宽度定律与谱几何
Width Laws and Spectral Geometry
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中文总结 AI 辅助
该研究构建了随机宽度定律等的统一框架，证明了d维正交体的宽度累积量奇偶性定律等结果，区分了通用聚合等内容，实现了正交体重构及相关几何性质的分析
中文摘要 AI 辅助
我们构建了随机宽度定律、谱总体与几何重构的统一框架。对于一个d维正交体，我们证明了各球形宽度累积量的最大π⁻¹阶的精确奇偶性定律，其在所有维度与阶数下均存在非抵消性与符号性。前d个标量宽度矩可恢复无序边长向量，而d-1个矩通常不足以实现该恢复。每个拉普拉斯模式会生成一条辅助宽度定律，其上端点满足M_{n,a} = λ_n(a)^{1/2}/π。在高能条件下，模态坐标划分收敛于通用狄利克雷分布，而未平滑的测度值截止展开则在面尺度保留首个几何记忆。其单形矩在一个显式非零因子与一个独立的非对角项的作用下，确定了一个与基无关的投影梯度外尔张量，该张量可重构正交体。真实的边尺度跳跃阻碍了总原始截止的第三个系数的获取；精确的混合边界莫比乌斯反演可分离每个坐标层，并恢复具有更小余项的递归体-边界展开。超出正交体范畴，我们证明了一类典范线性-二次模型的方向标记可识别性，以及在生成元界条件下通过方向敏感脊矩实现有限恢复的可行性。在三维空间中，全局大圆演算给出了约化 zonotope 宽度密度的精确步长、折叠、端点折叠与角系数，包括显式的非简单角抵消项。所得结果区分了通用聚合、可恢复的几何记忆与剩余的标量逆问题。
英文摘要
We develop a common framework for random width laws, spectral populations, and geometric reconstruction. For a $d$-dimensional orthotope, we prove an exact parity law for the maximal $π^{-1}$-grade of every spherical width cumulant, including noncancellation and sign in all dimensions and orders. The first $d$ scalar width moments recover the unordered side vector, and $d-1$ moments are generically insufficient. Each Laplace mode generates an auxiliary width law whose upper endpoint satisfies $M_{n,a} = λ_n(a)^{1/2}/π$. At high energy the modal coordinate partitions converge to a universal Dirichlet law, while an unsmoothed measure-valued cutoff expansion retains the first geometric memory at face scale. Its simplex moment determines, up to an explicit nonzero factor and a separate off-diagonal argument, a basis-independent projector-gradient Weyl tensor that reconstructs the orthotope. Genuine edge-scale jumps obstruct a third coefficient for the total raw cutoff; exact mixed-boundary Mobius inversion isolates every coordinate stratum and restores a recursive bulk-boundary expansion with a smaller remainder. Beyond orthotopes, we prove direction-labelled identifiability for a canonical linear-quadratic class and finite recovery from direction-sensitive ridge moments under a generator bound. In dimension three, a global great-circle incidence calculus gives the exact step, fold, endpoint-fold, and corner coefficients of reduced zonotopal width densities, including an explicit non-simple corner cancellation. The results distinguish universal aggregation, recoverable geometric memory, and the remaining scalar inverse problem.
