The paper investigates the scaling behavior of Rényi entanglement entropy (EE) with smooth boundaries at the phase transition point of the two-dimensional $J-Q_3$ model. Using a recently developed scaling formula, the authors find a subleading logarithmic term with a coefficient indicating four Goldstone modes, suggesting spontaneous symmetry breaking from an emergent $SO(5)$ to $O(4)$ symmetry in the thermodynamic limit, but restored in finite systems. This result indicates that the believed deconfined quantum critical point of the $J-Q_3$ model is actually a weak first-order transition. The method provides a new way to distinguish states with spontaneously broken continuous symmetry from critical states, particularly useful for identifying weak first-order phase transitions, which are challenging to determine using conventional methods. The study also explores the finite-size scaling behavior of spin stiffness and inertia moment density, supporting the validity of the $SO(5)$ symmetry at the transition point.The paper investigates the scaling behavior of Rényi entanglement entropy (EE) with smooth boundaries at the phase transition point of the two-dimensional $J-Q_3$ model. Using a recently developed scaling formula, the authors find a subleading logarithmic term with a coefficient indicating four Goldstone modes, suggesting spontaneous symmetry breaking from an emergent $SO(5)$ to $O(4)$ symmetry in the thermodynamic limit, but restored in finite systems. This result indicates that the believed deconfined quantum critical point of the $J-Q_3$ model is actually a weak first-order transition. The method provides a new way to distinguish states with spontaneously broken continuous symmetry from critical states, particularly useful for identifying weak first-order phase transitions, which are challenging to determine using conventional methods. The study also explores the finite-size scaling behavior of spin stiffness and inertia moment density, supporting the validity of the $SO(5)$ symmetry at the transition point.