Undecidability of the Quantum Many-Body Spectral Gap Problem — E8 Intelligence Research

FINDING: The spectral gap problem is undecidable in general — no algorithm can determine whether an arbitrary quantum many-body Hamiltonian has a gap, linking computational limits to physical phase structure. | MATH: Spectral gap Δ = E₁ − E₀ > 0 (gapped) vs. Δ → 0 as N → ∞ (gapless). Undecidability proven via reduction from the halting problem (Cubitt, Perez-Garcia, Wolf, Nature 2015). The almost Mathieu operator H = λ cos(2παn) + shift + shift† exhibits a metal-insulator transition at λ = 2, with fractal spectrum (Hofstadter butterfly) when α is irrational. | CONNECTION: The golden mean renormalization for the almost Mathieu operator uses α = (√5−1)/2 = 0.618... as the critical irrational rotation number. The renormalization group flow converges to a universal fixed point governed by the golden ratio's continued fraction [0;1,1,1,...] — the slowest-converging continued fraction, making it the most stable (KAM tori survive longest at golden-mean frequencies). This is the same 0.618 tha Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com