The Koizumi-Liu conjecture (Koizumi and Liu 2026) proposed eventual sign alternation for the magnitude of the tope graph of every real arrangement of hyperplanes.
The vertices of this graph are the chambers of the
arrangement, and two chambers are adjacent when they are separated by one hyperplane.
For a finite graph 
, let 
be the matrix with entries
,
where 
is the graph distance. Its magnitude is 
whenever the matrix
inverse exists, where 
is the all-ones vector. The conjecture
asserted that the coefficients of 
are eventually nonnegative.
Koizumi (2026) disproved the conjecture with a real rank-6 arrangement represented by 12 vectors. For its underlying matroid 
, 
has a pole of order 4 at 
and a pole of order 5 at 
. The higher-order pole at 
forces
![(-1)^l[q^l]Mag(M;q)<0](/images/equations/Koizumi-LiuConjecture/NumberedEquation1.svg) |
for infinitely many 
. The paper reports that GPT-5.6 Sol and GPT-6 Astra assisted
with arguments, computations, and drafting. Koizumi states that he independently
checked the resulting work. No external specialist review had been reported as of
Sep. 9, 2026.
