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 $\operatorname{eq}_{\mathbb{F}}(A)$ and weak equality rank $\operatorname{weq}_{\mathbb{F}}(A)$, proving $\operatorname{weq}_{\mathbb{R}}(D_d)\le\operatorname{bc}(D_d)\le 6^{\lceil d/6\rceil}\approx 1.3480^d$ on $2^d\times 2^d$ Set-Disjointness $D_d$. While Nederlof and W\k{e}grzycki (STOC 2021) and Alman and Li (FOCS 2025) gave $\widetilde{O}((2/\sqrt{3})^d n)$-time and $\widetilde{O}((2/\sqrt{3})^d n)$-space Orthogonal Vectors (OV) algorithms via non-blocky OR-circuits of covering degree $2/\sqrt{3}$---and D"urr et al.\ (ICALP 2026) gave $\widetilde{O}(1.25^d n)$ analytic and $O(1.16^d n)$ computer-aided bounds---the covering equality rank $\operatorname{bc}(D_d)$ ($Q_{\mathrm{OR}}(D_d)$) and $\operatorname{weq}_{\mathbb{F}}(D_d)$ remained at $O(6^{d/6})$. First, via a two-parameter H"older inequality on $k$-way disjointness $D_{d,k}$, we prove $\operatorname{bc}(D_{d,k})\ge\operatorname{fbc}(D_{d,k})\ge\gamma_k^d$ ($d\ge 1$), recovering $\gamma_2=2/\sqrt{3}$ and proving $\gamma_k>(k-1)^{(k-1)/k}$ ($k\ge 3$) and $\gamma_k>2$ ($k\ge 4$); 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 $(2/\sqrt{3})^d\le\operatorname{fbc}(D_d)\le 2d(d+1)^2(2/\sqrt{3})^d$ and $\operatorname{bc}(D_d)\le O(d^3\log d\cdot(2/\sqrt{3})^d)$ (beating $6^{d/6}$ for all $d\ge 51$), yielding $\operatorname{weq}_{\mathbb{F}}(D_d)\le\operatorname{bc}(D_d)$ over all infinite or $>3^d$-sized fields and deterministic $O(nd)$-space, $O_\eta((2/\sqrt{3}+\eta)^d n)$-time OV ($\forall\eta>0$). Second, we establish $(w_{\max}(D_d))_{d=1}^4=(2,6,14,36)$, prove $\operatorname{weq}_{\mathbb{F}}(D_d)=\operatorname{bp}(D_d)\in\{2,2,3\}$ ($d=1,2,3$) over all fields, and on $A_{r,Q}=\mathrm{OR}_r\circ\mathrm{EQ}_Q$ ($Q\ge Q_0(r)$) prove $\operatorname{weq}_{\mathbb{Q}}=\operatorname{weq}_{\mathbb{F}_{2^r}}=\operatorname{bc}=r$ while $\operatorname{eq}_{\mathbb{F}}=\operatorname{weq}_{\mathbb{F}_2}=2^r-1$. Third, over every field $\mathbb{F}$ and $\varepsilon\in[0,1)$, zero-false-positive probabilistic rank equals $\lceil(1-\varepsilon)2^d\rceil$. Fourth, on Promise Unique-SAT $\mathcal{U}_M$, $\operatorname{weq}_{\mathbb{F}}(f_{\mathrm{USAT}}|_{\mathcal{U}_M})=\Omega(\binom{M}{\lfloor M/2\rfloor})=\Omega(2^M/\sqrt{M})$ over every field $\mathbb{F}$.