Research of the Behavior of the Effective Potential in Systems with Phase Transitions through the Prism of A--D--E Type Singularities

Detecting a scalar singlet interacting through the Higgs portal demands a pivot from conventional particle detection strategies to a comprehensive examination of the effective potential’s landscape. The presence, intensity, and first-order nature of the electroweak phase transition are dictated by the critical manifold, with its universal traits encapsulated in the Milnor number $\mu$ – the dimensionality of the local Jacobian algebra. Throughout the parameter space consistent with experimental observations, the portal potential exhibits a non-simple singularity with $\mu =9$, maintaining topological stability amid substantial fluctuations in mixing angle, singlet mass, and cubic interactions. High-precision assessments of the Higgs trilinear self-coupling (${\kappa }_{\lambda }$), the uniform rescaling of Higgs couplings ($c_H$), and the stochastic gravitational-wave spectrum (${\mathrm{\Omega }}_{\mathrm{G}\mathrm{W}}$) collectively delineate the catastrophe, extending beyond mere mass matrix analysis. Projections for 2027–2040 collider and LISA capabilities indicate that no viable region supporting a strong first-order transition will evade scrutiny; thus, the singlet will either be identified or conclusively dismissed via direct interrogation of the electroweak vacuum’s critical structure.