Resolving Degeneracies in Complex R× S³ and $θ$-KSW

Lorentzian gravitational path integral for the Gauss-Bonnet gravity in $4D$ is studied in the mini-superspace ansatz for metric. The gauge-fixed path-integral for Robin boundary choice is computed exactly using { Airy}-functions, where the dominant contribution comes from No-boundary geometries. The lapse integral is further analysed using saddle-point methods to compare with exact results. Picard-Lefschetz methods are utilized to find the { relevant} complex saddles and deformed contour of integration thereby using WKB methods to compute the integral along the deformed contour in the saddle-point approximation. However, their successful application is possible only when system is devoid of degeneracies, which in present case appear in two types: { type-1} where the flow-lines starting from neighbouring saddles overlap leading to ambiguities in deciding the { relevance} of saddles, { type-2} where saddles merge for specific choices of boundary parameters leading to failure of WKB. Overcoming degeneracies using artificial { defects} introduces ambiguities due to choice of { defects} involved. Corrections from quantum fluctuations of scale-factor overcome degeneracies only partially (lifts { type-2} completely with partial resolution of { type-1}), with the residual lifted fluently by complex deformation of (G). { Anti-linear} symmetry present in various forms in the lapse action is the reason behind all the { type-1} degeneracies. Any form of { defect} or { deformation} breaking anti-linearity resolves { type-1} degeneracies, indicating complex deformation of (G) as an ideal choice. Compatibility with the KSW criterion is analyzed after symmetry breaking. Complex deformation of (G) modifies the KSW criterion, imposing a strong constraint on the deformation if No-boundary geometries are required to be always KSW-allowed.