A certificate that removes outliers sees the data only through its low-degree moments, and an adversary exploits exactly this, hiding corruption where the clean data already looks typical, in the blind spot no bounded-degree test resolves. That blind spot turns out to have an exact size: the Christoffel function of the clean marginal, the very quantity modern data analysis thresholds to detect outliers, here read from the adversary's side as the corruption a bounded-degree certificate cannot remove. We turn this inversion into the organizing principle of the reweighted-hinge approach to robustly learning $γ$-margin halfspaces under malicious noise (Shen, 2025; Zeng and Shen, 2025): the governing resource is the Sum-of-Squares degree of the outlier-removal certificate, and the resolution principle states that the maximal corruption mass which can hide at a center $c$ from a degree-$2t$ certificate is exactly the Christoffel function $λ_{t+1}(c)$ of the clean marginal. Three consequences follow, all against the certificate method (not information-theoretic). A margin-degree tradeoff: certifying the dense pancake to error $ε$ costs SoS degree $Ω(\log(1/ε))$ or margin $Ω(\sqrt{\log(1/ε)}/\sqrt{d})$, explaining why the $\log(1/ε)$ margin Shen (2025) records is forced, with a weighted-Chebyshev reduction making the threshold $2t=Θ((|c|/s)^2)$ tight modulo one classical weighted-extremal estimate. A degree-$2$ outlier barrier: the resolution principle realized as an explicit instance on which degree $2$ is stuck at $η^{1/2}$ while degree $4$ escapes, locating the method's small breakdown rate in the degree, not the analysis. And a degree-$2t$ algorithm tracing the frontier $η^{1-1/2t}$ (recovering Shen (2025) at $t=1$), whose gain is an explicit constant, capped by the pancake density and shown unimprovable by the degree-$2$ barrier.

翻译：一种通过仅观察低阶矩来剔除异常值的证书方法，其盲区恰被对手利用：在干净数据本已典型的区域隐藏污染，而该区域恰好是有界度检验无法分辨的。该盲区存在精确的规模：即干净边缘分布的Christoffel函数——这个现代数据分析中用于检测异常值的阈值量，在此处从对手视角被重新解读为有界度证书无法剔除的污染量值。我们将这种视角转换确立为重加权铰链法在恶意噪声下鲁棒学习γ-间隔半空间问题中的组织原则（Shen, 2025; Zeng and Shen, 2025）：其核心约束是异常值剔除证书的平方和度，而解构原理表明，在中心点c处能够躲过2t度证书的最大污染质量正好等于干净边缘分布的Christoffel函数λ_{t+1}(c)。由此推导出三个针对证书方法（非信息论意义）的结论：第一，间隔-度权衡——为达到误差ε需对致密饼状分布进行认证，所需平方和度为Ω(log(1/ε))或间隔为Ω(√log(1/ε)/√d)，这解释了Shen（2025）记录中log(1/ε)间隔的必然性，并通过加权切比雪夫约化使得阈值2t=Θ((|c|/s)^2)在经典加权极值估计范围内达到紧致。第二，2度异常值障碍——通过显式实例证明：2度方法被困于η^{1/2}水平，而4度方法可突破此界限，从而将方法的小崩溃率定位在度数层面而非分析层面。第三，描述2t度算法的前沿边界η^{1-1/2t}（在t=1时恢复Shen（2025）结果），其增益显式常数受致密饼状分布密度上限约束，且被证明无法通过2度障碍突破。