Resolution Width Lifts to Near-Quadratic-Depth Res(⊕) Size

Abstract

We show that for any unsatisfiable CNF formula φ that requires resolution refutation width at least w, and for any 1-stifling gadget g (for example, g = MAJ₃), (1) every resolution-over-parities (Res(⊕)) refutation of the lifted formula φ∘g of size at most S has depth at least Ω(w²/log S); (2) every Res(⊕) refutation of the lifted formula φ∘g has size Ω(w²).
The first result substantially extends and simplifies all previously known lifting theorems for bounded-depth Res(⊕). The lifting result of Itsykson and Knop [Dmitry Itsykson and Alexander Knop, 2026] requires gadgets of logarithmic size and applies only to refutations of depth at most O(nlog n), whereas our result applies to nearly quadratic depth. The liftings of Bhattacharya and Chattopadhyay [Sreejata Kishor Bhattacharya and Arkadev Chattopadhyay, 2025] and of Byramji and Impagliazzo [Farzan Byramji and Russell Impagliazzo, 2025] apply to nearly quadratic depth as well, but rely on a much stronger assumption of (Ω(n),Ω(n))-DT-hardness, which is far less standard than large resolution width.
Our proof combines the random-walk-with-restarts method of Alekseev and Itsykson [Yaroslav Alekseev and Dmitry Itsykson, 2025] with a new idea: the random walk is defined relative to the structure of the refutation graph, rather than by a distribution on inputs induced by the formula.
Using this technique, we substantially strengthen the supercritical size-depth tradeoff of Itsykson and Knop [Dmitry Itsykson and Alexander Knop, 2026], both by improving the depth lower bound and by reducing the size of the separating formulas to polynomial in the number of variables, with the latter resolving an open question posed in [Dmitry Itsykson and Alexander Knop, 2026]. In particular, we construct a family of polynomial-size formulas that admit polynomial-size resolution refutations, while any Res(⊕) refutation of depth o(n²/log⁴ n) necessarily has superpolynomial size.
Our second result yields a pure quadratic lower bound on the size of Res(⊕) refutations, improving upon the previously known near-quadratic lower bound of [Farzan Byramji and Russell Impagliazzo, 2025].