Exact Blocky-Cover Exponents, Weak Equality Rank, and One-Sided Probabilistic Rank Asymmetries for Set-Disjointness, Orthogonal Vectors, and Promise Unique-SAT

Williams (FOCS 2024) introduced equality rank eqF(A) and weak equality rank weqF(A), proving weqR(Dd)(Dd)≤ 6d/6≈ 1.3480ᵈ on 2ᵈ× 2ᵈ Set-Disjointness Dd. While Nederlof and W{e}grzycki (STOC 2021) and Alman and Li (FOCS 2025) gave O((2/√3)ᵈ n)-time and O((2/√3)ᵈ n)-space Orthogonal Vectors (OV) algorithms via non-blocky OR-circuits of covering degree 2/√3---and D\"urr et al.\ (ICALP 2026) gave O(1.25ᵈ n) analytic and O(1.16ᵈ n) computer-aided bounds---the covering equality rank bc(Dd) (QOR(Dd)) and weqF(Dd) remained at O(6d/6). First, via a two-parameter H\"older inequality on k-way disjointness Dd,k, we prove bc(Dd,k)(Dd,k)≥γₖᵈ (d≥ 1), recovering γ₂=2/√3 and proving γₖ>(k-1)(k-1)/k (k≥ 3) and γₖ>2 (k≥ 4); conversely, via single-valued random s-separators (uniform and min-wise), unit-Jacobian double-integral saddle monotonicity, and diagonal slice aggregation, we prove without computer search (2/√3)ᵈ(Dd)≤ 2d(d+1)²(2/√3)ᵈ and bc(Dd)≤ O(d³log d·(2/√3)ᵈ) (beating 6d/6 for all d≥ 51), yielding weqF(Dd)(Dd) over all infinite or >3ᵈ-sized fields and deterministic $O(nd)$-space, O_η((2/√3+η)ᵈ n)-time OV (∀η>0). Second, we establish (wₘₐₓ(Dd))d=1⁴=(2,6,14,36), prove weqF(Dd)=bp(Dd)∈\2,2,3\ ($d=1,2,3$) over all fields, and on Ar,Q=ORᵣQ (Q≥ Q₀(r)) prove weqQ=weq_F2ʳ=bc=r while eqF=weqF₂=2ʳ-1. Third, over every field F and ε∈[0,1), zero-false-positive probabilistic rank equals (1-ε)2ᵈ. Fourth, on Promise Unique-SAT UM, weqF(fUSAT|UM)=Ω(M M/2)=Ω(2M/√M) over every field F.