Testing for admissible sequences
Brachylog, 25 24 19 bytes

5 bytes thanks to Karl Napf.


lybb'(eM-yA,?:[M]z:%aodA)
l:2'(eM-yA,?:[M]z:%aodA)
l:2'(eMg:?rz:%adlM)


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Verify all testcases!

l:2'(eMg:?rz:%adlM)
l:2                  Temp = [2:length(input)]
   '(             )  true if the following cannot be proven:
     eM                  M is an element of the interval
                         indicated by Temp, i.e. from 2
                         to the length of input inclusive,
       g:?rz:%adlM       every element of input modulo M
                         de-duplicated has length M.