AI 中文总结
研究布尔格伪根有理生成问题，通过分析与\(P_4\)相关种子映射的形式逆，证明其非自由有理，利用矩阵次数论证得出矛盾，将障碍扩展到含诱导\(P_4\)的图。
AI 中文摘要
对于与四顶点路径\(P_4\)相关的邻域种子，菱形运算无法恢复所有布尔格伪根。自由斜域中无限制有理运算的相应问题更微妙：种子映射有可逆线性化，因此在每个一般标量点附近有唯一形式逆。我们证明此形式逆不是自由有理的。\(2×2\)矩阵输出的对称单参数曲线有形式逆，其系数域包含在\(\mathbb{Q}(t)\)上次数为三的元素。二次泡利代数中的精确消元产生不可约三次式。其共轭逆分支无分歧，迫使种子映射在每个\(n\geq2\)大小中的一般矩阵次数至少为三。这与任何自由有理逆的一次数结果相矛盾。相同的矩阵次数论证将障碍扩展到每个包含诱导\(P_4\)的图。
英文摘要
For the neighborhood seed associated with the four-vertex path $P_4$, the diamond operations do not recover all Boolean-lattice pseudo-roots. The corresponding question for unrestricted rational operations in the free skew field is subtler: the seed map has an invertible linearization and therefore a unique formal inverse near every generic scalar point. We prove that this formal inverse is not free rational. A symmetric one-parameter curve of $2 \times 2$ matrix outputs has a formal inverse whose coefficient field contains an element of degree three over $\mathbb{Q}(t)$. An exact elimination in a quadratic Pauli algebra produces the irreducible cubic. Its conjugate inverse branches are unramified, forcing the generic matrix degree of the seed map to be at least three in every size $n \ge 2$. This contradicts the degree-one consequence of any free rational inverse. The same matrix-degree argument, without specializing a hypothetical inverse, extends the obstruction to every graph containing an induced $P_4$.
Comments12 pages, ancillary Python file, corrected notation errors
