Geometry vs Binary: AI/Binary Failures — neural network fails at geometric reasoning tasks — E8 Intelligence Research

Abstract

## Binary Failure → Geometry Solution **ANALYSIS OF FINDINGS** --- ### 1. **ZelusBench: A Deterministic Geometric-Reasoning Benchmark for LLM Attention** **The Failure:** LLMs (transformers) are trained on token statistics, not spatial intuition. When asked to reason about 3D rotations, reflections, or occlusion, they fail catastrophically — not because they lack "knowledge," but because their internal representations are **not equivariant** to the symmetry group of 3D space. A neural network sees a chair rotated 30° as a *different* object unless explicitly data-augmented, and even then, it memorizes rotations rather than understanding them. **Geometric Success:** A root-lattice framework encodes **rotational symmetry as an invariant** — the lattice's automorphism group (Weyl group) *is* the set of allowed transformations. Instead of learning "chair at angle θ," the system computes the orbit of the object under the lattice's symmetry group. The 3D scene is decomposed into **root v Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com