\begin{abstract}
Alman and Song showed that attention with head dimension $O(\log n)$ can be approximated in almost linear
time when the entries are $o(\sqrt{\log n})$, and needs quadratic time at $\Theta(\sqrt{\log n})$ under the Strong
Exponential Time Hypothesis. In October 2026 the Exponential Time Hypothesis was refuted, and with it SETH. The
quadratic lower bound now rests on the hardness of Orthogonal Vectors in dimension $c\log n$ (LDOV), which no
open hypothesis about satisfiability is known to imply. We measure an instance by its spread $W$, the largest gap
between two logits of a row. At every nontrivial constant precision and in every dimension between $\omega(\log n)$
and $n^{o(1)}$, we show that under LDOV the time exponent of attention is exactly $1+\min\{W/\ln n,1\}$: uniform
sampling of keys is optimal, the cost is $\min\{n\,e^{W},n^2\}$ up to factors $n^{o(1)}$, and the transition to
quadratic time sits at $W=\ln n$, the spread at which one key can outweigh all others. At polynomial precision and in
dimension $O(\log n)$ the threshold becomes $\Theta(\log n)$, with a constant that tends to zero as the precision grows,
and the dimension moves it: at precision $n^{-1-O(1/\log\log n)}$, spread $O(\log n/\log\log n)$ is hard in dimension
$(\log n)^{1+\Omega(1)}$ but easy in dimension $O(\log n)$. From satisfiability we obtain hardness at spread
$(1+\theta)\ln n$ in dimension $n^{o(1)}$, provided the hard CNF formulas have $N^{c_0}$ clauses for some $c_0<2$, and we
show that the construction behind this bound, which builds one instance for each proof of a Merlin--Arthur protocol
for disjointness, needs $c_0<2$ for every such protocol. For the entry bound, every
reduction that embeds Orthogonal Vectors vector by vector, with polynomial blowup and logit gap $\Omega(\log n)$,
yields a Merlin--Arthur protocol for disjointness; by Klauck's lower bound, such reductions cannot derive
$B=\Theta(\sqrt{\log n})$ from Orthogonal Vectors in dimension $(\log n)^{1+\Omega(1)}$. We also map the conditional lower
bounds for attention in the literature to the hypotheses that now support them, and find two families whose
hypotheses are false, one of them in part of its parameter range.
\end{abstract}