A Hermitian 12-Direction Topological Dirac Operator: Number-Theoretic Doubler Sequestering, Preserved Chiral Symmetry, and a Single Massless Mode

The bond-direction Dirac operator on the face-centered cubic (FCC) lattice, D_FCC(k) = Σ_{j=1}^{12} (γ·n̂_j) e^{i k·n̂_j} summed over the twelve nearest-neighbor unit bonds, is maximally symmetric: its structure tensor is S^{μν} = 4δ^{μν}, the long-wavelength limit is an isotropic Dirac cone, and it anticommutes with γ₅ at finite lattice spacing. It is also doubled. We give the doubler census in the operator's own Brillouin zone and show that every doubler sits at a rational momentum that ordinary even-sided lattices sample exactly, including a one-dimensional set of nodal lines. We then construct a mass term that removes them. Second-order hopping around the non-bipartite triangular faces of the cuboctahedral coordination shell generates, through the 120° closure angle and the cancellation of opposite face orientations, a purely scalar correction ∝ 1 (the identity). Summed over the shell it is the isotropic Wilson term W(k) = r Σ_j [1 − cos(k·n̂_j)]. Because the operator is anti-Hermitian, this scalar adds in quadrature, |eig|² = W² + |V|², and lifts every doubler, every nodal line included, to a gap that is uniform in [12r, 16r] and independent of lattice size, while leaving the cone at Γ untouched. The cost is explicit chiral-symmetry breaking, as the Nielsen–Ninomiya theorem requires. The same scalar-mass construction transfers to the D₄ root lattice in four dimensions (24 bonds, S^{μν} = 6δ^{μν}, 96 triangular 2-faces). We note a structural overlap with the triangular-face confinement channels proposed in companion FCC-vacuum papers. All numerical claims are reproducible from short scripts.