A Bijection Between Refinement-Comparable Partition Pairs and Pre-Bifurcation Crystal Classes at n = 3

We establish a bijection between the six refinement-comparable ordered pairs of integer partitions of 3 and the six pre-bifurcation crystal classes of three-dimensional Euclidean space. The map sends each pair (μ, α) with μ ≤ α — where μ is the metric partition on the lattice axes and α is the angular partition on the interaxial angles — to the unique crystal class whose conventional unit cell realizes those partitions under the crystallographic restriction theorem. The load-bearing result is the Selectivity Lemma: for each refinement-comparable pair at n = 3, the partition data plus the crystallographic restriction (allowed rotational orders {1, 2, 3, 4, 6}) plus the geometry of three-dimensional Euclidean space jointly select a unique angular structure up to setting choice. The bijection extends to the seven crystal systems by composition with the orthogonality bifurcation that splits the (3) | (3) class. We prove that the bijection preserves the natural poset structure: the pointwise refinement order on partition pairs corresponds to the Bravais-class containment order on crystal classes, with cover relations matching one-to-one. We document why the selectivity property fails at n ≥ 4 and the bijection therefore does not extend, citing the established four-dimensional enumeration. The result promotes the explicit pair-by-pair correspondence in Ness (2026a, Section 2.4) from structurally argued to mathematically proven.I'm