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Quantum Spectral Gap Provably Undecidable, Linking Physics to Halting Problem — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 26, 2026• 0 citations
Publisher

Abstract

FINDING: The spectral gap problem — determining whether a quantum Hamiltonian's energy spectrum has a gap above the ground state — is provably undecidable in general, linking quantum many-body physics to the halting problem and Gödelian limits. | MATH: The key construction (Cubitt, Perez-Garcia, Wolf, 2015) reduces the halting problem to a family of 2D spin-lattice Hamiltonians \( H(n) \) on a fixed lattice, where the spectral gap (difference between ground state energy \( E_0 \) and first excited state \( E_1 \)) is either \( \Delta > 0 \) (gapped) or \( \Delta = 0 \) (gapless), depending on whether a Turing machine halts. The Hamiltonian is built from translationally invariant local terms with frustration-free ground states; the gap closes as \( \Delta \sim 1/\text{poly}(n) \) in the gapless case. No closed-form equation for \( \Delta \) exists — it is algorithmically uncomputable. | CONNECTION: The construction uses a lattice structure (square lattice, 2D) with local interaction ter Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.23658245

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