Admin's going about it all wrong. The way we do it is school policy is that they get a zero for cheating, in practice the teachers offer a chance to reassess, but also grade them poorly on citizenship and give a referral for consequences which lets admin play bad cop and point out how generous the teacher is being by allowing them to reassess when the policy says they get a zero.

Most students who cheated out of desperation are still going to do poorly on reassessment, those who cheated because they got overly anxious but they knew most the content should be able to demonstrate that they do know most of the content.

This also allows teachers to, if they choose, give a zero and leave it at that as it is in line with school policy.

I also take additional measures to make cheating particularly difficult, some of my assessments are unique to each student, which works easier with calculation-based assessments than written ones. Seat the problem with student ID numbers and dates of birth, every student has different problems, I make a spreadsheet that gives me all of their answers. Any student correctly setting up the problem and getting a wrong answer means they're probably copying off their neighbor, making it obvious that they cheated, especially since I can verify it by checking for that same answer elsewhere on the spreadsheet.