A Multiscale Hilbert-Fourier Variety Approach to Hilbert's 16th Problem: Establishing the Upper Bound H(2)=7 for Quadratic Systems

This article proves a global upper bound for the maximum number of limit cycles in planar quadratic vector fields (n=2): it is exactly H(2)=7. Unlike traditional perturbative methods, such as Melnikov analysis or local Hopf bifurcation theory—which are inherently restricted to the vicinity of a singular point or a period annulus—this work maps the system's complete dynamics onto an infinite-dimensional algebraic variety in a Hilbert space. By treating the existence of periodic orbits as a global structural identity of the Fourier-coefficient equations, we identify a multiscale hierarchy of isolated solutions organized by the scaling parameter \mu. Our analysis reveals that the quadratic variety is composed of three structurally independent scales: a primary scale (X_{1,0}\neq0) that hosts 4 cycles by integrating local bifurcations with global energy balance; an emergent scale (X_{1,1}\neq0) providing 2 additional solutions; and a third-order scale (X_{1,2}\neq0) that contributes a final isolated cycle before algebraic saturation occurs. The total capacity H(2)=4+2+1=7 represents a definitive algebraic ceiling that transcends the classical Bautin limit of 3 cycles for a single focus. The theory is validated through the construction of a specific quadratic field that explicitly distributes these 7 cycles across local and global topological configurations. This result provides a rigorous resolution to the second part of Hilbert's 16th problem for the quadratic case, closing the gap between known lower bounds and the system's total algebraic capacity.