  In this work we study a sequence of perfect t-embeddings of uniformly
weighted Aztec diamonds. We show that these perfect t-embeddings can be used to
prove convergence of gradients of height fluctuations to those of the Gaussian
free field. In particular we provide a first proof of the existence of a model
satisfying all conditions of the main theorem of arXiv:2109.06272. This
confirms the prediction of arXiv:2002.07540. An important part of our proof is
to exhibit exact integral formulas for perfect t-embeddings of uniformly
weighted Aztec diamonds.
  In addition, we construct and analyze perfect t-embeddings of another
sequence of uniformly weighted finite graphs called tower graphs. Although we
do not check all technical assumptions of the mentioned theorem for these
graphs, we use perfect t-embeddings to derive a simple transformation which
identifies height fluctuations on the tower graph with those of the Aztec
diamond.
