Hochster's small Cohen-Macaulay module conjecture for complete local domains

Theorem 1.1: there is a three-dimensional complete Noetherian normal local domain $R\supset\mathbb C$ with residue field $\mathbb C$ having no small Cohen-Macaulay module; the uncompleted vertex localization of the same section ring, essentially of finite type over $\mathbb C$, also has none. The mechanism is Theorem 1.2: if the completed cone $R(X,H)$ over a smooth projective surface has a small Cohen-Macaulay module, then $K_X^2-2c_2(X)\le6H^2$, applied to a Hirzebruch-Kummer surface with $K_X\equiv5H$. It does not treat positive or mixed characteristic, the non-domain forms, or dimension three with other residue fields. Priority: Liang Chen posted a claimed counterexample of the same kind on 21 September 2026, two days before this manuscript (arXiv:2609.24142): the completed section ring of a degree-six Hirzebruch-Kummer surface with $K_Y\sim4H$, resting on his stated Main claim.

Theorem 1.1: there is a three-dimensional complete Noetherian normal local domain $R\supset\mathbb C$ with residue field $\mathbb C$ having no small Cohen-Macaulay module; the uncompleted vertex localization of the same section ring, essentially of finite type over $\mathbb C$, also has none. The mechanism is Theorem 1.2: if the completed cone $R(X,H)$ over a smooth projective surface has a small Cohen-Macaulay module, then $K_X^2-2c_2(X)\le6H^2$, applied to a Hirzebruch-Kummer surface with $K_X\equiv5H$. It does not treat positive or mixed characteristic, the non-domain forms, or dimension three with other residue fields. Priority: Liang Chen posted a claimed counterexample of the same kind on 21 September 2026, two days before this manuscript (arXiv:2609.24142): the completed section ring of a degree-six Hirzebruch-Kummer surface with $K_Y\sim4H$, resting on his stated Main claim.