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 \(μ\) and a product-like proxy \(u\), we study the low-degree discrepancy \(Δ_k(μ,u)\), 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 \(Δ_k\) (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 \(Δ_k(μ,u) \ge n^{-β}\), then any polynomial-calculus or SoS refutation separating \(μ\) from \(u\) requires degree \(Ω(k)\). Instantiating \(k = c \log n\) gives an explicit \(Ω(\log n)\) SoS degree lower bound whenever \(Δ_k \ge n^{-η}\). 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.

41 pages, 2 tables. Part II (IECZ-II) of a series; complements IECZ-I (calibration/blueprint) and a planned IECZ-III (lifting to families). Compiles with pdfLaTeX; bibliography provided as .bbl. A minimal Python helper for packaging sources accompanies the submission; an archived code bundle for reproducibility will be released on Zenodo (DOI to be added in a revised version)