Rational Cycles of Periods Three and Four for Quadratic Hénon Maps

For the maps (x, y) ↦ (y, y² + c + bx), this manuscript classifies rational marked 3-cycles for every fixed rational b and proves the existence of infinitely many distinct rational parameters c, treating b=-1 separately and distinguishing the noninvertible case b=0. The explicit period-three section contradicts the finite-exception formulation of Conjecture 4 in Patrick Ingram's section of Hénon maps: a list of open problems (2024). At period four and b=1, it gives the three marked components, their genus-zero projective normalizations, rational lifts, exceptional fibers and arithmetic lifting conditions on normalized rotation quotients, with classical equations and invariant-field results attributed. Period four at other coefficients and higher-period classifications are not claimed. Selected exact-algebra and Lean certificates cover only the statements described in the reproduction notes; the entire paper is not formally verified. No global priority or peer-review claim is made. The deposit contains the article, supplement, a scientific scope and correction note, and selected reproducibility sources.