  This note draws conclusions that arise by combining two recent papers, by
Anuj Dawar, Erich Gr\"adel, and Wied Pakusa, published at ICALP 2019 and by
Moritz Lichter, published at LICS 2021. In both papers, the main technical
results rely on the combinatorial and algebraic analysis of the invertible-map
equivalences $\equiv^\text{IM}_{k,Q}$ on certain variants of
Cai-F\"urer-Immerman (CFI) structures. These
$\equiv^\text{IM}_{k,Q}$-equivalences, for a number $k$ and a set of
primes~$Q$, refine the well-known Weisfeiler-Leman equivalences used in
algorithms for graph isomorphism. The intuition is that two graphs
$G\equiv^\text{IM}_{k,Q}H$ cannot be distinguished by iterative refinements of
equivalences on $k$-tuples defined via linear operators on vector spaces over
fields of characteristic $p \in Q$. In the first paper it has been shown, using
considerable algebraic machinery, that for a prime $q \notin Q$, the
$\equiv^\text{IM}_{k,Q}$ equivalences do not distinguish between non-isomorphic
CFI-structures over the field $\mathbb{F}_q$. In the second paper, a similar
but not identical construction for CFI-structures over the rings
$\mathbb{Z}_{2^i}$ has, again by rather involved combinatorial and algebraic
arguments, been shown to be indistinguishable with respect to
$\equiv^\text{IM}_{k,\{2\}}$. Together with earlier work, this second result
separates rank logic from polynomial time. We unify the two approaches and
prove that CFI-structures over the rings $\mathbb{Z}_{2^i}$ are
indistinguishable with respect to $\equiv^\text{IM}_{k,\mathbb{P}}$, for the
set $\mathbb{P}$ of all primes. This implies the following two results.
  1. There is no fixed $k$ such that the invertible-map equivalence
$\equiv^\text{IM}_{k,\mathbb{P}}$ coincides with isomorphism on all finite
graphs.
  2. No extension of fixed-point logic by linear-algebraic operators over
fields can capture polynomial time.
