Spectral Gap of Quantum Many-Body Hamiltonians Is Undecidable — E8 Intelligence Research

FINDING: The spectral gap of a quantum many-body Hamiltonian is undecidable — no algorithm can determine whether a given local Hamiltonian has a gap or is gapless. | MATH: Cubitt–Perez-Garcia–Wolf (Nature 528, 207–211, 2015): constructs a family of frustration-free, translation-invariant, nearest-neighbor Hamiltonians H(φ) on a 2D lattice such that the spectral gap Δ(H(φ)) encodes the halting problem. Formally: the ground-state energy density and gap are non-computable; the "gap function" Δ: ℕ→ℝ is not Turing-computable. The construction reduces from Wang tilings / aperiodic tilings (Berger's theorem) to a quantum Hamiltonian whose ground state is a tiling configuration. | CONNECTION: The construction relies on aperiodic tilings (Wang tiles) which are governed by local matching rules — a discrete analog of crystallographic symmetry breaking. The undecidability arises from the absence of a periodic ground state, i.e., the breakdown of translational symmetry at the quantum level. No dire Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com