The stable forking conjecture asserts that forking in a simple first-order theory can always be witnessed by a stable formula. The formulation is attributed to Hart, Kim, and Pillay by Freitag and Mutchnik (2026).
A formula 
is stable when it has no order property: there are no infinite sequences 
and 
with 
holding precisely when 
. Forking describes dependence
of a tuple on parameters over a base set. If the type of 
over 
forks over 
, the conjecture asks for a
stable formula in that type which itself forks
over 
.
The stability requirement applies to the formula with its parameter places left as
variables, rather than to a single instance.
Freitag and Mutchnik (2026) constructed a counterexample using a vector space over a division ring equipped with additional relations. GPT-5.6 Sol supplied the counterexample and central arguments during interactive research, and the authors developed and checked the proof. Independent external verification had not been reported as of Sep. 7, 2026.
