AI 中文总结
本文通过代数枚举局部删除规则，证明Thompson群$F$的密度下界大于3.50074529，并证明固定窗口最优规则等价于加权最密子图问题。
AI 中文摘要
我们研究Thompson群$F$的标准Cayley图中Belk和Brown的标记森林集合上的局部删除。一个区间选择规则后接不相交的三顶点和两顶点删除，其资格取决于树根处的分裂和合并操作。我们通过一个九状态表和初等首达方程来评估该构造。特别地，根敏感的上下文频率允许闭式表达式，而无需枚举大型乘积自动机。所得的有限子图给出\\[ \operatorname{dens}(F;\{x_0,x_1\})>3.50074529. \\] 证明中使用的所有量都是$\mathbb Q(\sqrt3,\sqrt{2\sqrt3-1})$的显式元素。我们还给出了粗森林类别的联合根-子定律，并证明了在固定窗口上所有保留规则的最优值是一个精确的加权最密子图问题。其上限具有初等边分配证书。一个显式证书证明，读取标记类别、其直接右邻以及左侧任意固定数量的类别的规则不能改进极限密度$7/2$。
英文摘要
Let $Γ$ be the Cayley graph of Thompson's group $F$ with its standard generators. Belk and Brown constructed finite marked-forest subgraphs of limiting density $7/2$, and Guba showed that deleting certain low-degree vertices gives density greater than $3.5004$. We refine this approach in two directions. First, a deterministic interval-deletion rule followed by root-sensitive triple and pair deletions yields finite induced subgraphs with \[ \operatorname{dens}(Γ)>3.50074529. \] The interval rule is evaluated by a nine-state recurrence, while the simultaneous deletion conditions created by split and merge operations are computed exactly by a common-suffix first-passage argument. Second, we determine the joint distribution of the categories of a tree and its two children and reduce optimization over every fixed category-window retention rule to a weighted densest-subgraph problem. An explicit edge-allocation certificate shows that, for every fixed $a\ge0$, the optimal limiting density of a rule depending on the window $[-a,1]$ is exactly $7/2$. Thus arbitrary finite left context together with one right-hand category does not improve the Belk--Brown limit, whereas the root-sensitive whole-segment construction does. All constants are explicit elements of $\mathbb Q(\sqrt3,\sqrt{2\sqrt3-1})$.
Comments21 pages
