The Superfluid Necessity Theorem: Why Emergent Lorentz Isotropy Requires a Condensate, Not a Crystal Solid

We prove that the acoustic isotropy underlying emergent Lorentz invariance in the Body-Centred Tetragonal (BCT) Superfluid Lattice Model is a consequence of the superfluid character of the vacuum, and would fail for any elastic-solid realisation of the same lattice. Two dynamical models are compared on the identical BCT lattice at the unique axial ratio c/a=√2: a scalar phase field (superfluid phonon) and a central-force elastic solid (vector phonons). For the scalar field, the long-wavelength sound speed is exactly isotropic in every direction, because the second-moment tensor of the twelve nearest-neighbour vectors satisfies Σδᵢδⱼ = 4a²δᵢⱼ identically at c/a=√2, and cubic point symmetry forces every second-rank invariant tensor to be a multiple of the identity. For the elastic solid, terminal verification shows directional sound-speed variations of order 15% even at c/a=√2, because cubic symmetry constrains second-rank tensors but not the fourth-rank elastic tensor. An elastic-solid vacuum on this lattice is therefore excluded by observed Lorentz invariance, while the superfluid vacuum passes exactly: superfluidity is not a modelling convenience of the BCT programme but a physical necessity. Residual anisotropy enters only at O(k²) in the phase velocity, yielding a Planck-suppressed Lorentz-violation prediction class. Status: PROVEN; terminal-verified 16 July 2026. Companion technical appendix: HO6.