The problem of determining the existence of a spectral gap in a lattice quantum spin system was previously shown to be undecidable for one [J. Bausch et al., "Undecidability of the spectral gap in one dimension", Physical Review X 10 (2020)] or more dimensions [T. S. Cubitt et al., "Undecidability of the spectral gap", Nature 528 (2015) and Forum of Mathematics Pi, 10 (2022)]. In this work, we focus on the 1-dimensional result, showing that the constructed family with undecidable behavior is extremely sensitive to perturbations. In particular, for any
ε>0, there exists a 1-local, rank 1, perturbation with norm
O(ε), such that the spectral gap problem for the family in [J. Bausch et al., "Undecidability of the spectral gap in one dimension", Physical Review X 10 (2020)] now becomes decidable.
FINDING: Undecidability of the spectral gap in quantum many-body systems is proven via reduction from the undecidable tiling problem, linking quantum physics to Turing's halting problem. MATH: Reduction: Halting problem ≤ Turing machine ≤ tiling problem ≤ spectral gap problem. Key constants: none directly; relies on logical undecidability, not numerical constants. CONNECTION: Tiling problem uses Wang tiles, which are directly related to quasicrystalline tilings and Penrose tilings (golden ratio φ = 1.618, 1/φ = 0.618). The undecidability proof for the spectral gap uses a Hamiltonian whose ground state encodes a tiling; the spectral gap is zero iff the tiling is aperiodic. Aperiodic tilings are intimately linked to the golden ratio and fivefold symmetry (crystallographically forbidden). DEPTH: 9 — This result shows that fundamental physical properties (energy gaps) can be logically undecidable, implying limits to what can be computed or known about quantum systems. It bridges comp Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com