In this work, we present a geometrical reconstruction of the critical points of the spinfoam amplitude for a 4D Lorentzian model with a nonzero cosmological constant. By establishing the correspondence between the moduli space of $\mathrm{SL}(2,\mathbb{C})$ flat connections on the graph-complement 3-manifold $S^3\backslash {\mathrm{\Gamma }}_5$ and the geometry of a constantly curved 4-simplex, we demonstrate how the critical points encode discrete curved geometries. The analysis extends to 4-complexes dual to colored graphs, aligning with the improved spinfoam model recently introduced in Han and Pan [Complex Chern-Simons theory with $k=8\mathbb{N}$ and an improved Spinfoam model with cosmological constant, Phys. Rev. D 111, DD13861 (2025).]. Central to this reconstruction are translating the geometry of constantly curved 4-simplices into Fock-Goncharov coordinates and spinors, which translate the geometry data into holonomies and symplectic structures, thereby defining the critical points of the spinfoam amplitude. This framework provides an algorithmic foundation for computing quantum gravity corrections and opens avenues for applications in quantum cosmology and black hole physics, where the cosmological constant plays a pivotal role.

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