A purely inseparable obstruction to binomial expansions of rational symbolic powers

We exhibit homogeneous prime principal ideals in disjoint polynomial variables whose rational symbolic powers do not satisfy the binomial expansion. The ground field has positive characteristic and is imperfect. For every prime power q >= 2 we construct a mixed sum K such that all ordinary powers of K are primary and integrally closed, and compute its entire rational symbolic filtration by one explicit valuation. The binomial expansion at a positive rational exponent u holds exactly when ceil(qu) = q ceil(u). Thus every integer exponent satisfies the formula, whereas in each interval (n-1,n] the formula fails precisely on (n-1,n-1/q]. We also give a reducedness criterion for the formula at all exponents in (0,1] when the two input ideals are radical. The counterexample answers the unrestricted ground-field formulation of Question 3.17 of Dey-Ha-Mahato-Mantero negatively; it does not address the corresponding question over algebraically closed fields. We verify the relevant Rees-package input data and identify a failure of the arbitrary-field tensor-valuation gluing assertion for this inseparable family.