The simple solution to resolving the dispute is to read the statutes in question using the standard rules of statutory construction that wherever possible, statutes should be read to give meaning to all provisions, to be consistent, and to avoid absurd results. Using these rules, it is clear that White is obligated to sign the Certificate of Appointment. For whatever their reasons, none of the parties has chosen to focus on these principles.
The two key provisions of the Illinois Statutes, 15 ILCS 305/5, are as follows:
According to White, the Certificate of Appointment is not a "commission" so all White need do is make a register of the appointment, which he has done. White's interpretation, however, does not give meaning to the distinction drawn in the statute between the Secretary of State's need to countersign and affix the seal to a document (the commission) versus the record keeping obligation to register the appointment. The Secretary of State's register is the official record of the State of an appointment, but the certified commission is the proof of the appointment. These are two distinct functions, and the only consistent way to read the statute is to honor this distinction, not read the "commission" provision out of existence as White proposes.
Sec. 5. It shall be the duty of the Secretary of State:
1. To countersign and affix the seal of state to all commissions required by law to be issued by the Governor.
2. To make a register of all appointments by the Governor, specifying the person appointed, the office conferred, the date of the appointment, the date when bond or oath is taken and the date filed. If Senate confirmation is required, the date of the confirmation shall be included in the register. [Italics added.]
Additionally, White's interpretation creates a potentially absurd situation (which in fact is what we have here). If there is an appointment, there must be an official state document reflecting the act of appointment. That document is the commission, in this case titled the Certificate of Appointment. The Secretary of State's register is just a State record, not the document creating the appointment. But if the Secretary of State will not sign and seal the document reflecting the appointment, despite having registered the appointment, the Secretary of State may create doubt as to the validity of the appointment despite having no power to do so. To read the statute as White proposes requires circular reasoning which turns a ministerial function into a substantive function.
I make no predictions as to what the Illinois Supreme Court will do. But simple statutory construction seems the key to me as to how to resolve this dispute.