Improved Blocky Partition, Cover, and Equality-Rank Bounds for Set Disjointness via Kronecker Rebalancing

A binary matrix B∈\0,1× n is blocky (an equality matrix) if its $1$-entries form pairwise row- and column-disjoint rectangles. For Set Disjointness Dd=D₁⊗ d∈\0,1\2ᵈ× 2ᵈ, Williams (FOCS 2024) bounded blocky cover number bc(Dd) and weak equality rank weqR(Dd) by O(6d/6)≈ O(1.3480ᵈ) and blocky partition number bp(Dd) and exact equality rank brR(Dd) by O(5d/5)≈ O(1.3797ᵈ). While multi-bucket separator circuits give Õ(n·(2/√3)ᵈ)-time and -space Orthogonal Vectors (Nederlof and W{e}grzycki, STOC 2021; Alman and Li, FOCS 2025) and unrestricted depth-$2$ circuits solve #OV in o(n· 1.25ᵈ) time (Alman and Li, 2025; Ye, 2026), O(5d/5) and O(6d/6) remained the best published bounds on bp(Dd) and bc(Dd) (with the exact cover base 2/√3 established in a companion paper). First, via nonnegative LP dual vertex potentials we prove fbc(Dd)=bp^*(Dd) for 1≤ d≤ 5 with values $(3/2,7/4,23/10,35/13,205/62)$ (showing bc(D₅),bp(D₅)∈\4,5\, alongside bp(D₄)=4); Kronecker submultiplicativity and greedy cover rounding yield bc(Dd)=O(d·(205/62)d/5)=O(d· 1.27021ᵈ) (and the same bound on weqF(Dd) over every infinite or >3ᵈ-sized field F). Second, we introduce a palette Kronecker rebalancing LP τbp(M) over exact rectangle partitions and blocky packings, proving τbp(D₁,D₂,D₃)=(3/2,11/6,23/10), bp(A⊗ D₃)≤ 276(A)/120 (refined to 92(A)/40), and bp(Dd)<3·(23/10)d/3<3· 1.320007ᵈ (<11·(23/10)d/3 from 120→ 276 alone), which bounds brR(Dd) (R≠ 0) and sprF(Dd),weqF(Dd) over every field F.