A new mathematical result shows that adding a sufficiently strong Euclidean penalty to any multivariate polynomial makes its minimum exactly solvable with a single sum‑of‑squares (SoS) semidefinite program (SDP). The method defines a regularized polynomial m_{ heta}(s)=P(s)+ frac{ heta}{2d} orm{s}_2^{2d}, where P has degree at most p, the integer 2d is even and greater than p, and P contains a nonzero linear term. For large enough θ the penalty dominates lower‑degree perturbations. The shifted polynomial q_{ heta}=m_{ heta}-ar m_{ heta} (ar m_{ heta}=m_{ heta}(s_{ heta}^*)) admits a positive‑definite Gram matrix on centered non‑constant monomials and is a degree‑d sum‑of‑squares. Consequently, the global minimizer and its value can be recovered exactly from one degree‑d SDP, whose size grows only polynomially with the dimension when d is fixed. The proof builds on a strict Gram margin derived from the Bregman remainder of the Euclidean regularizer; this margin increases with θ and eventually overcomes any lower‑degree disturbance. The authors also show that the same exact SDP regime applies to optimization over small Euclidean balls, linking large regularization to trust‑region problems. However, the technique does not extend to coordinate‑separable regularizers: although such regularizers are coercive and SoS‑convex, their Bregman remainder fails to generate mixed monomial directions, preventing a positive‑definite Gram representation. This advance resolves a long‑standing difficulty: non‑negativity of a polynomial does not imply an SoS representation, and higher levels of the SoS hierarchy are normally needed. The new mechanism removes that requirement, providing a practical single‑level SDP approach for a broad class of polynomial optimization problems.