Discriminant Control of Saddle Selection and Bifurcation in Lorentzian Minisuperspace Quantum Cosmology

This preprint presents an analytic study of the saddle-point structure of the Lorentzian no-boundary path integral in minisuperspace quantum cosmology. We show that a single discriminant, D = 3k(3k − Λq₁), linear in the cosmological constant Λ, governs the saddle selection, the Picard–Lefschetz thimble topology, and the saddle bifurcation — all as direct algebraic consequences of the no-boundary condition (q₀ = 0). The key results are: (i) an analytic derivation of Re(Δτ) = 0 from assumptions (A0)–(A5) without invoking the perturbative sector; (ii) an analytic proof that the effective intersection number n(Nₛ) = 1 holds over the entire parameter range, including the D < 0 regime where the contributing saddle becomes complex (Proposition 2); and (iii) a description of the saddle bifurcation at the critical value Λc = 3k/q₁, with leading-order scaling Φ ∝ (Λ − Λc)1/2. These results complement the framework of Feldbrugge, Lehners, and Turok at the algebraic level, showing that the saddle behavior they observed pointwise through numerical path-tracing is governed uniformly across the full parameter range by the sign of D.