Undecidable Spectral Gaps in Quasicrystals Link Computation and Topology — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Quantum many-body topology of quasicrystals reveals undecidable spectral gaps linked to quasicrystalline order, bridging computational limits and physical structure. | MATH: The spectral gap problem for quasicrystalline Hamiltonians is undecidable (reduction from the halting problem); key constants emerge from quasicrystalline diffraction patterns (e.g., scaling factors like 1.618, 0.618 for Penrose tilings). | CONNECTION: Quasicrystals exhibit 5-fold rotational symmetry (forbidden in periodic crystals), with inflation/deflation ratios exactly 1.618 (golden ratio) and 0.618; these ratios govern the self-similarity of the tiling and appear in the Fourier transform as Bragg peaks at positions involving φ. The undecidability result implies that the spectral gap—a topological invariant—cannot be algorithmically determined for such systems, linking computational undecidability to geometric harmony. | DEPTH: 9 — This finding directly ties the limits of computation (undecidability) t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Materials Chemistry
Field: Materials Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7171107785
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