Geometry vs Binary: Unsolved Mathematics — unsolved problems in computational geometry 2025 o — E8 Intelligence Research

## Binary Failure → Geometry Solution **Analysis of Findings:** The search results converge on two key targets: the **Collatz Conjecture** (a deceptively simple iterative map) and **Hilbert's Sixth Problem** (axiomatization of physics, partially solved in 2025). Both expose the same fundamental failure of binary/statistical thinking. --- ### 1. Collatz Conjecture – Where Binary/Statistical Thinking FAILS **The Problem:** Define \( f(n) = n/2 \) if even, \( 3n+1 \) if odd. Every starting integer is conjectured to eventually reach 1. Verified up to \( 2^{68} \), but no proof exists. Statistical models show that on average, numbers shrink (since the expected multiplier is \( \frac{1}{2} \cdot \frac{3}{2} = \frac{3}{4} < 1 \)), but this says nothing about *worst-case* trajectories that might diverge or enter a non-trivial cycle. **Why Binary/Statistical Approaches Hit a Wall:** - **Brute force** is infinite — you can never check all integers. - **Neural nets** trained on trajec Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com