An SoS Entropy Dichotomy via Windowed Hypercontractivity

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 ${\mathrm{\Delta }}_k(\mu ,u)$, 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 ${\mathrm{\Delta }}_k$ (up to constants) to the squared ${\mathrm{\ell }}_2$ 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 ${\mathrm{\Delta }}_k(\mu ,u)\ge n^{-\beta }$, then any polynomial-calculus or SoS refutation separating $\mu$ from $u$ requires degree $\mathrm{\Omega }(k)$. Instantiating $k=c\log n$ gives an explicit $\mathrm{\Omega }(\log n)$ SoS degree lower bound whenever ${\mathrm{\Delta }}_k\ge n^{-\eta }$. 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.