This paper presents an improvement to the four-dimensional spinfoam model with cosmological constant ($\mathrm{\Lambda }$-SF model) in loop quantum gravity. The original $\mathrm{\Lambda }$-SF model, defined via $\mathrm{SL}(2,\mathbb{C})$ Chern-Simons theory on graph-complement 3-manifolds, produces finite amplitudes and reproduces curved 4-simplex geometries in the semiclassical limit. However, extending the model to general simplicial complexes necessitated ad hoc, nonuniversal phase factors in face amplitudes, complicating systematic constructions. We resolve this issue by redefining the vertex amplitude using a novel set of phase space coordinates that eliminate the extraneous phase factor, yielding a universally defined face amplitude. Key results include: (1) The vertex amplitude is rigorously shown to be well defined for Chern-Simons levels $k\in 8\mathbb{N}$, compatible with semiclassical analysis ($k\to \infty$). (2) The symplectic structure of the Chern-Simons phase space is modified to accommodate $\mathrm{SL}(2,\mathbb{C})$ holonomies, relaxing quantization constraints to $\mathrm{Sp}(2r,\mathbb{Z}/4)$. (3) Edge amplitudes are simplified using constraints aligned with colored tensor models, enabling systematic gluing of 4-simplices into complexes dual to colored graphs. (4) Stationary phase analysis confirms consistency of critical points with prior work, recovering Regge geometries with curvature determined by $\mathrm{\Lambda }$. These advancements streamline the spinfoam amplitude definition, facilitating future studies of colored group field theories and continuum limits of quantum gravity. The results establish a robust framework for 4D quantum gravity with nonzero $\mathrm{\Lambda }$, free of previous ambiguities in face amplitudes.

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• Cosmological constant
• Loop quantum gravity