Undecidability of the Quantum Spectral Gap Limits Material Property Prediction — E8 Intelligence Research

FINDING: Undecidability in quantum spectral gaps proves that the existence of a spectral gap in many-body systems is not algorithmically decidable, implying fundamental limits to predicting material properties. | MATH: The spectral gap problem reduces to the halting problem; no algorithm can determine if a Hamiltonian's ground state energy is separated from the excited states by a finite gap. Key constants: spectral gap Δ = E₁ - E₀, where Δ > 0 indicates an insulator, Δ = 0 a conductor. | CONNECTION: The undecidability result is tied to the structure of Projected Entangled Pair States (PEPS), which encode lattice symmetries and tensor network geometries. The spectral gap's undecidability mirrors the non-computability of certain geometric invariants in crystallographic root systems (e.g., Coxeter groups). | DEPTH: 9 — This is a profound link between computation theory and quantum many-body physics, showing that geometric order (e.g., lattice gaps) can be algorithmically inaccessible, ec Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com