persistent-nilpotence no-go verification
critical algebra basis: 1, e, e^2
rank over the parameter base: 3
x^3(1+t*x) and x^3 define the same completed local ideal: unit factor
e^3 = 0 and e^2 != 0: PASS
Hessian determinant j = 3e^2 is nonzero nilpotent: PASS
Norm(j) = det(multiplication by j) = 0: PASS
Trace-pairing discriminant = 0: PASS
Reduced support of every fiber is one point: follows from rad(e^3)=(e)
Pairwise Poisson commutativity of separated Hamiltonians: PASS
Conclusion: finite-flat persistent nilpotence and Liouville integrability
do not force a positive-dimensional critical locus in a fiber.