Reverse mode computes the exact gradient of the real-arithmetic graph

For each input or operation node $i$, define a directed path from $i$ to the output by choosing one parent slot at each successive operation; when $i=o$, include the length-zero path. Associate to an edge from parent slot $k$ of node $j$ the linear map $D_k{\varphi }_j$ at the stored primal arguments. Repeated slots give separate edges.

Fix a node $i$ and form the downstream map $F_i(u)$ by replacing its value with an independent $u$ while holding earlier node values fixed and then evaluating later operations in topological order. Repeated use of [F4] shows that the derivative contribution along a fixed path from $i$ to $o$ is the ordered product of its edge maps. At each operation, its derivative is linear in the concatenated parent direction, so contributions from distinct parent slots add. If $i=o$, the output is $u$ and the length-zero path contributes the identity; if there is no path from $i$ to $o$, $F_i$ is locally constant and the sum is zero. Thus $D{F}_i(z_i)$ is the sum of the path products over all paths from $i$ to $o$, and its cotangent is the transpose of that sum applied to $1$.

The output-node path sum is the identity scalar $1$, matching the unit seed in [F3], whether the output is an input or an operation. For every other node, each path to $o$ begins with exactly one outgoing parent-slot edge $i\to j$, followed by a path from $j$ to $o$. Hence its path-sum recurrence is ${\bar{z}}_i={\sum }_{j,k:p_{j,k}=i}{D}_k{\varphi }_j^T{\bar{z}}_j$, with an additional identity seed only when $i=o$; this is exactly the initialization and accumulation in [F3]. Reverse induction gives ${\bar{z}}_i=D{F}_i(z_i{)}^T$ at operation nodes and, at inputs, $\bar{x}=Df(x{)}^{T}1$.

The algorithm returns precisely those input cotangents, proving the statement.