In many States, where a legal adoption proceeding was defective under State law or where a contemplated legal adoption was not completed, a claimant may be considered to be an equitably adopted child. A claimant will have the relationship of an equitably adopted child for annuity and lump-sum payment purposes if, in addition to meeting the other requirements of this part—

(a) The employee had agreed to adopt the claimant; and

(b) The natural parents or the person legally responsible for the care of the claimant agreed to the adoption; and

(c) The employee and the claimant lived together as parent and child; and

(d) The agreement to adopt is recognized under applicable State law such that, if the employee were to die without leaving a will, the claimant could inherit a share of the employee's personal estate as the child of the employee.