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

• URL: http://arxiv.org/abs/2510.11278v2
• Date: Thu, 16 Oct 2025 09:21:06 GMT
• Title: ENIGMA: The Geometry of Reasoning and Alignment in Large-Language Models
• Authors: Gareth Seneque, Lap-Hang Ho, Nafise Erfanian Saeedi, Jeffrey Molendijk, Ariel Kuperman, Tim Elson,
• Abstract summary: We present Entropic Mutual-Information Geometry Large-Language Model Alignment (ENIGMA)<n>It improves reasoning, alignment and robustness by treating an organisation's policies/principles as directions to move on a model's information manifold.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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

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