Unsolved Problems and Tropical Mathematics: From Collatz to Poincaré — E8 Intelligence Research

FINDING: Compilation of unsolved competition-level problems, including Collatz Conjecture, Poincaré Conjecture (solved by Perelman), and tropical/idempotent mathematics linking to mathematical physics. | MATH: Collatz Conjecture: f(n) = n/2 if n even, 3n+1 if n odd; Poincaré Conjecture: every simply connected closed 3-manifold is homeomorphic to S³; tropical mathematics: (a ⊕ b = max(a,b), a ⊗ b = a+b) with idempotent semiring structure. | CONNECTION: No direct geometric ratios (0.382, 0.618, etc.) or base-60 found. However, tropical geometry connects to toric varieties and lattice polytopes, which relate to root systems and crystallographic symmetries (e.g., Aₙ, Bₙ, Cₙ, Dₙ lattices). The Collatz dynamics exhibit scaling patterns that may hint at fractal self-similarity, but no proven harmonic ratio. | DEPTH: 6 — The Collatz problem is deceptively simple yet reveals deep connections to dynamical systems and number theory; tropical mathematics provides a framework for piecewise-linear s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com