A previous paper proposed a matter-first route to loop quantum gravity: interaction vertices define the graph, fermion-line segments carry local SL(2,C) frames, and geometry is reconstructedfrom the resulting spacetime-algebra comparators. Its central open gap was the relation betweenthose matter-built bivectors and the simplicity-constrained boundary data of spin-foam dynamics. This paper closes that kinematical gap and adds a finite-cutoff vertex algorithm. First, we express the EPRL linear simplicity constraint in the invariant plane of the bivector classification B^2 = s + pI: up to the standard sign and normalisation conventions for SL(2, C) Casimirs, the EPRL embedding Y_γ : j → (ρ, k) = (γj, j) selects a ray p/s = 2γ/(γ^2 − 1). Second, we sharpen the closure bridge: the area-vector closure condition at a tetrahedron is equivalent to the Minkowski polygon inequality, and this is exactly the condition that the four-valent intertwiner space be non-empty. Thus classical closure and quantum admissibility are two presentations of the same Gauss constraint. Third, we construct the Lorentzian vertex from the same data: the boost sector is represented by principal-series generators certied by the so(1, 3) algebra and Casimirs, and the node booster B^γ_4 is evaluated as a radial boost integral. Finally, by absorbing each booster and intertwiner into a rank-four node tensor, the finite-cutoff EPRL four-simplex amplitude becomes a tensor-network contraction on K_5 rather than an explicit state sum. The result is not a continuum limit; it is a reproducible bridge between the matter determinant,bivector simplicity, and a computational Lorentzian EPRL vertex.