Spectral Gap Undecidability Dominates Quantum Many-Body Physics — E8 Intelligence Research

FINDING: Undecidability of the spectral gap in quantum many-body systems (Cubitt et al.) is the dominant mathematical result; other findings are peripheral (hexaquark SU(3) multiplet, graph recoloring, spectral line precision, traffic models, leaf counting). | MATH: The spectral gap problem: given a translationally-invariant Hamiltonian \( H = ∑_i h_i \) on a 2D lattice (finite local Hilbert space), determining whether the gap \( Δ = E_1 - E_0 \) (difference between ground state and first excited energy) is zero or positive is undecidable. This maps the halting problem onto the gap via a construction where the gap closes iff a Turing machine halts. Key constants: no specific numeric constants, but the proof uses a lattice of qudits with local dimension \( d ≥ 2 \) (often \( d=2 \) suffices for the 2D case), and relies on the existence of a "gadget" Hamiltonian whose spectrum encodes the machine's computation. | CONNECTION: The undecidability proof constructs a Hamiltonian wh Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com