A lifted rational page descends through the antipodal cover

Let $\varphi$ be a continuous complete flow on a selected Levi-Civita component $\Sigma_{\mu,c}$. Suppose the component is invariant under the antipodal map, that this map has no fixed points, and that the flow commutes with it. Let $\gamma$ be a least-period closed orbit reaching its antipode at half-period.

If $\gamma$ admits a lifted rational page, then it induces a rational two-disk global surface of section in the antipodal quotient:

The conclusion includes jointly continuous quotient dynamics, primeness of the half-period quotient binding, continuity of the quotient page, embedding of its interior, precisely the two-fold boundary identifications, and unbounded positive and negative return times for every nonbinding quotient trajectory. Transversality is measured on the smooth lift by the Levi-Civita Hamiltonian vector field.

This is a cover-descent lemma for the rational-page encoding used in the mission. It assumes neither a near-equal-mass restriction nor the existence of any page; it applies whenever the stated cover and page data are supplied.