Structural Theorem BCT Letter 18: The BCT Uniqueness Theorem — Why D4 at c/a = sqrt(2) Is the Only Vacuum Lattice Consistent with Physics

The Body-Centred Tetragonal (BCT) Superfluid Lattice Model derives 109 Standard Model observables from a specific vacuum geometry: a BCT lattice with axial ratio c/a = √ 2. We prove the BCT Uniqueness Theorem: the BCT lattice at c/a = √ 2 is the only crystalline vacuum state satisfying five simultaneous physical requirements: (1) Lorentz invariance (isotropic phonon propagation); (2) three fermion generations (Z3 outer automorphism); (3) maximum packing density in the parent dimension; (4) modular invariance (self-dual lattice); and (5) the correct Standard Model gauge group. No other lattice in any dimension satisfies all five. The axial ratio c/a = √ 2 is uniquely determined by requirement (1). The vacuum has no choice: if it crystallises, it must crystallise into the BCT lattice, and the 109 predictions follow necessarily. Of these, 33 are sub-0.1% — including all three fundamental SM scales (me, ΛQCD, v) now derived from geometry alone (Letters 31–33).