From Intertwiners to the Oriented Tetrahedron: Algebraic Origin of the Factor K_node = 2 within Tetrahedral Emergent Gravity (TEG) .

This technical report provides a self-contained, rigorous, and machine-readable account of the algebraic steps leading from the SU(2) invariant singlet subspace of a 4-valent spin-network node to the structural factors utilized in Tetrahedral Emergent Gravity (TEG vH3.2). Specializing the general intertwiner dimension formula via Clebsch–Gordan multiplicities to the fundamental spin j_i = 1/2 regime, we explicitly construct the orthonormal basis {|ι₀⟩, |ι₁⟩} using Levi-Civita tensor contractions. Through the Minimal Simplex Principle (MSP) and a quaternionic projection of the 4D 5-cell from S³ to R³, these states are mapped 1:1 onto the two discrete spatial chiralities of the regular tetrahedron (z_fund = 4). We explicitly formalize the boundary between rigorous representation theory and the physical hypotheses required for macrostructure scaling, documenting the causal rigidity constraints of the Lorentzian EPRL vertex amplitude and the Bethe-lattice return probability heuristic that yields the effective branching factor b_eff = 8 and the spectral dimension d_s = ln 8.