Curve lower bound from a homogeneous section test

Let $L$ be ample on $S$, $r>0$, $p$ a configuration, $U$ affine with a trivialization of $L$, and suppose that for all $k,m>0$ with $k\sqrt{L^2/r}<m$ some $\mathbb C$-points $\rho_i$ of $\Gamma(U)$ over $p_i$ have the property that every section of $L^k$ with coefficient in $(\ker\rho_i)^m$ for all $i$ vanishes. Then $\sqrt{L^2/r}\operatorname{mult}_p(C)\le\deg_CL$ for each integral curve $C$.

Notation. $L^2$ is selfIntersection, namely $\chi(L^2)-2\chi(L)+\chi(\mathcal O_S)$. $\deg_CL$ is curveDegree, namely $\chi(L|_C)-\chi(\mathcal O_C)$. $\operatorname{mult}_p(C)$ is totalMultiplicity, the sum over the points of curveMultiplicity, the order of the kernel ideal of the local ring map of $C$ at the point. IsAmple means that for each point $x$ and open $V\ni x$, some positive power has a global section with nonvanishing open containing $x$, inside $V$, and affine. A Surface is an integral scheme smooth of relative dimension 2 over $\operatorname{Spec}\mathbb C$ with a closed immersion into some complex projective space over $\mathbb C$. An IntegralCurve on a surface is an integral closed subscheme of topological Krull dimension one, given by a closed immersion. Configuration S r is the set of $r$-tuples of complex points with distinct images, with the Zariski topology induced from the product.

Formalization Note The statement is verbatim from OpenAI's Lean development, with Definitions files of this development imported in place of the upstream modules; these are supporting code for the statement.