Quantum Spectral Gap Provably Undecidable, Linking Physics to Halting Problem — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
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 E0 and first excited state E1) is either Δ > 0 (gapped) or Δ = 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 Δ ∼ 1/poly(n) in the gapless case. No closed-form equation for Δ 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
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BibTeX
@misc{caldin2026quantum,
title = {{Quantum Spectral Gap Provably Undecidable, Linking Physics to Halting Problem — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23658246},
url = {https://zenodo.org/records/23658246},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum Spectral Gap Provably Undecidable, Linking Physics to Halting Problem — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23658246
UR - https://zenodo.org/records/23658246
AB - 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 Δ > 0 (gapped) or Δ = 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 Δ ∼ 1/poly(n) in the gapless case. No closed-form equation for Δ 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
ER -