BCT Letter 1: Emergent Lorentz Invariance from a Planck-Scale BCT Lattice — k4 Anisotropy Eighteen Orders Below Fermi-LAT Bounds

Any theory proposing discrete structure at the Planck scale must confront stringent observational bounds on Lorentz invariance violation (LIV). We compute the leading LIV correction for a bodycentred tetragonal (BCT) lattice with axial ratio c/a = √ 2 and Planck-scale spacing. The lattice's centrosymmetry (inherited from D4 root lattice parentage) forbids all odd-order dispersion corrections, so the first LIV term enters at O(k 4 ). We derive the full rank-4 dispersion tensor by explicit summation over 12 nearest neighbours, decompose it into isotropic and anisotropic parts, and extract the Lorentz-violating coefficient ξ (LV) 2 = 1/480 ≈ 2.08×10−3 . This is 4×1018 times below the Fermi-LAT bound from GRB 090510, and ∼ 1019 below the LHAASO bound from GRB 221009A. The anisotropy arises entirely from the 4D → 3D projection of the D4 root lattice; the parent 4D lattice is fully isotropic. The BCT lattice thus provides a concrete, calculable example of how a genuine Planck-scale discrete structure can be consistent with all existing Lorentz invariance tests by many orders of magnitude. The specific angular pattern of the k 4 anisotropy constitutes a falsifiable signature distinguishing this lattice from competing discrete-spacetime proposals.