Local Uniform Finite Cyclicity of the $H_{14}^{3}$ Semihyperbolic Hemicycle in Quadratic Systems

Abstract

We prove local uniform finite cyclicity of the labelled $H_{14}^{3}$ hemicycle in a full neighborhood of its base field in the twelve-dimensional space of planar quadratic vector fields. Thus there is one fixed two-sided annular neighborhood of the compactified graphic in which the number of isolated limit cycles is bounded uniformly for every sufficiently close quadratic field. A local analytic slice theorem is part of the result: the displayed five-parameter source-normalized family is transverse at the $B=0$ field to the seven-dimensional action of affine phase changes and positive constant time rescalings. This removes the normalization and transfers the same cyclicity bound to the full quadratic coefficient space. The normalized problem combines a noncompact period annulus, two semihyperbolic endpoints at infinity, and a degeneration at the upper equatorial point. We replace the unavailable global Poincare map by finitely many stopped transition maps and prove an exact multiplicity-preserving correspondence between collar cycles and displacement zeros. At the noncompact source a matched nonlinear return on a common physical domain, controlled through six derivatives, gives a two-zero bound by Rolle's theorem. At the two saddle-nodes an exhaustive physical incidence and scale decomposition reduces all limits to source, mixed, hyperbolic, central, and lips estimates. A finite specialization argument then extends these bounds across coefficient and identity strata. The resulting bound is existential and is not claimed to be sharp. In the terminology of the quadratic finite-cyclicity program, this completes the labelled $H_{14}^{3}$ open case.