We introduce a quadratic form Q on the space of functions on the gap poset G of the numerical semigroup \(\langle a,b\rangle \) . We prove combinatorially that when evaluated on the indicator function of an upward closed subset D, this quadratic form precisely recovers the Gorsky–Mazin \(\texttt {dinv} \) statistic of D, viewed as a Young subdiagram of G. Furthermore, we prove Theorem 1.2 that when evaluated on a pair of subdiagrams of G, the symmetric bilinear form associated with Q is equal to a novel cross- \(\texttt {dinv} \) statistic, which is non-negative. Combining these, we prove the inequality \(\begin{aligned} Q(\mathbf {\textit{n}})\ge \dfrac{1}{|G|}\,\Vert \mathbf {\textit{n}}\Vert _\infty ^2 \end{aligned}\) if \(\mathbf {\textit{n}}\) is a real-valued decreasing function on G, showing an effective positive definiteness of Q on the corresponding cone. Theorem 1.2, the main engine of the paper, was autoformalized in Lean/Mathlib by AxiomProver.
