Tokareva's conjecture states that, for every even positive integer 
,
every Boolean function 
of polynomial
degree at most 
is the sum of two bent functions, with addition
in the finite field 
(Tokareva 2014). The degree is that of the unique multilinear
polynomial representing 
. The conjecture holds for 
but fails for every even 
(Feng et al. 2026).
Feng et al. (2026) gave the explicit counterexamples
 |
where 
.
These functions have degree 
and cannot be expressed as sums of two bent
functions. In particular, the first counterexample is 
.
Feng et al. (2026) also gave explicit counterexamples to a derivative formulation. For every even 
,
the function 
is balanced, has degree 
, and has a nonzero period, but is not a derivative of
any bent function in any direction. Here balanced means taking each value in 
equally often; a nonzero period is
a vector 
satisfying 
;
and the derivative of 
in direction 
is 
. Nonconstructive counterexamples to this
derivative formulation were already known (Feng et al. 2026).
Feng et al. (2026) report that MiniMax M3.1-Flash-Preview agents discovered the counterexamples and generated proofs. The authors checked the arguments and report accompanying Lean 4 formalizations of both counterexample theorems.
