# The Geometric Price of Discrete Logic: Context-driven Manifold Dynamics of Number Representations

arXiv: https://arxiv.org/abs/2603.23577 · HTML: https://mlsanity.com/paper/2603.23577/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 11 of its 13 references that are in the map.
- First posted on arXiv 24 Mar 2026.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)

## Builds on (most cited first, 11 of 11)

- [A Survey on Hallucination in Large Language Models: Principles, Taxonomy, Challenges, and Open Questions](https://mlsanity.com/paper/2311.05232/) (2023)
- [Root Mean Square Layer Normalization](https://mlsanity.com/paper/1910.07467/) (2019)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [Projecting Assumptions: The Duality Between Sparse Autoencoders and Concept Geometry](https://mlsanity.com/paper/2503.01822/) (2025)
- [The Geometry of Reasoning: Flowing Logics in Representation Space](https://mlsanity.com/paper/2510.09782/) (2025)
- [Unifying Attention Heads and Task Vectors via Hidden State Geometry in In-Context Learning](https://mlsanity.com/paper/2505.18752/) (2025)
- [Geometry of Decision Making in Language Models](https://mlsanity.com/paper/2511.20315/) (2025)
- [LSSF: Safety Alignment for Large Language Models through Low-Rank Safety Subspace Fusion](https://mlsanity.com/paper/2602.00038/) (2026)
- [Low-Dimensional Execution Manifolds in Transformer Learning Dynamics: Evidence from Modular Arithmetic Tasks](https://mlsanity.com/paper/2602.10496/) (2026)
- [The Representational Geometry of Number](https://mlsanity.com/paper/2602.06843/) (2026)
- [Human-like Social Compliance in Large Language Models: Unifying Sycophancy and Conformity through Signal Competition Dynamics](https://mlsanity.com/paper/2601.11563/) (2025)

## Built on by (most cited first, 1 of 1)

- [Authority Inversion in LLM-Mediated Ubiquitous Systems: When Models Trust Users Over Sensors](https://mlsanity.com/paper/2605.23938/) (2026)

## Abstract (arXiv)

Large language models (LLMs) generalize smoothly across continuous semantic spaces, yet strict logical reasoning demands the formation of discrete decision boundaries. Prevailing theories relying on linear isometric projections fail to resolve this fundamental tension. In this work, we argue that task context operates as a non-isometric dynamical operator that enforces a necessary "topological distortion." By applying Gram-Schmidt decomposition to residual-stream activations , we reveal a dual-modulation mechanism driving this process: a class-agnostic topological preservation that anchors global structure to prevent semantic collapse, and a specific algebraic divergence that directionally tears apart cross-class concepts to forge logical boundaries. We validate this geometric evolution across a gradient of tasks, from simple mapping to complex primality testing. Crucially, targeted specific vector ablation establishes a strict causal binding between this topology and model function: algebraically erasing the divergence component collapses parity classification accuracy from 100% to chance levels (38.57%). Furthermore, we uncover a three-phase layer-wise geometric dynamic and demonstrate that under social pressure prompts, models fail to generate sufficient divergence. This results in a "manifold entanglement" that geometrically explains sycophancy and hallucination. Ultimately, our findings revise the linear-isometric presumption, demonstrating that the emergence of discrete logic in LLMs is purchased at an irreducible cost of topological deformation.