Symmetry-Constrained Spectra: Bloch Bands and Non-Computable Thermodynamic Limits — E8 Intelligence Research

FINDING: Translational invariance constrains Hamiltonian spectral structure via Bloch theorem and crystallographic group theory, linking discrete symmetry to conserved quantum numbers and non-computable spectral classifications in the thermodynamic limit. | MATH: Bloch theorem: ψ_k(r) = eik·r u_k(r), with k in first Brillouin zone; translational operator T_a commutes with H: [T_a, H] = 0 → simultaneous eigenstates |n,k⟩ with eigenvalues E_n(k). For 2D crystallographic groups: 17 wallpaper groups, each with specific glide/rotation constraints; rotationally invariant H: [J, H] = 0 → eigenstates labeled by total angular momentum j, degeneracy 2j+1. | CONNECTION: Crystallographic point groups are finite subgroups of O(2) — orders {1,2,3,4,6} only (crystallographic restriction theorem). This yields allowed rotation angles: 60° (hexagonal, ratio √3), 90° (tetragonal), 120° (trigonal). The golden ratio φ = 1.618 is *forbidden* in 2D periodic crystals (5-fold symmetry impossible) — but appe Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com