Detained at Munich airport for accidental 24-hour overstay due to day count error

I am writing to objectively document a procedural error that resulted in a significant border incident yesterday. My background is in physics, and I generally rely on precise calculations; however, I appear to have fundamentally misunderstood the variable definition of the Schengen “90-day” limit.

The Facts:
I entered the Schengen zone (Germany) on June 1st. I booked my return for August 30th, operating under the assumption that a three-month duration was colloquially equivalent to the 90-day allowance. I failed to account for the fact that July and August both possess 31 days. Consequently, my stay totaled 91 days.

The Incident:
Upon attempting to exit through Munich, I was flagged by the automated gate. I was taken to a side room by the federal police. I was informed that I had overstayed my visa validity. Despite my explanation that this was a mathematical oversight and not a malicious attempt to reside illegally, I was processed, fined, and warned about a potential entry ban. I was eventually allowed to depart, but the status of my record is unclear.

Inquiries:

  1. Does a 1-day overstay automatically trigger a SIS (Schengen Information System) ban, or is it discretionary?
  2. How can I verify if a ban was actually imposed versus just a warning?
  3. Is there a formal mechanism to appeal this based on the “good faith” nature of the mistake (i.e., I had a booked return flight consistent with my misunderstanding)?