The Complex Spectral Flow: Spectral Conditions for Two-Parameter Equivariant Bifurcation Guarantees

Abstract: We introduce the \emph{complex equivariant spectral flow} -- a virtual $G$-representation assembling the eigenvalue winding numbers of the linearization at an isolated two-parameter critical point for $G = S^1 \times Γ$ bifurcation problems -- and prove that, for maximal twisted orbit types, the coefficient of the local bifurcation invariant in $A_1^t(G)$ reduces to a closed-form dimension formula, bypassing the standard pipeline of basic degree factorization and Burnside ring multiplication entirely. When the eigenvalue dependence is holomorphic, topological cancellation among winding numbers is impossible, yielding unconditional local and global bifurcation guarantees. As applications, we establish the macroscopic escape of symmetric Hopf branches in $Γ$-equivariant systems and of patterned relative equilibria for the complex Ginzburg--Landau equation directly from spectral data.