= Solution
A <Lagrangian submanifold> of a $2n$-dimensional <symplectic manifold> is an embedded $n$-dimensional submanifold on which the <symplectic form> restricts to zero. The dimension requirement distinguishes it from a lower-dimensional <isotropic submanifold>.
A <differential one-form> $\sigma$ on $L$ gives an embedded section of its <cotangent bundle>, since $\pi\circ\sigma=\mathrm{id}$. Directly from the <canonical one-form on a cotangent bundle>,
$$
\sigma^*\lambda=\sigma,\qquad\sigma^*\omega_{\mathrm{can}}=-d\sigma.
$$
Its graph already has half the ambient dimension. Therefore
$$
\boxed{\operatorname{graph}(\sigma)\text{ is Lagrangian}\iff d\sigma=0.}
$$
This proves the <graph of a closed one-form is Lagrangian> criterion, in both directions.
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