AI 中文总结
研究错排熵算子，证明其是变量单调变化下绝对连续定律的等变变换且动力学可解，阐述其在迭代、连续流等方面的特性，通过变分原理等得出相关结论，还涉及提升到维拉索罗代数对偶及多维度依赖性等内容。
AI 中文摘要
错排熵算子通过自身累积分布函数的固定轮廓对概率密度进行重新加权，仅通过秩起作用。我们证明这些算子恰好是在变量单调变化下绝对连续定律的等变变换，且它们通过区间映射复合，其动力学完全可解。迭代将每个定律凝聚到中位数上，具有通用的柯尼希斯极限定律；连续流可用封闭形式求解；分布函数服从过阻尼正弦 - 戈登方程，其唯一稳态定律双曲正割是全局稳定的扭结。变分原理选择规范核，更新是贝叶斯后验，对每个定律恰好花费一位成本，酉提升为每个定律配备等谱斯特姆 - 刘维尔系统和量子地毯。大数定律证明了具有跳跃的任意实有界变差数据的薛定谔密度的分形性。提升到维拉索罗代数对偶，归一化后，定律与非共轭希尔势、尾部与共形权重、柯西族与特殊轨道相关。随机相位产生精确采样器和乘法混沌，在弱 tempered 区域对数相关；在多个维度中，依赖性分层为守恒相互作用、由最大相关性驱动的扭转和平坦的Sinkhorn 传输。
英文摘要
A derangetropy operator reweighs a probability density by a fixed profile of its own cumulative distribution function, acting through ranks alone. We prove that these operators are precisely the transformations of absolutely continuous laws equivariant under monotone changes of variable, and that they compose through interval maps, making their dynamics exactly solvable: iteration condenses every law onto its median with a universal Koenigs limit law, the continuous flow is solvable in closed form with the Cauchy family as invariant hyperbolic manifold, and balanced against diffusion the distribution function obeys an overdamped sine-Gordon equation whose unique steady law, the hyperbolic secant, is a globally stable kink. A variational principle selects the canonical kernel, the squared ground state of the Dirichlet Laplacian on the unit interval, whose update is a Bayesian posterior costing exactly one bit for every law, and a unitary lift equips each law with an isospectral Sturm-Liouville system and a quantum carpet: a strong law of large numbers for the critical Sobolev mass, with sharp constant given by Wiener's jump statistic, proves the fractality of the Schrödinger density for arbitrary real bounded-variation data with a jump, previously known only for rational step data. A lift to the dual of the Virasoro algebra, at central charge one, a normalization, identifies laws with disconjugate Hill potentials, tails with conformal weights, and the Cauchy family with the exceptional orbit. Randomized phases yield an exact sampler and a multiplicative chaos, log-correlated in the weakly tempered regime; in several dimensions, dependence stratifies into a conserved interaction, a torsion driven by maximal correlation, and a flat Sinkhorn transport.
