Lemma 54 ([CM disk coordinates]). The conductor- s CM points are canonical lifts. In their Serre--Tate disks, the points z=ζ₂^mj , for j odd, have endomorphism order O₂^ms and are defined over H₂^ms with the specified cyclic level. …

The conductor- s points are canonical lifts because reduction is ordinary at the split prime, the endomorphisms split the 2 -divisible directions, and the Serre--Tate parameter must be one by the endomorphism lifting relation. Consequently the geometric special end ring is exactly O_s (all such endomorphisms lift at the trivial parameter). Make the disk evaluations over the completed unramified base, enlarging constants if needed for the form coefficients. Choose Serre--Tate generators in both directions by polarization. For each a use a prime-to- 2s and prime-to-level ideal action as representative, viewed also at higher 2-conductors. Write δ_ a ∈ℤ_2^× for its Serre--Tate exponent from the base (connected divided by étale factor); take the identity at the trivial label. Points at z=ζ_ 2^m ^ \,j , j odd for a primitive ζ_ 2^m , have endomorphisms O_ 2^m s : among endomorphisms of the reduction the two local actions must agree modulo 2^m . Canonical cyclic 2 -level is preserved by the order, as are the oriented tame groups. Thus by the main theorem of CM these X_0 -points are defined over H_ 2^m s : the units of the completed endomorphism order stabilize the cyclic level. Prime-to-level ideal actions transport the groups (including the canonical group at 2) and the disk parameters by the isogeny exponent. In particular the kernel of lowering the order class map to conductor s , the unit ratios (ℤ/2^m)^× , acts simply transitively on the primitive roots in a disk. For an ideal principal below, extending to (γ) , the exponent is the local ratio γ_w/γ_ w up to reversing conventions (action by the corresponding endomorphism on the underlying conductor- s reduction). Thus in the inverse limit the exponent on the order-class kernel maps to Γ by one fixed η:ℤ_2^×→Γ , independent of s by the order class sequence. Here isogeny actions in all labels are identified by their actual Artin actions on the CM orbit, inverting ideals consistently if required by reciprocity. The cyclic 2 -level throughout is the canonical connected subgroup; there is no independent variation of a 2 -level structure along the orbit.