Coverage gap between arrival and enrolment

The standard resolution:

1. The chicken-and-egg is solved by "intent": Krankenkassen issue membership confirmations based on your admission letter and intended enrolment date — you don't need to be enrolled yet. Apply online, state your enrolment date as the start date, receive the confirmation the university needs. This is routine; their onboarding teams do it daily.
2. For the physical gap (landing → coverage start): a travel/incoming policy covers those days for a few euros. Many visa applications required one anyway — check what you already have.
3. Backdated coverage: statutory insurance legally starts at enrolment even if the card comes later — treatment in between gets billed retroactively to the Krankenkasse once membership finalises. Keep receipts.
4. Workers: your insurance starts with your employment start date automatically — the gap is only ever the pre-start days; a travel policy handles them.

Emergency during the gap? You will always be treated — sort billing later with the insurer; hospitals know this situation well.