If cellularBinomialIsPrimary does not return true it can either return 'false' or two associated primes. If this option is set then two partial characters of distinct associated primes are returned. If ReturnPrimes is set too, then partial characters will be returned.
i1 : R = QQ[x]
o1 = R
o1 : PolynomialRing
i2 : I = ideal (x^2-1)
2
o2 = ideal(x - 1)
o2 : Ideal of R
i3 : cellularBinomialIsPrimary (I,ReturnPChars=>true)
The radical is not prime, as the character is not saturated
o3 = {{{x}, | 1 |, {1}}, {{x}, | 1 |, {-1}}}
o3 : List