Pure hafnian relations in a mixed matching ideal

For three independently weighted edge colors on six vertices, let the mixed matching ideal J_6 be generated by the coefficients of all nonconstant vertex colorings. Over any field of characteristic different from two and three, its contraction to the polynomial algebra of the three pure hafnians h_0, h_1, h_2 is generated by the exponent patterns (2,2,1), (2,1,2) and (1,2,2). The new membership step is an explicit 32-term symmetrized integer certificate. For nonmembership, the paper gives an elementary explanation of a previously recorded separating functional and its fixed-matching specialization. In the same characteristics, a separate 86-term certificate of degree twelve puts the fourth power of the common pure amplitude in the ideal obtained by adjoining the balance equations; with a prior lower bound, this gives nilpotence index exactly four and minimum normalized certificate degree exactly twelve. On eight vertices, an analytic four-orbit functional proves first-power nonmembership in the monochromatic ideal in characteristic zero or prime characteristic at least eleven. Files. paper.pdf is the manuscript (16 pages). hafnian-certificates-v5.zip is the supplement: the certificates as data and standard-library Python programs that replay every computational claim. Its SHA-256 is printed in Section 7 of the paper. Prior work. The closest earlier results are credited in Section 1 and in the references of the paper. Disclosure. The mathematics, code and text were produced with OpenAI Codex and Anthropic Claude at the author's direction, as the paper's disclosure section sets out. The author publishes under a pseudonym. Every computational claim is machine-checkable with the supplement, but no independent human mathematical review has taken place and none is claimed. Corrections are welcome at [email protected].