The Hopfield model is a binary spin model whose interaction strengths are constructed from stored patterns (Hopfield 1982, Amit et al. 1985).
In its statistical-mechanical form, the configurations are 
, and the columns of an 
matrix 
are the stored patterns. A convention for the energy that
retains the diagonal terms is
 |
At inverse temperature 
, the Gibbs
measure assigns weights proportional to 
. The free energy
in the convention of Ali et al. (2026) is the natural logarithm
of the statistical-mechanical
partition function per spin with the uniform measure as reference,
![F_N(beta)=1/NlnE_sigma[e^(-betaH_N(sigma))],](/images/equations/HopfieldModel/NumberedEquation2.svg) |
where 
is the expectation value for uniform 
at fixed 
.
For independent entries of 
taking each of 
with probability 
and 
,
Ali et al. (2026) report a deterministic variational limit of 
for every 
. The stated convergence
is almost sure and in 
with respect to the patterns. Ali reports GPT-6 Astra assistance with the existence
proof. The authors report checking the proofs
by hand.
