Quadratic Hénon Maps over Q with Rational Cycles of Length Three and Four for Every Jacobian

In a list of open problems on Hénon maps (Arnold Math. J. 10 (2024)), P. Ingram conjectured that for all but finitely many \(\delta\in\mathbb{Q}\) every \(\mathbb{Q}\)-rational periodic point of every map \(f(x,y)=(y,\,y^2+c-\delta x)\) with \(c\in\mathbb{Q}\) has period dividing \(2\). We show that the statement, as printed, is false for every \(\delta\in\mathbb{Q}\), in each reading the text supports (finitely many exceptional \(\delta\); for fixed \(\delta\), finitely many exceptional \(c\); finitely many exceptional pairs). For \(\delta\ne1\) and \(c=-(29\delta^2-38\delta+29)/16\) the point \(\bigl(\frac{5-7\delta}{4},-\frac{1+\delta}{4}\bigr)\) has period \(3\); for \(\delta=1\) and \(c=-1-s^2\), \(s\in\mathbb{Q}\), there is a rational \(3\)-cycle; for every \(\delta\ne0\) an explicit \(c\) gives a rational \(4\)-cycle. Such cycles are essentially implicit in the source: the interpolation construction that opens the same section yields rational \(4\)-cycles for every \(\delta\notin\{0,1\}\), and our \(4\)-cycles are its normal form; so the printed conjecture contradicts its own section and is probably not the intended statement. We also describe all rational \(3\)-cycles: shifted by \(\frac12\), their \(x\)-coordinates are the rational points, off the line \(u=v=w\), of the cone over the nodal cubic \(u^3+v^3+w^3-u^2v-v^2w-w^2u=0\). Hence every \(\delta\ne1\) has infinitely many \(c\) with a rational \(3\)-cycle, and for \(\delta=1\) one exists if and only if \(-(1+c)\) is a rational square. Computations, which are not theorems, find for \(\delta=-1\) exactly Ingram's periods \(1,2,3,4,6,8\), and a rational \(5\)-cycle for \(\delta=-4\) and, by a duality \(\delta\leftrightarrow1/\delta\), for \(\delta=-\frac14\). The case \(\delta=0\) is the classical map \(z^2+c\) (Morton; Morton and Raianu), the case \(\delta=-1\) is Ingram's, and the novelty of the note is modest. This is an unrefereed note.