Sum-of-Squares Degree Barriers for the Reweighted-Hinge Method in Robust Halfspace Learning: A Christoffel-Function Characterization

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 gamma-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 lambda 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 epsilon costs SoS degree ((1/epsilon)) or margin (sqrt (1/epsilon) /sqrt d), explaining why the (1/epsilon) margin Shen (2025) records is forced, with a weighted-Chebyshev reduction making the threshold 2t= ((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 eta 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 eta 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.