Rigidity of the three-variable mixed kernel line

Determine whether the polynomial kernel line associated with a determinant-free mixed self-normalizing polynomial on the covariance space of three variables is constant, thereby completing the unconditional three-variable converse or producing a self-normalizing denominator outside the flag-power class.

Background

The paper classifies exactly self-normalizing covariance polynomials completely in dimension two and reduces the dimension-three problem to rank-one, pure plane-minor, and mixed branches. In the mixed branch, the relative gradient has a fixed spectrum and a polynomial kernel vector, but the associated projective kernel line may vary with the covariance matrix.

Constancy of this kernel line is equivalent to the mixed polynomial having the nested flag form consisting of a rank-one variance factor and a plane-minor factor. Proving constancy would establish the full three-variable converse; finding a nonconstant example would yield a self-normalizing denominator not generated by nested covariance minors. The paper notes that Hessian degeneracy alone is insufficient, so additional integrability, cofactor, or boundary arguments are required.

References

The main open algebraic problem is the rigidity of the mixed kernel line. A proof that the kernel line in Theorem~\ref{thm:d3-reduction} (iii) is constant would complete the unconditional three-variable converse, while a counterexample would produce a self-normalizing denominator outside the flag class.

— Self-Normalizing Denominators in Rational Causal Estimation  (2608.20223 - Tamano, 20 Aug 2026) in Section 7, Discussion; Remark 4.8, “The mixed-kernel problem”

The principal distributional question is self-normalization beyond the Gaussian covariance operator. Ellipticity only rescales the constant, so the first open case is a semiparametric family with unrestricted fourth cumulants. A characterization of polynomials whose sandwich variance is proportional to their square over such a family would determine when the standardized denominator remains a samplewise constant under a matched studentizer.

— Self-Normalizing Denominators in Rational Causal Estimation  (2608.20223 - Tamano, 20 Aug 2026) in Section 7, Discussion