Class-Number-One Cubic Fricke Companions on ${\mathrm{\Gamma }}_0(2{)}^{+}$ and an Exact Chudnovsky Bridge

A structural classification-and-bridge result is proved for RS series on ${\mathrm{\Gamma }}_0(2{)}^{+}$. Among the fundamental Heegner discriminants with $h(D)=1$, the irreducibly cubic Fricke-cusp companions occur exactly for $$D=-11,-19,-43,-67,-163.$$ The proof gives an explicit bridge between $z=1728\mathrm{/}j$ and $X=256t\mathrm{/}(t+64{)}^2$, a hypergeometric gauge identity, and a CM interpretation of the fibre. For $D=-163$, the cubic Fricke RS series is the exact level-two transport of the Chudnovsky linear form.