Causally Confined Euclidean Saddles in Spin-Foam Quantum Gravity

We investigate the semiclassical structure of spin-foam transition amplitudes for boundary data that do not admit a real Lorentzian Regge geometry. For a fixed triangulation with a single internal vertex, we show that when boundary tetrahedra carry mutually incompatible causal orientations, the closure equations have no real solution and the path integral is dominated by a complex Euclidean saddle of the Regge action. The vertex amplitude then acquires a non-oscillatory factor exp(−Sₑ/ħ), where Sₑ is the Euclidean action at the complex saddle. We introduce a causal-obstruction criterion based on the convexity of the future timelike cone in ℝ³,¹ and classify boundary data into three types according to the existence and nature of the saddle-point solutions. For the causally obstructed type, Sₑ scales linearly with the spin parameter j: Sₑ = ħ·j·C(α)/(8πG), where C(α) is a dimensionless geometric constant computed without free parameters. Non-degeneracy of the Hessian is verified after gauge fixing, and stability of the suppression under triangulation refinement is established.