Dated copies of the official program page and FAQ as they read when each decision was made, all payment records and confirmations, the identity and background material supplied for vetting, and any written correspondence about the underlying classification. Nothing should rest on a summary provided by an intermediary.
Keep the version of the rules you relied on
Where a framework is new and has already been amended, the most valuable thing in a file is a dated record of what it said when a decision was taken. Print or archive the official page and the FAQ each time a step is reached, with the date visible. Keep the receipt and confirmation for every payment, and keep the processing fee entirely separate in the records from any gift, because they are different in character and only one of them relates to adjudication.
Retain everything provided for vetting, including identity documents, background disclosures and the correspondence acknowledging them. Where the application proceeds under a particular employment-based classification, keep the material supporting that classification too, since it is what the underlying process will examine. Finally, keep the written scope of any adviser's engagement.
A household committing a seven-figure sum should be able to show later exactly what it was told, by whom, and against which published version of the terms.