# ENIGMA: The Geometry of Reasoning and Alignment in Large-Language Models

arXiv: https://arxiv.org/abs/2510.11278 · HTML: https://mlsanity.com/paper/2510.11278/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 38 of its 60 references that are in the map.
- First posted on arXiv 13 Oct 2025.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)
- LLM alignment / RLHF / preference optimisation / [LLM preference optimization methods](https://mlsanity.com/alignment/llm-preference-optimization-methods/)

## Builds on (most cited first, 30 of 38)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Constitutional AI: Harmlessness from AI Feedback](https://mlsanity.com/paper/2212.08073/) (2022)
- [TruthfulQA: Measuring How Models Mimic Human Falsehoods](https://mlsanity.com/paper/2109.07958/) (2021)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [LIMA: Less Is More for Alignment](https://mlsanity.com/paper/2305.11206/) (2023)
- [GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks](https://mlsanity.com/paper/1711.02257/) (2017)
- [Training Large Language Models to Reason in a Continuous Latent Space](https://mlsanity.com/paper/2412.06769/) (2024)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [ORPO: Monolithic Preference Optimization without Reference Model](https://mlsanity.com/paper/2403.07691/) (2024)
- [On Variational Bounds of Mutual Information](https://mlsanity.com/paper/1905.06922/) (2019)
- [Beyond the 80/20 Rule: High-Entropy Minority Tokens Drive Effective Reinforcement Learning for LLM Reasoning](https://mlsanity.com/paper/2506.01939/) (2025)
- [The Platonic Representation Hypothesis](https://mlsanity.com/paper/2405.07987/) (2024)
- [Orca: Progressive Learning from Complex Explanation Traces of GPT-4](https://mlsanity.com/paper/2306.02707/) (2023)
- [Chain of Thought Monitorability: A New and Fragile Opportunity for AI Safety](https://mlsanity.com/paper/2507.11473/) (2025)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (2018)
- [Lessons from the Trenches on Reproducible Evaluation of Language Models](https://mlsanity.com/paper/2405.14782/) (2024)
- [Harnessing the Universal Geometry of Embeddings](https://mlsanity.com/paper/2505.12540/) (2025)
- [Distributional Preference Alignment of LLMs via Optimal Transport](https://mlsanity.com/paper/2406.05882/) (2024)
- [Natural gradient via optimal transport](https://mlsanity.com/paper/1803.07033/) (2018)
- [Self-Supervised Alignment with Mutual Information: Learning to Follow Principles without Preference Labels](https://mlsanity.com/paper/2404.14313/) (2024)
- [It's a (Blind) Match! Towards Vision-Language Correspondence without Parallel Data](https://mlsanity.com/paper/2503.24129/) (2025)
- [Kernelized Wasserstein Natural Gradient](https://mlsanity.com/paper/1910.09652/) (2019)
- [Representation Potentials of Foundation Models for Multimodal Alignment: A Survey](https://mlsanity.com/paper/2510.05184/) (2025)

## Built on by (most cited first, 2 of 2)

- [ATLAS: Constitution-Conditioned Latent Geometry and Redistribution Across Language Models and Neural Perturbation Data](https://mlsanity.com/paper/2604.17663/) (2026)
- [Order Is Not Control: Driven-Dissipative Response Laws Across Artificial and Biological Systems](https://mlsanity.com/paper/2606.12923/) (2026)

## Abstract (arXiv)

We present Entropic Mutual-Information Geometry Large-Language Model Alignment (ENIGMA), a novel approach to Large-Language Model (LLM) training that jointly improves reasoning, alignment and robustness by treating an organisation's policies/principles as directions to move on a model's information manifold. Our single-loop trainer combines Group-Relative Policy Optimisation (GRPO), an on-policy, critic-free RL method with Chain-of-Thought (CoT)-format only rewards; a Self-Supervised Alignment with Mutual Information (SAMI)-style symmetric InfoNCE auxiliary; and an entropic Sinkhorn optimal-transport regulariser on hidden-state distributions to bound geometry drift. We also introduce infoNCE metrics that specialise to a standard MI lower bound under matched negatives to measure how strongly a model's CoT encodes these policies. These metrics include a Sufficiency Index (SI) that enables the selection and creation of principles that maximise downstream performance prior to training. In our experiments using small (1B) LLMs, high-SI principles predict steadier training dynamics and improved benchmark performance over GRPO ablations. Our information-geometry analysis of trained models validates desirable structural change in the manifold. These results support our hypothesis that reasoning, alignment, and robustness are projections of a single information-geometric objective, and that models trained using ENIGMA demonstrate principled reasoning without the use of a reward model, offering a path to trusted capability