Reverse-mode work is a fixed multiple of primal work under explicit primitive-cost bounds

For a finite $C^1$ graph on a fixed strict branch, fix a finite set of scalar/vector primitive types with at most $r$ parent slots per operation and bounded dimensions. Suppose each operation $j$ has primal work $w_j\ge 1$, its local vector-Jacobian product costs at most $c{w}_j$ operations, and accumulating each parent-slot contribution costs at most $a$ operations. Assume each invocation's retained local tape and peak pullback scratch use at most a fixed number of scalar entries. Let $S$ be the scalar storage in the primal tape and input-gradient buffer, counting all input and node-value coordinates, per-node and per-parent metadata, branch labels, and retained local-tape entries. Then the exact reverse-mode algorithm for a scalar output uses at most $(1+c+ar){\sum }_{j=1}^T{w}_j$ operations for the primal evaluation, reverse sweep, and parent accumulations, and stores $O(S)$ scalar entries. Opaque solver calls are excluded unless their primal and derivative costs satisfy the same hypotheses.

Baur–Strassen proves a separate nonscalar arithmetic-circuit complexity theorem over an infinite field. The local bound above is proved directly for a fixed smooth primitive set with explicit primal/pullback cost ratios; it does not extend the arithmetic-circuit theorem to arbitrary opaque procedures or claim that every differentiable primitive has a cheap derivative.