Undecidability of Spectral Gaps via Turing Machine Encoding in Tilings — E8 Intelligence Research

FINDING: The halting problem is undecidable for Turing machines, proven by diagonalization; recent work encodes this undecidability into spectral gaps of translation-invariant Hamiltonians via periodic tilings, linking computation to quantum many-body physics. | MATH: Halting problem undecidability (Turing 1936); spectral gap problem for 2D translation-invariant Hamiltonians is undecidable (Cubitt, Perez-Garcia, Wolf 2015); encoding uses aperiodic tilings (e.g., Wang tiles) to simulate Turing machine tape and state transitions, with Hamiltonian energy gap encoding halting condition. Key constants: none directly, but tiling rules involve local constraints with no simple ratio. | CONNECTION: The tiling construction inherently uses lattice structures (square lattice) and local constraint patterns reminiscent of crystallographic symmetry breaking. The aperiodic tilings that encode computation often exhibit self-similarity and scaling ratios (e.g., Penrose tilings have golden ratio φ = 1.61 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com