A Sharp Curvature Threshold for GLMY Path Homology

Let $G$ be a finite simple graph with at least one edge. We prove the sharp vanishing theorem $${\kappa }_{\min }^{}(G)>\frac{1}{2}⟹{\text{PathH}}_1(G;\mathbb{R})=0.$$ Equivalently, nonzero first GLMY path homology forces an edge of Lin–Lu–Yau curvature at most $1\mathrm{/}2$. The threshold $1\mathrm{/}2$ is sharp and is attained by $C_5$. The proof combines the cycle-space description of first GLMY path homology with the limit-free Laplacian characterization of Lin–Lu–Yau curvature. As a secondary consequence of the curvature-preserving universal-cover method, we prove that if $G$ is connected and ${\kappa }_{\min }^{}(G)>0$, then ${\pi }_1(\text{Xshort}5(G),o)$ is finite, where $\text{Xshort}5(G)$ is obtained by filling every simple cycle of length at most five. Equivalently, the normal subgroup generated by based simple $5$-cycle loops has finite index in ${\pi }_1^{\mathrm{G}\mathrm{L}\mathrm{M}\mathrm{Y}}(G,o)$. In higher degrees the situation is different: for each integer $r\ge 1$, the Cartesian product $T_r=C_5^{\mathrm{\square }r}$ has curvature $1\mathrm{/}(2r)$ on every edge and, for every field $\text{F}$, $${\text{PathH}}_p(T_r;\text{F})\cong {\text{F}}^{\left(\genfrac{}{}{0px}{}{r}{p}\right)}(0\le p\le r),$$ so strict positivity of Lin–Lu–Yau curvature does not force higher-dimensional GLMY path homology to vanish.