Majority and Torus Polynomials: Bounded-Lift Lower Bounds and Explicit Constructions

Abstract

Bhrushundi et al. (2019) introduced torus polynomial approximation as an approach to separating Majority from $ACC^0$. In this model, approximation is measured modulo integers. Krishan and Vishwanathan (2026) developed a duality framework for lower bounds and posed five open problems. We resolve two of these, concerning integer lifts and explicit polynomial approximations. An integer lift records the integer shifts that turn a torus approximation into an ordinary real approximation. We show that a degree-d torus polynomial approximating Majority with error at most 1/8 and lift contained in an interval of width $B$ must satisfy $d(B + 1) = Ω(n)$. This rules out low-degree approximating polynomials with lifts in {-1, 0, 1}, answering their Open Problem 2. We also answer Open Problem 5 by giving, for every Boolean function on $n \geq 1$ variables, an explicit torus polynomial approximation of degree at most $n - 1$. Its error is optimal: zero when the accepting set has even cardinality and $2^{-n-1}$ otherwise. The results presented were discovered and formalized in Lean using GPT-6 Astra.