FCC Uniqueness Among All Bravais Lattices and the Continuum Field Theory Limit of the SU(2) Neighbourhood Operators

This paper resolves two open problems stated in Papers 4 and 1 of the Field of Resonance (FoR) programme. OP10 (FCC uniqueness): all 14 Bravais lattices in three dimensions are classified by the Clifford grade of their nearest-neighbour directions in C₃,₀. Theorem 3.1 proves that the face-centred cubic (FCC) lattice is the unique Bravais lattice satisfying simultaneously: (a) nearest-neighbour directions lie in grade-2 bivector planes of C₃,₀; (b) all nearest neighbours are equidistant from the origin; and (c) the close-packing condition (neighbour-to-neighbour distance equals origin-to-neighbour distance). Every other Bravais lattice fails at least one condition — simple cubic fails condition (a), BCC fails (a), face-centred orthorhombic fails (b), rhombohedral fails (c). Corollary 3.2 identifies the three cubic lattices with the three pure Clifford grades: SC↔grade-1, BCC↔grade-3, FCC↔grade-2. OP11 (continuum limit): the six FCC neighbourhood operators introduced in Paper 4 are shown to converge to the Clifford-algebra Dirac derivative operator in the limit of vanishing lattice spacing a → 0, with O(a²) corrections verified numerically to machine precision across three decades of lattice spacing. This extends the one-dimensional Bohm-Davies-Hiley (1982) result to the full three-dimensional FCC lattice. The symmetric difference [Nᵢⱼ+ − Nᵢⱼ−]/(a√2) converges to (∂_i + ∂_j), and the sum over all six operators recovers the Clifford Dirac derivative D = e₁∂_x + e₂∂_y + e₃∂_z. The continuum limit of the FCC lattice action yields the standard kinetic term of the SU(2) sigma model — a known, mathematically consistent quantum field theory — confirming the FoR Lagrangian has a well-defined field-theoretic interpretation. The speed ratio c_L/c_T = √2 is shown to be a topological invariant of the cuboctahedral coordination geometry, arising from cos²(π/4) = ½ and preserved exactly in the continuum limit.