Fermion Chirality from Non-Bipartite Topology: Geometric Doubler Lifting on the FCC Lattice via Holographic U(1)/Z₂ Phase Projection

We construct and analyse the bond-direction Dirac operator on the Face-Centred Cubic (FCC) lattice using all 12 nearest-neighbour unit bond directions: D_SSM(k) = Σ_{j=1}^{12} (γ·n̂_j) exp(ik·n̂_j). The operator satisfies {γ₅, D_SSM} = 0 exactly at finite lattice spacing (exact chiral symmetry, proved algebraically).

Our main analytical result is the Irrational Doubler Theorem: every non-Γ zero of D_SSM has at least one irrational FCC fractional coordinate f_i = ±1/(2√2), placing it permanently outside any finite rational momentum grid. Two types of doubler zero exist: isolated Z₂ zeros (Type-1, all bond phases ∈ {+1,−1}) and flat-band U(1) zeros (Type-2, generically complex bond phases); both types share the irrational fractional coordinate property. This is proved analytically via the V-form structure and confirmed by a dense 32³ FCC BZ scan: among 32,768 sampled momenta, no non-Γ mode has E = 0.

This Z₂/U(1) distinction provides a geometric basis for the holographic projection of the SSM (Sparse-Simplex Matrix) framework: physical electrons carry U(1) electric charge and require genuinely complex (propagating) bond phases; staggered Z₂ configurations are static, carry no U(1) charge, and are identified as non-propagating codespace states rather than physical particles. After this projection, all zone-boundary high-symmetry modes (X, W, L, K) carry U(1) phases and are lifted to UV energies E ≈ 1–5/a, while a single massless Dirac mode with exact chiral symmetry persists at Γ.

The Nielsen-Ninomiya theorem is satisfied globally on the continuous torus: all non-Γ zeros reside at irrational FCC fractional coordinates and are inaccessible to any finite integer-L simulation grid. In the SSM physical sector, the unique zero is at Γ.