I'll try using solar noon as a fixed duration. We know that the duration between solar noons is the same every day (24hrs + a little bit). So let's slice that

I'll try using solar noon as a fixed duration. We know that the duration between solar noons is the same every day (24hrs + a little bit). So let's slice that up into day and night. Consider the two solar noons that adjoin the maximum night duration. I claim one of those two noons happens during the shortest day. We have midnight -> Sunrise -> noon -> sunset -> midnight -> sunrise -> noon -> sunset. We know noon .. noon is fixed at 24hrs +a bit. Same with midnight..midnight. We know that midnight..noon is about 12 hours (+ half a bit). This is just orbital mechanics. We know sunset..midnight..sunrise is periodic over the year, with one maximum duration. Therefore we know that noon..midnight..noon contains a periodic amount of dark time with one maximum. Let that time be |D|. Let |L| be the "light time" either the preceeding or following midnight..noon..midnight period - whichever has less light. Since midnight..noon..midnight..noon has 36 hrs (ish), then |D|+|L| = 36 (ish), so maximum darkness must have a minimum light time.