# Counterexample verification

The proof gives two certificates. An integer polynomial identity
places xy^3 in the scheme theoretic fourth jet closure. A local
homomorphism to k[s]/(s^10) kills the Jacobian ideal plus the fifth
maximal-ideal power and sends xy^3 to nonzero -s^9. The same proof
works for the irreducible four-variable variant.

The exact checker passes 104 checks in both normal and optimized
modes. Independent rational row reduction gives ranks 19 versus 20
in dimension 35, and 29 versus 30 in dimension 70. A deliberately
incorrect membership certificate is rejected.

The algebraic proof establishes the result over the source's
characteristic-zero field. Finite tests are supplementary, not an
extrapolation to a general theorem. Formal irreducibility is proved
using Gauss's lemma and positive weighted initial forms.

The counterexamples are nonisolated. The assertion refuted is the
general conjecture as printed, not an isolation-restricted variant.
The known cusp defect is credited to de Fernex-Ein-Ishii. The original
OWR all-orders equality is separately credited to Mallory.

Independent expert review and formal proof-assistant verification
have not been performed. No such approval or novelty certificate is
implied by successful compilation or package validation.