An SoS Entropy Dichotomy via Windowed Hypercontractivity

Abstract

We prove an entropy versus degree dichotomy for low-degree tests and the Sum-of-Squares (SoS) hierarchy on a calibrated window after a gadget layer. For a target distribution \(\mu\) and a product-like proxy \(u\), we study the low-degree discrepancy \(\Delta_k(\mu,u)\), defined as the optimal distinguishing advantage of degree \(\le k\) polynomial tests. Using a bias-orthonormal Walsh basis and a test-moment equivalence on the window, we relate \(\Delta_k\) (up to constants) to the squared \(\ell_2\) mass of signed low-degree moments. Calibrated pseudoexpectations match \(u\) on all moments of degree \(\le k\), hence test discrepancy equals SoS pseudoexpectation deviation. Under bias, product, and width assumptions along a switching path, a windowed Bonami--Beckner inequality yields hypercontractive tail bounds. Combining these with moment matching, we obtain a discrepancy-to-degree theorem: if \(\Delta_k(\mu,u) \ge n^{-\beta}\), then any polynomial-calculus or SoS refutation separating \(\mu\) from \(u\) requires degree \(\Omega(k)\). Instantiating \(k = c \log n\) gives an explicit \(\Omega(\log n)\) SoS degree lower bound whenever \(\Delta_k \ge n^{-\eta}\). All constants are explicit and depend only on calibrated window parameters. This work provides the SoS/low-degree core and complements a prior calibration blueprint; a companion paper lifts the windowed statements to full distribution families.