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

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-15
DOI
https://doi.org/10.5281/zenodo.22762285
Primary Topic
Quantum many-body systems
Type
preprint
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preprint

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

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
Quantum many-body systems
preprint

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

Andrew Stewart Caldin
preprint en

Abstract

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 \( H = \sum_i h_i \) on a 2D lattice (finite local Hilbert space), determining whether the gap \( \Delta = E_1 - E_0 \) (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 \( d \geq 2 \) (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

Zenodo (CERN European Organization for Nuclear Research)
Quantum many-body systems
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Spectral Gap Undecidability Dominates Quantum Many-Body Physics — E8 Intelligence Research — Andrew Stewart Caldin · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS