Heilbronn's triangle problem: a power-saving lower bound n^(-2+eta), refuting the almost n^-2 upper bound

Theorem 1.1: absolute constants $\eta,c_1>0$ and $n_0$ with $\Delta(n)\ge c_1n^{-2+\eta}$ for every $n\ge n_0$. Hence the almost $n^{-2}$ upper-bound formulation is false. The construction samples integer columns in a box with finite-field norm congruence conditions and deletes the few small triangles; $\eta$ is explicit but extremely small. It does not determine the order of $\Delta(n)$: the gap to the upper bound $n^{-7/6+o(1)}$ remains, and no conjecture for the true exponent is made.

Theorem 1.1: absolute constants $\eta,c_1>0$ and $n_0$ with $\Delta(n)\ge c_1n^{-2+\eta}$ for every $n\ge n_0$. Hence the almost $n^{-2}$ upper-bound formulation is false. The construction samples integer columns in a box with finite-field norm congruence conditions and deletes the few small triangles; $\eta$ is explicit but extremely small. It does not determine the order of $\Delta(n)$: the gap to the upper bound $n^{-7/6+o(1)}$ remains, and no conjecture for the true exponent is made.