloomington

LLM Reasoning LLM Reasoning with Graphs Efficient Language Model Reasoning Multi-Hop Reasoning

Open paper AskPDF arXiv

Existing prompting paradigms structure LLM reasoning in limited topologies: Chain-of-Thought (CoT) produces linear traces, while Tree-of-Thought (ToT) performs branching search. Yet complex reasoning often requires merging intermediate results, revisiting hypotheses, and integrating evidence from multiple sources. We propose Network-of-Thought (NoT), a framework that models reasoning as a directed graph with typed nodes and edges, guided by a heuristic-based controller policy. Across four benchmarks (GSM8K, Game of 24, HotpotQA, ProofWriter) and three models (GPT-4o-mini, Llama-3.3-70B-Instruct, Qwen2.5-72B-Instruct), we investigate when network topology outperforms chain or tree structures, whether LLM-generated heuristics can guide graph-based reasoning search, and the computation-accuracy tradeoff across topologies, evaluating each method on accuracy, topology simplicity, and token efficiency. Our results show that CoT remains effective for sequential tasks with GPT-4o-mini (89.5% on GSM8K), while NoT surpasses ToT on multi-hop reasoning (91.0% vs.\ 88.0% on HotpotQA with LLM-as-Judge). With 72B open-source models, NoT achieves the highest accuracy on GSM8K (91.5%), and Qwen2.5-72B achieves the best multi-hop QA result overall (91.7% on HotpotQA). Self-generated controller heuristics outperform fixed and random strategies on logical reasoning, with uncertainty-only weighting achieving 57.0% on ProofWriter. We also find that evaluation methodology significantly impacts method rankings: string-match underestimates all methods on open-ended QA, with the largest gap for NoT, a pattern consistent across all three models (14--18 percentage point gap on HotpotQA).

Abstract

Language Model Reasoning Evaluation

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