The incompressible porous media equation, abbreviated IPM equation, transports a scalar field 
representing density by the divergence-free Darcy velocity
that it generates. On the two-dimensional torus, the forced
equation can be written
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is a zero-order linear operator
specified through the Fourier transform by Darcy's
law. It is an active-scalar model for the motion of incompressible fluids through
porous media.
Córdoba and Martínez-Zoroa (2024) proved finite-time blow-up from smooth initial data on 
with a force that is smooth
in space and bounded in time. Alpöge et al. (2026) claim the corresponding
result on 
with 
smooth jointly in space
and time. More precisely, they construct a solution smooth
for every 
which converges in 
for every 
, while
 |
(3)
|
Alpöge et al. (2026) report using Claude to reconstruct the earlier argument and Claude and Codex to prepare the manuscript, handle inductive orders and constants, and formalize the argument in Lean under the authors' direction. As of Sep. 16, 2026, the corresponding IPM Lean files had not been publicly released (Buckmaster 2026a), and independent specialist review of the complete argument had not been reported.
