One-variable main conjecture for CM elliptic curves (${\mathrm{char}}_{\Lambda }{X}_{\chi }=({\mathcal{L}}_{\chi })$)

At a good split prime, the one-prime CM division tower has a pro-$p$ Galois group isomorphic to ${\mathbb{Z}}_p$. Its prime-ramified Iwasawa module is torsion on each finite-character component. The main conjecture identifies its characteristic ideal with the corresponding elliptic-unit p-adic L-function. The class-field unit sequence and equality of characteristic ideals of global-units modulo elliptic-units and of class groups prove the identification. The cyclotomic Selmer formulation requires the appropriate CM twist and two-variable specialization; the one-prime tower is not the cyclotomic tower.