We give a characteristic-function formulation of Kunchenko's stochastic-polynomial construction for settings in which raw moments may fail to exist. In the finite-variance trigonometric case, the coefficients of the Kunchenko normal system are expressed through the characteristic function and its derivative. In the moment-free case, empirical characteristic functions on a fixed finite frequency grid define a bounded discrepancy geometry that remains meaningful for Cauchy, symmetric stable, and other heavy-tailed laws. We prove well-definedness and finite-grid almost sure consistency of this empirical characteristic-function geometry. We introduce the associated minimum-CF-distance estimator and establish its identifiability, strong consistency, and asymptotic normality on a fixed grid, with a covariance built from bounded trigonometric moments that stays finite even for Cauchy and stable laws; refining the grid increases the optimal-weight information monotonically to the Fisher information, so the estimator is asymptotically efficient in the dense-grid limit. We also relate bounded sine scores to weak stochastic-polynomial estimating equations. A small Lean 4 / Mathlib supplement checks selected deterministic identities underlying the bounded-score construction; convergence arguments and statistical interpretation remain outside the formalization.

翻译：我们给出了昆钦科随机多项式构造的特征函数公式，适用于原始矩可能不存在的情形。在有限方差三角情形下，昆钦科正规系统的系数通过特征函数及其导数表示。在无矩情形下，固定有限频率网格上的经验特征函数定义了一个有界差异几何结构，该结构对柯西分布、对称稳定分布及其他重尾分布仍然有意义。我们证明了该经验特征函数几何结构的良定义性及有限网格上的几乎必然一致性。我们引入相应的最小特征函数距离估计量，并建立了其在固定网格上的可识别性、强相合性及渐近正态性，其协方差由有界三角矩构成，即使对柯西分布和稳定分布仍然有限；细化网格会使最优权重信息单调增加至费舍尔信息，因此该估计量在稠密网格极限下是渐近有效的。我们还将有界正弦得分与弱随机多项式估计方程联系起来。一个简短的 Lean 4 / Mathlib 补充内容验证了支撑有界得分构造的若干选定确定性恒等式；收敛性论证与统计解释不在形式化验证范围内。