AI 中文总结
该研究对金几乎完美非线性函数的迹积切换在各偶维度下分类，证明了维度刚性定理，确定非平凡切换仅存在于4、6、8维，排除了更大偶维度的该切换机制。
AI 中文摘要
我们对金几乎完美非线性函数 $x\mapsto x^3$ 在每个偶维度下的自然标量迹积切换进行了完整分类。非平凡切换仅发生在 $n=4,6,8$：可允许系数分别为非零迹零元、乘法阶为9的6个元素以及 $\mathbb{F}_4^{*}$。对于所有偶 $n\geq10$，不存在非零可允许系数。该无限范围通过费马三次型上的加法特征估计排除，且对 $n=10,12$ 有精确有限桥接。$n=6,8$ 的原始系数列表较早出现在Arshad的学位论文中；我们的贡献是对其进行内在描述、证明维度一致的定理以及由此得到的维度刚性定理。我们还在8维下通过 $\mathbb{P}^{1}(\mathbb{F}_4)$ 对归一化秩2扩展进行分类。二元迹选择器在每个非基点射影点接受两个系数值，且8个被接受的标记切换恰好形成两个扩展仿射类，因此对应两个CCZ类。与中心无关的低秩导数准则将每个秩-$r$候选简化为对预计算禁集中的 $2^r-1$ 个成员资格测试。所得到的全局APN类是已知的；该结果描述了它们在金中心周围的局部组织，并排除了该切换机制在所有更大偶维度下的应用。
英文摘要
We completely classify a natural scalar trace-product switching of the Gold almost perfect nonlinear function $x\mapsto x^3$ in every even dimension. Nontrivial switchings occur only for $n=4,6,8$: the admissible coefficients are, respectively, the nonzero trace-zero elements, the six elements of multiplicative order nine, and $\mathbb{F}_4^{*}$. For every even $n\geq10$, no nonzero coefficient is admissible. The infinite range is excluded by additive-character estimates on a Fermat cubic, with exact finite bridges for $n=10,12$. The raw coefficient lists for $n=6,8$ appeared earlier in Arshad's dissertation; our contribution is their intrinsic description, a proof uniform in the dimension, and the resulting dimension-rigidity theorem. We also classify normalized rank-two extensions in dimension eight by $\mathbb{P}^{1}(\mathbb{F}_4)$. A binary trace selector accepts two coefficient values at each non-base projective point, and the eight accepted marked switchings form exactly two extended-affine, hence two CCZ, classes. A centre-independent low-rank derivative criterion reduces each rank-$r$ candidate to $2^r-1$ membership tests in precomputed forbidden sets. The global APN classes reached are known; the results describe their local organization around the Gold centre and rule out this switching mechanism in all larger even dimensions.
