The upcoming Euclid mission will map the geometry of the dark universe with unprecedented precision. Forecasts indicate that instead of resolving current tensions (like $H_0$ and $S_8$), Euclid's high-fidelity data might amplify them to undeniable statistical significance ($>10\sigma$) if the underlying $\Lambda$CDM model is incorrect. Specifically, the cross-correlation of weak lensing and galaxy clustering is predicted to tighten constraints on dark energy parameters ($w_0, w_a$) and modified gravity, potentially exposing a fundamental mismatch between early-universe (CMB) and late-universe (lensing/clustering) physics that cannot be fixed by simple systematics (Euclid Collaboration 2025; Di Valentino et al. 2024).