Coexisting Flux String Vacua from Numerical K\"ahler Moduli Stabilisation

We present a comprehensive study of K\"ahler moduli stabilisation in Type IIB flux compactifications, combining advanced numerical techniques with analytical methods. Our JAX-based computational framework enables efficient scanning of the UV parameter space, while incorporating $\alpha'$ corrections, loop and non-perturbative effects, as well as uplift contributions to the scalar potential. The implementation features rigorous vacuum validation protocols derived from analytic results. We apply our methods to explicit flux compactifications on more than 80,000 Calabi-Yau threefolds with $h^{1,1}\leq 6$ K\"ahler moduli. By systematically scanning over a wide range of values of the flux superpotential $W_0$ and the string coupling $g_s$, we find explicit realisations of every established K\"ahler moduli stabilisation scenario: for $10^{-15} \leq |W_0| \leq 10^{-2}$ we obtain both KKLT-like and K\"ahler uplifted vacua, while for the broader range $10^{-1} \leq |W_0| \leq 10^2$ we recover LVS as well as LVS-like hybrid solutions. Notably, we discover significant parameter regions where multiple vacua coexist within a single flux potential, including novel configurations pairing AdS, Minkowski, and dS minima with different volume hierarchies. These findings enable, for the first time, the analysis of vacuum decay processes within fixed flux configurations, complementing the established theory of transitions between distinct flux vacua and decays towards decompactification.