AI 中文总结
该研究针对树的控制数与拉普拉斯特征值的关系，解答了阶缺陷、等式结构等问题，得出阶数29为比值9/7首次可超的情况，明确次立方树的结构规律与有限机制。
AI 中文摘要
对于树T，令γ(T)为其控制数，μ(T)为区间[0,1)内的拉普拉斯特征值个数。严格不等式γ(T)/μ(T)<4/3留下三个关联问题：精确的有限阶缺陷、等式附近的结构，以及固定缺陷是否仅允许有限个局部障碍。我们回答了所有三个问题。对于每个n≥2，当|V(T)|=n时，max(7γ(T)-9μ(T))=floor((n-20)/9)。因此阶数29是比值9/7首次可被超过的情况。证明基于将Φ(T)=|V(T)|-20-9(7γ(T)-9μ(T))精确分解为非负整数项，结合清洁-收缩恒等式。对于次立方树，等式情况是具有完美匹配的树的典范提升。我们对Φ=42的每一层进行分类，在20、21、40、41、42处定位连续的相变。有限机制表述简单：移除可重复的匹配骨架片段后，固定松弛仅留下有界的异常核心。形式上，局部紧性定理在固定填充盈余和剩余负指数下，界定了每个非正则加权深分量。因此，对于每个固定k，每个Φ(T)=k的次立方树由有界端口缺陷核、任意兼容的正则块森林，以及至多floor(k/21)个清洁扩张组成。所有无界陈述通过符号证明；精确计算仅用于显界原子列表和独立有限验证。
英文摘要
For a tree T, let gamma(T) be its domination number and let mu(T) count the Laplacian eigenvalues in [0,1). The strict inequality gamma(T)/mu(T)<4/3 leaves open three linked questions: the exact finite-order defect, the structure near equality, and whether a fixed defect permits only finitely many local obstructions. We answer all three. For every n>=2, max_{|V(T)|=n} (7 gamma(T)-9 mu(T)) = floor((n-20)/9). Thus order twenty-nine is the first at which the ratio 9/7 can be exceeded. The proof is based on an exact decomposition of Phi(T)=|V(T)|-20-9(7 gamma(T)-9 mu(T)) into nonnegative integer terms, together with a clean-contraction identity. For subcubic trees, equality cases are canonical lifts of trees with perfect matchings. We classify every layer through Phi=42, locating successive phase transitions at 20, 21, 40, 41, and 42. The finiteness mechanism is simple to state: after the repeatable matched-skeleton pieces are removed, fixed slack leaves only a bounded exceptional core. Formally, a local compactness theorem bounds every nonordinary weighted deep component at fixed packing surplus and residual negative index. Consequently, for each fixed k, every subcubic tree with Phi(T)=k consists of a bounded ported defect kernel, an arbitrary compatible forest of ordinary tiles, and at most floor(k/21) clean expansions. All unbounded statements are proved symbolically; exact computation is used only for explicitly bounded atom lists and independent finite verification.
Comments54 pages, 10 figures. Exact fixed-order extremal law, canonical equality structures, fixed-defect finiteness, and an explicit phase atlas through slack 42. Reproducibility supplement: https://doi.org/10.5281/zenodo.22131333
