From Tetrahedral Geometry to Binary Tetrahedral Holonomy: A Geometric Realization of the Central Element −I on Diamond-Lattice Chair Hexagons

This paper constructs an explicit geometric realization of the central element −I of the binary tetrahedral group 2T directly from the diamond lattice. Starting from the canonical orientation structure of the two bipartite sublattices (det(M_A) = +1/16, det(M_B) = −1/16), the geometry induces a canonical oriented face cycle, which is identified with a generator of the local C₃ symmetry. The corresponding SO(3) rotation is uniquely fixed to 120°, whose SU(2) lift is U(d) = exp(−iπ/3 · d·σ). Numerical enumeration confirms that all 24 physical chair hexagons of the diamond lattice yield holonomy W = −I. The work establishes that a spinorial topological ℤ₂ structure emerges from tetrahedral geometry without assuming SU(2) a priori. The order-parameter space SO(3)/A₄ is assumed, not derived. No claims are made regarding fermions, particle statistics, or Standard Model gauge structure.