Generally, I always felt like this is a pretty confusing gap in the commentary of 1002.9, when the only thing(s) missing are not in the applicant's control. Don't have a complete application trigger to 1002.9(a) clock and don't have anything in borrower control to trigger 1002.9(c). What I fall back on is commentary for the definition of an application 1002.2(f). Lender has to act with reasonable diligence to complete the application.
6. Completed application - diligence requirement. The regulation defines a completed application in terms that give a creditor the latitude to establish its own information requirements. Nevertheless, the creditor must act with reasonable diligence to collect information needed to complete the application. For example, the creditor should request information from third parties, such as a credit report, promptly after receiving the application. If additional information is needed from the applicant, such as an address or a telephone number to verify employment, the creditor should contact the applicant promptly. (But see comment 9(a)(1)-3, which discusses the creditor's option to deny an application on the basis of incompleteness.)
More specific to your situation, "the lending department would like to close this file as incomplete" sounds a lot like "A refusal to grant credit in substantially the amount or on substantially the terms requested in an application," which is the definition of an adverse action, and you have to provide notice "(ii) 30 days after taking adverse action on an incomplete application, unless notice is provided in accordance with paragraph (c) of this section;"