A Common Selection Principle for Crystallographic Class Structure and Phonon Band Degeneracy Version 2

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. Both halves have prior art: the lattice realization recovers the form spaces of Bravais groups from arithmetic crystallography (Opgenorth, Plesken & Schulz 1998; Michel 1995), and the phonon realization descends from Wigner (1930) through Maradudin & Vosko (1968); the contribution is the explicit cross-domain statement and the exact location of its limits. The lattice case at n = 3 carries an additional value-pinning property absent in the phonon case — every angle the partition constrains is pinned at a special value, while the phonon partition forces eigenvalue equalities without fixing a single frequency. We trace the asymmetry to the readout: entry readouts inherit forced zeros from Schur’s vanishing of maps between non-isomorphic irreducibles, spectral readouts inherit forced equalities without values. The result restates the structural analogy in Ness (2026a, §4) as a proven theorem: the components are Tier 1, the unification reading is Tier 2, with a path to Tier 1 via the categorical refinement posed as an open problem