AI 中文总结
该研究证明了SO(n)上的坐标平面Kac游走的全变差混合时间为Θ(n²)，通过离散Malliavin耦合等方法改进了此前的O(n²log n)估计，移除了对数项。
AI 中文摘要
令N=二项式系数(n,2)=SO(n)的维度。我们证明，SO(n)上的坐标平面Kac游走的全变差混合时间为N阶：对每个固定的0<ε<1，t_mix^(n)(ε)=Θ_ε(n²)。下界是N步前的维度奇异性障碍。上界将此前已知的O(n²log n)估计中的最终对数项移除。证明结合了离散Malliavin耦合、低度伪混合输入以及对导数壳的无对数新分析。其静态核心是针对物理五素数的电路锚定、任意谱根/传递恒等式：为每个原始电路保留一个归一化基，可在不产生人工切割代价的情况下同时实现标量重粘。其时间核心是精确的时序演算：被动单元素运行获得上同调/黎曼增益，根中断分量在取绝对值前由顶点标记分组分配。曲率被分解为纯外尔（Weyl）部分和里奇（Ricci）部分，并保持其物理张量类型。全外尔分组保留完整的N⁻¹资源；混合分组包含类型化的O(n⁻¹/²)里奇亏空；最终无分组里奇单元通过对其两个根命中电路的联合不变列估计以及标记根时间的精确因果恢复来闭合。这些估计得出O(n)平方的一阶导数壳，并通过对数阶数得到可求和的全阶标记壳展开。在cN步后，所得得分能量为O(n/c²)。随后，加权浸没分部积分论证和哈尔（Haar）对数索伯列夫不等式给出均匀全变差上界。未断言存在截断剖面或截断窗口。
英文摘要
Let $N=\binom n2=\dim\mathrm{SO}(n)$. We prove that the coordinate-plane Kac walk on $\mathrm{SO}(n)$ has total-variation mixing time of order $N$: for every fixed $0<\varepsilon<1$, \[ t_{\mathrm{mix}}^{(n)}(\varepsilon)=Θ_\varepsilon(n^2). \] The lower bound is the dimensional singularity obstruction before $N$ steps. The upper bound removes the final logarithm from the previously known $O(n^2\log n)$ estimate. The proof combines the discrete Malliavin coupling and low-degree pseudo-mixing inputs with a new log-free analysis of the derivative shells. Its static core is a circuit-anchored, arbitrary-spectrum root/pass identity for the physical five-box prime. Keeping one normalization base per original circuit permits simultaneous scalar regluing without paying for artificial cuts. Its temporal core is an exact chronological calculus: passive singleton runs acquire a coboundary/Riesz gain, while root-interrupted components are allocated by vertex-labelled packets before absolute values are taken. The curvature split into pure-Weyl and Ricci parts is kept at its physical tensor type. All-Weyl packets retain a full $N^{-1}$ resource; mixed packets contain a typed $O(n^{-1/2})$ Ricci debit; and the final packetless Ricci cell is closed by a joint invariant-column estimate on its two root-hit circuits and an exact causal restoration of the marked root time. These estimates yield an $O(n)$ squared first-derivative shell and a summable all-order marked-shell expansion through logarithmic degree. The resulting score energy is $O(n/c^2)$ after $cN$ steps. A weighted submersion integration-by-parts argument and the Haar log-Sobolev inequality then give the uniform total-variation upper bound. No cutoff profile or cutoff window is asserted.
CommentsWithdrawn by author: The main result requires further rigorous verification before publication
