Morse–Bott Structure of the Global Lorentzian 1-to-5 EPRL Critical Manifold: Transverse Nondegeneracy and Square-Root Null-Boundary Density

The Lorentzian EPRL 1-to-5 amplitude possesses a continuous four-dimensional family of flat geometric critical points associated with displacements of the interior vertex at fixed boundary data. Building on the identification of this critical family in previous numerical studies, this work investigates its transverse Hessian structure and the behavior of the complete leading Morse–Bott density near a null-tetrahedron boundary. Differentiation of the full critical equations yields four exact tangent zero modes. After removing 60 microscopic gauge directions, the complete 280-dimensional gauge-quotiented Hessian is numerically found to have precisely these four tangent zero directions at the tested regular configurations. The resulting 276-dimensional transverse Hessian is nondegenerate, providing local numerical evidence for a clean Morse–Bott critical manifold. The analysis then examines four generic approaches to a null internal tetrahedron. Along these approaches, the regular geometric saddle escapes to infinite rapidity, while a longitudinal boost mode softens linearly with the null-boundary parameter. An exact Schur–Morse–Bott determinant identity relates the full transverse determinant to auxiliary and spin-Schur contributions. Including the induced critical-family measure, gauge-orbit conversion, projective-spinor measure, and explicit spin dimensions, the complete leading Morse–Bott density exhibits a square-root boundary law, proportional to δ^(1/2), on all four tested rays. Finally, the work analyzes the possible breakdown of uniform stationary-phase asymptotics near the null boundary. If the next-to-leading-order to leading-order ratio behaves as 1/(λδ^p), with p > 3/2, boundary-layer matching produces a relative correction of order λ^(-3/(2p)). In particular, p = 2 would imply a nonanalytic λ^(-3/4) correction. The asymptotic exponent p is not independently established here. The results provide a local finite-complex characterization of the Lorentzian EPRL critical manifold and a quantitative account of its non-uniform null-boundary behavior. Continuum refinement, exact-null gluing, parity-branch uniformization, and the physical two-helicity graviton sector remain open.