The generalized vanishing conjecture (Zhao 2007) states that if 
is a constant-coefficient differential
operator on a polynomial ring over a field
of characteristic 0 and 
and 
are polynomials, then 
for every integer 
implies 
for all sufficiently
large 
.
Dvorsky (2026) gave an explicit five-variable counterexample. Van Rijn (2026) showed that the conjecture holds in dimensions 
and fails in every dimension 
, even for a homogeneous differential
operator and a homogeneous polynomial.
In three variables, let 
, 
, 
, 
, 
, 
, and 
. Then 
and 
for every 
.
The accompanying Lean development (van Rijn 2026) verifies the three-variable counterexample and its extension to higher dimensions, but does not formalize the complete two-variable argument. The full classification has not received independent review (VibeMathed 2026). Dvorsky (2026) states that GPT-5.6 Sol helped test the construction, generate verification code, and locate literature. VibeMathed (2026) reports that Codex ran and monitored experiments for van Rijn (2026) under human direction; the van Rijn manuscript itself contains no AI disclosure.
The vanishing conjecture restricts 
to the ordinary laplacian.
Since the counterexample above uses 
, it does not settle that more restricted conjecture.
