Decidable Spectral Gap for Frustration-Free Spin-1/2 Chains — E8 Intelligence Research

FINDING: Frustration-free spin-1/2 chains admit a complete classification into gapped and gapless phases, with the spectral gap question being decidable for this restricted class despite general undecidability results for translationally invariant Hamiltonians. | MATH: Hamiltonian H = Σᵢ hᵢ with hᵢ ≥ 0 (frustration-free condition: ground state |ψ⟩ satisfies hᵢ|ψ⟩ = 0 ∀i). Spectral gap Δ = E₁ − E₀. For spin-1/2 chains, the relevant algebra is SU(2) with root system A₁ (rank 1, single simple root α, Cartan matrix [2]). The MPS (matrix product state) representation classifies phases via injectivity of the transfer matrix. Key result: gapped iff the MPS is injective (finite correlation length ξ < ∞); gapless iff the MPS is non-injective (algebraic decay, ξ → ∞). | CONNECTION: Root lattice A₁ = 1D chain of spins; the Dynkin diagram of A₁ is a single node (trivial). The classification of gapped/gapless phases maps to whether the ground state lives in a trivial or non-trivial superselection s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com