AI 中文总结
该研究探讨树级MHV引力分子的唯一性，通过有限维计算分析七点、八点情形的解空间，结合玻色对称性、因子化条件等确定Hodges分子，相关计算经Lean验证。
AI 中文摘要
我们研究树级MHV引力分子是否由其次数以及对每一对满足⟨ij⟩=[ij]=0的条件所确定。利用旗簇标准单项式基和Sₙ分解的限制映射，该问题可简化为精确的有限维计算。在七点情形，我们得到W₇,ℚ≅S⁽²,1⁵⁾⊕S⁽1⁷⁾，Hodges分子张成符号分支，而六维hook空间给出额外的代数解，因此对理想条件无法确定唯一代数解，但玻色对称性选出Hodges直线。在八点情形，对理想条件与玻色对称性留下二维交替空间，同 helicity的BCW标度、归一化共线因子化及领头软系数施加相同线性条件，选出Hodges直线。我们还证明，在任意多重度下，交替固定次数的分子由其在一条共线边界上的完整值确定，该边界上标记腿及其旋量比固定，结合标准因子化，这在固定公共分母ansatz内确定分子至多相差归一化。所有秩和理想成员计算均使用精确整数或有理算术，其有限维结果在Lean中单独验证。
英文摘要
We study whether a tree-level MHV gravity numerator is determined by its degree and by vanishing on $\langle ij\rangle=[ij]=0$ for every pair. A flag-variety standard-monomial basis and an $S_n$-resolved restriction map reduce the problem to exact finite-dimensional calculations. At seven points we find $W_{7,\mathbb{Q}}\simeq S^{(2,1^5)}\oplus S^{(1^7)}$. The Hodges numerator spans the sign summand, while the six-dimensional hook gives additional algebraic solutions. The pair-ideal conditions therefore do not determine a unique algebraic solution, but Bose symmetry selects the Hodges line. At eight points, pair-ideal conditions and Bose symmetry leave a two-dimensional alternating space. Same-helicity BCFW scaling, normalized collinear factorization, and the leading soft coefficient impose the same linear condition and select the Hodges line. We also prove that, at arbitrary multiplicity, an alternating fixed-degree numerator is determined by its full value on one collinear boundary with the marked legs and their spinor ratio fixed. Together with standard factorization, this determines the numerator up to normalization within the fixed-common-denominator ansatz. All rank and ideal-membership calculations use exact integer or rational arithmetic, and their finite-dimensional consequences are checked separately in Lean.
Comments44 pages, 3 figures. Reproducibility materials and a Lean 4 companion are available at https://github.com/ybzhang-nxu/autoNMHV
