Sparse 1% agreement testers without ℓ∞-expansion

Develop agreement testers in the 1% regime on sparse complexes without assuming ℓ∞-expansion by completing the argument under reverse hypercontractivity and spectral gap of the down-up walk alone.

Background

The paper gives an optimal 1% regime test (Z-test) under the stronger ℓ∞-expansion condition, which holds for dense complexes (e.g., products, random complexes). The core argument, however, only needs reverse hypercontractivity and a spectral gap for the down-up walk.

Removing the ℓ∞ assumption would extend optimal soundness to sparse HDX and strengthen connections with recent topological characterizations of testers.