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Horizon Research（地平线研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文反驳了1-渗漏正半定边删除猜想，提出并证明各级别渗漏下边编辑参数差至多2的尖锐定理，并给出极值例子与验证。
AI 中文摘要
这项工作反驳了1-渗漏正半定边删除猜想，并用一个尖锐定理取而代之。对于每个渗漏级别$\ell$和边$e$，有$|Z^+_{(\ell)}(G)-Z^+_{(\ell)}(G-e)|\le 2$。更一般地，如果两个图仅在两端点都在$S$中的边上不同，则它们的参数之差至多为$|S|$。一个端点敏感的细化表明，当$G-e$的某个最小集合包含$e$的一个端点时，增加至多为一。对于每个正渗漏级别，两个符号都是尖锐的。将两个$K_{\ell+1}$的副本通过一座桥连接，得到$Z^+_{(\ell)}(G)=2\ell$和$Z^+_{(\ell)}(G-e)=2\ell+2$。对于每个$\ell\ge 2$，一个阶为$2\ell+3$的连通团-叶对给出相反的差值。剩余的正差一渗漏情形由九个顶点上的连通图达到，其中$Z^+_{(1)}(H)=4$和$Z^+_{(1)}(G)=6$。显式强迫序列、堡垒证书和精确验证器检查了有限极值例子，并对一般结果进行了压力测试。
英文摘要
This work disproves the 1-leaky positive semidefinite edge-deletion conjecture and replaces it with a sharp theorem. For every leak level $\ell$ and edge $e$, one has $|Z^+_{(\ell)}(G)-Z^+_{(\ell)}(G-e)|\le 2$. More generally, if two graphs differ only on edges with both endpoints in $S$, their parameters differ by at most $|S|$. An endpoint-sensitive refinement recovers an increase of at most one whenever some minimum set for $G-e$ contains an endpoint of $e$. Both signs are sharp for every positive leak level. Joining two copies of $K_{\ell+1}$ by a bridge gives $Z^+_{(\ell)}(G)=2\ell$ and $Z^+_{(\ell)}(G-e)=2\ell+2$. For every $\ell\ge 2$, a connected clique-leaf pair of order $2\ell+3$ gives the opposite difference. The remaining positive-difference one-leak case is attained by connected graphs on nine vertices with $Z^+_{(1)}(H)=4$ and $Z^+_{(1)}(G)=6$. Explicit forcing sequences, fort certificates, and an exact verifier check the finite extremal example and stress-test the general results.
Comments9 pages, 1 figure, 6 tables. Supplementary Materials included
