  The theory of holographic algorithms introduced by Valiant represents a novel
approach to achieving polynomial-time algorithms for seemingly intractable
counting problems via a reduction to counting planar perfect matchings and a
linear change of basis. Two fundamental parameters in holographic algorithms
are the \emph{domain size} and the \emph{basis size}. Roughly, the domain size
is the range of colors involved in the counting problem at hand (e.g. counting
graph $k$-colorings is a problem over domain size $k$), while the basis size
$\ell$ captures the dimensionality of the representation of those colors. A
major open problem has been: for a given $k$, what is the smallest $\ell$ for
which any holographic algorithm for a problem over domain size $k$ "collapses
to" (can be simulated by) a holographic algorithm with basis size $\ell$? Cai
and Lu showed in 2008 that over domain size 2, basis size 1 suffices, opening
the door to an extensive line of work on the structural theory of holographic
algorithms over the Boolean domain. Cai and Fu later showed for signatures of
full rank that over domain sizes 3 and 4, basis sizes 1 and 2, respectively,
suffice, and they conjectured that over domain size $k$ there is a collapse to
basis size $\lfloor\log_2 k\rfloor$. In this work, we resolve this conjecture
in the affirmative for signatures of full rank for all $k$.
