Mubayi's Polynomial-Ideal Conjecture and Cover-Ideal Tur\'{a}n Methods

We revisit a conjecture of Mubayi that was proposed as a hypergraph analogue of the Li--Li algebraic proof of Tur\'{a}n's theorem. The conjecture compares a polynomial ideal generated by multipartite 3-graphs with a differentiated diagonal-vanishing ideal. We show that the proposed equality fails for every non-vacuous choice of parameters. The obstruction is structural: diagonal vanishing does not remember the missing codegree-star condition that drives Mubayi's hypergraph problem. We then give a replacement in edge-variable rings using monomial cover ideals. For ordinary forbidden-family Tur\'{a}n problems, the cover ideal converts extremal edge counting into an initial-degree computation. For generalized Tur\'{a}n numbers, the same cover ideal encodes the forbidden condition, while the objective becomes a quotient rank on the space spanned by the target-copy monomials. For Mubayi's core-pair family $\mathcal{K}_{\ell}^{(r)}$, this cover ideal has an explicit missing codegree-star form. A Hilbert-function symmetrization theorem for square-zero quadratic monomial quotients computes its initial degree and recovers Mubayi's hypergraph Tur\'{a}n theorem.