Finite Cyclicity of the Graphic $I_{16a}^2$ in Quadratic Systems

We prove that every graphic in the I₁₆ₐ² family has cyclicity at most two in the normalized quadratic family of Dumortier–Roussarie–Rousseau. A fixed base field carries a continuum of such graphics, each determined by a regular connecting orbit from D₂ to D₃. For every r>0 and every finite value k>0 of the connection parameter, the corresponding graphic satisfies this bound. The coefficient and Hausdorff neighborhoods, and the analytic estimates, may depend on (r,k). The proof works locally at an arbitrary fixed positive ratio. It retains the actual Dulac exponent r_h, treats resonances near that fixed ratio in parameter-local compensator charts, and projectivizes the two exponentially small saddle-node multipliers without assigning them a relative order. The identity return on the center component then reduces the remaining zero count to a complete-period-annulus first variation. Including both moving endpoints of the degenerating annulus yields a strictly concave limiting profile for every fixed r>0. A relative C² estimate and an extended complete Chebyshev argument transfer the two-zero bound to the nonlinear displacement. We formulate this projective–Chebyshev passage as an abstract two-zero criterion for three normalized variations. This criterion separates the projective–Chebyshev mechanism from the asymptotic and localization arguments specific to the present graphic. Finally, a six-box localization argument shows that every nearby periodic orbit is represented by the same displacement equation. The upper bound is not asserted to be sharp, and no uniform neighborhood is claimed as the fixed graphic approaches either bordering graphic.