A Common Selection Principle for Crystallographic Class Structure and Phonon Band Degeneracy

Two selectivity statements appear analogous: in three-dimensional crystallography, the partition-refinement structure on lattice parameters plus the crystallographic restriction theorem uniquely selects the realized geometry of each crystal class (Ness 2026a); in condensed matter physics, the symmetry of a crystal lattice selects which phonon modes are degenerate at high-symmetry wavevectors (Maradudin & Vosko 1968). This paper proves that both selectivities are instances of a single mathematical mechanism: the isotypic decomposition of a finite-dimensional vector space under a finite group action, formalized via Maschke’s theorem and Schur’s lemma. We define the forced-equality partition Π(G, V) of a parameter index set induced by a G-action and prove a Bridge Theorem stating that both the lattice (μ, α) partition at n = 3 and the phonon mode-degeneracy partition at high-symmetry k-points are instances of Π(G, V) for appropriate (G, V). Worked examples in both domains — the cubic, tetragonal, and monoclinic lattice systems on the metric-tensor side, and the diamond structure at Γ on the phonon side — verify that the bridge is non-vacuous. The lattice case at n = 3 carries an additional value-determination property absent in the phonon case; we identify the source of this asymmetry and pose it as an open problem. The result promotes the structural analogy in Ness (2026a, §4) from a Tier 2 architecture-match to a stated unification theorem at Tier 2 with paths to Tier 1 via categorical refinement.