从问题出发，走到证明¶
探索算法的边界，记录成立的证明，也留下让猜想失败的反例。
Algorithm problems and research notes, with precise assumptions, complete proofs, and reproducible evidence. Read in Chinese or English.
重点研究 / Featured research¶
已给出证明 · Proof provided
四邻接网格森林平移匹配的三次方复杂度上界¶
A cubic time bound for translation matching in four-neighbor grid forests
对只在新位移处筛选的候选列表算法，完整历史的收费证明去掉了对数因子，得到一般森林上的 O(n³+m) 上界。
Complete-history charging removes the logarithmic factor for candidate-list filtering at new displacements, giving O(n³+m) time on general forests.
已给出证明 · Proof provided
诱导四邻接网格森林不一定能通过补点扩充成树¶
Induced four-neighbor grid forests need not extend to trees by adding vertices
保持原位置与全部四邻接边时，补点不一定能把森林连成树；这种障碍可以扩展到线性数量、每棵线性大小的路径。
With positions and all induced edges preserved, adding vertices need not connect a forest into a tree—even for linearly many paths of linear size.
已给出证明 · Proof provided
仅对角邻接的网格森林：奇偶拆分与三次方界¶
Diagonal-only grid forests: parity decomposition and cubic bounds
只允许四个对角方向时，两种奇偶类各自同构于四邻接网格，保留原检查次数上界与三次方最坏量级；补树障碍也可迁移。
With diagonal directions only, each parity class is isomorphic to an orthogonal grid, preserving the check bound, cubic worst-case order, and tree-extension obstructions.
已给出证明 · Proof provided
八邻接诱导森林：三次方复杂度仍然成立¶
Eight-neighbor induced forests retain the cubic bound
同时允许正交和对角方向时，秩四方向词保留完整历史收费与有限第二轴引理，得到相同的检查次数上界和三次方最坏量级。
With both orthogonal and diagonal moves, rank-four direction words preserve complete-history charging and the finite second-axis lemma, giving the same exact check bound and cubic worst-case order.
已给出证明 · Proof provided
顶点与路径障碍：四邻接、八邻接仍有三次方界¶
Vertex and edge obstacles: cubic bounds for four and eight neighbors
只要求实际开放边构成森林时，修改重复位置规则并使用公共开放边交域，仍保留原精确检查次数上界和三次方最坏量级。
When only the actual open-edge graph is a forest, revised revisit handling and common open-edge intersections preserve the exact check bound and cubic worst-case order.
选择你的阅读路径 / Find your starting point¶
想了解结论：从成果摘要开始，先看模型和适用边界，再进入完整证明。
想复核推导：沿证明中的引理与算例阅读，再运行附带的验证程序。
想继续研究：在成果目录查看已否定猜想和历史探索，避免重复走过的弯路。
Start with a result summary, follow its proof, then inspect the evidence. The research directory distinguishes current proofs, refuted conjectures, and historical experiments.
继续扩展 / Continue the research¶
新的结论、推广或开放问题使用同一套文档与索引结构。每项结果保留稳定标识，关联原问题、证明和验证材料。
新增成果与维护说明 · 引用与固定版本 · Agent 阅读导航 · 机器可读索引