Rudolf Peierls (1955): a 1D half-filled metal is always unstable to dimerization. Alternating bond lengths (t₁, t₂ = t ± δ) fold the Brillouin zone to [−π/2a, π/2a] and open a gap ΔE = 4δ at k=±π/2a. The electronic energy gain scales as δ² ln(1/δ) (logarithmic Kohn anomaly), while the elastic cost is Kδ². Since ln(1/δ)→∞ as δ→0, ANY δ lowers the total energy — spontaneous symmetry breaking. Example: polyacetylene (CH)ₙ shows alternating single/double bonds.