  This report explores the theory that explains the high sparsity phenomenon
\citep{tosato2023emergent} observed in the forward-forward algorithm
\citep{hinton2022forward}. The two theorems proposed predict the sparsity
changes of a single data point's activation in two cases: Theorem
\ref{theorem:1}: Decrease the goodness of the whole batch. Theorem
\ref{theorem:2}: Apply the complete forward forward algorithm to decrease the
goodness for negative data and increase the goodness for positive data. The
theory aligns well with the experiments tested on the MNIST dataset.
